University of Southern California Dissertations and Theses

Motivic cohomology of BG₂

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Motivic cohomology of BG₂

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Content MOTIVIC COHOMOLOGY OF B G ₂ by Alexander Maxwell Port A Dissertation Presen ted to the F A CUL TY OF THE USC GRADUA TE SCHOOL UNIVERSITY OF SOUTHERN CALIF ORNIA In P artial F ulllmen t of the Requiremen ts for the Degree DOCTOR OF PHILOSOPHY (MA THEMA TICS) Decem b er 2021 Cop yrigh t 2021 Alexander Maxw ell P ort Dedication I dedicate this w ork all m y friends and family , esp ecially m y lo v ely wife Hadassah and m y paren ts Da vid and Doroth y . Y ou ha v e supp orted me immensely in m y academic endev ours. Thank y ou, I couldn’t ha v e done it without y ou! ii Acknowledgments I’d lik e to thank m y adviser, Professor Ara vind Asok, for his guidance in this pro ject. I’d lik e to thank m y committee for their in v olv emen t in m y w ork. I’d lik e to thank the USC Mathematics Departmen t for the six y ears of funding. iii T able of Contents Dedication ii A c kno wledgmen ts iii List of T ables vi List of Figures vii Abstract viii 1 Con v en tions and Summary of Results 1 1.1 Con v en tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Classifying Spaces and G 2 5 2.1 Principal Bundles and Classifying Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Principal Bundles and Basic F acts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Classifying Spaces and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.3 P ath and Lo op Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Basics and Constructions of BG 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Classical bac kground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.2 Zorn’s Octonions and Its Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 G 2 and SO 4 as Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Motivic Homotop y 31 3.1 Homotop y Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 The tale T op ology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Nisnevic h T op ology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Construction of Unstable Motivic Homotop y . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4.1 A dding Colimits for Homotop y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4.2 Unstable Homotop y Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.5 F rom Motivic Stable Homotop y to Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6 T ransfers and Euler Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.7 The Six Op erations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Motivic Cohomology 52 4.1 Motivic Cohomology in the Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1.1 Prop erties of Motivic Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.1.2 Blo c h’s Higher Cho w Groups as an Alternate Mo del . . . . . . . . . . . . . . . . . . . 53 4.1.3 V anishing Theorems and Knneth F orm ula . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Classifying Spaces in Motivic Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 Steenro d Op erations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.3.1 Classical Steenro d Square Op erations . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 iv 4.3.2 Cohomological P o w er Op erations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3.3 Steenro d in the Motivic Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5 Results 63 5.1 Statemen t of Theorem in Characteristic 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2 Discussion of SO 2n Before Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 In tegral Motivic Cohomology of BSO 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.4 In tegral Motivic Cohomology of BG 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.5 Minor Result Ab out Steenro d Op erations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6 Conclusions and F uture Directions 98 Bibliograph y 99 7 App endix 104 7.1 SL 3 ⋊Z 2 Computations for Pro of of Lemma 2.2.4 . . . . . . . . . . . . . . . . . . . . . . . . 104 7.2 Isomorphisms Bet w een F orms of SO n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.3 Supp orting W ork for H ∗ ,∗ ′ (BSO 4 ;Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.4 Supp orting W ork for H ∗ ,∗ ′ (BG 2 ;Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.5 Co de W ritten for Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 v List of T ables 2.1 Multiplication table of the split o ctonions O ′ . Here is base eld has c haracteristic other than 2. 13 2.2 Matrices of the forms dening m ultiplication in O Z in standard basis. . . . . . . . . . . . . . 17 2.3 Multiplication table of the split o ctonions O Z in standard basis. . . . . . . . . . . . . . . . . . 19 5.1 The elemen ts of H ∗ ,∗ ′ (BSO 4 ;Z) fall in to nine families. Eigh t of these ha v e lifts as ab o v e but the nin th do es not. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2 The elemen ts of H ∗ ,∗ ′ (BG 2 ;Z) fall in to nine families. Eigh t of these ha v e lifts as ab o v e but the nin th do es not. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3 A list of the p olynomial terms of degree 7 in no more than 7 v ariables along with the results of applying the Steenro d homomorphism Sq 1 . Coun ts the n um b er of em b eddings of the terms in to a xed c hoice of 7 v ariables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 A list of the results of applying the Steenro d homomorphism Sq 2 to the f i ’s determined previously . Applying Sq 1 to these f i mak es them 0 and the notes describ e wh y or wh y not it is p ossible to do the same with Sq 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.1 Simplied m ultiplication, realization, and degree tables for generators of H ∗ ,∗ ′ (BSO 4 ;Z 2 ) as presen ted b y Corollary 7.3.2. Here the realization map is the canonical one H ∗ ,∗ ′ (BSO 4 ;Z 2 )→ H ∗ (BSO 4 ;Z 2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 vi List of Figures 4.1 Images represen ting the corresp ondence b et w een motivic cohomology groups and higher Cho w groups. Cy an regions are areas that v anishing according to the Beilinson-SoulØ v anishing conjecture; green areas v anish b ecause m>2n; y ello w areas v anish b ecause m>dim(X)+n; gra y areas v anish b ecause r <0; the gradien ts from blue to red are areas that can p oten tially b e non-trivial (assuming Beilinson-SoulØ). The region in the images on the left ll the image b ecause these are domains; the shap es in the righ t side images are the corresp onding ranges. 55 vii Abstract Motivic cohomology is a p o w erful to ol in algebraic geometry that is notoriously dicult to use but so univ ersal that it’s w orth the eort. Its asso ciated realization maps directly giv e imp ortan t information ab out the relations b et w een dieren t cohomological in v arian ts of sc hemes and their classifying spaces. The problem of computing general cohomological in v arian ts of classifying spaces is ongoing. In [T ot99],T otaro constructed the Cho w ring of a classifying space in general and used this to study symmetric groups. In [Gui07], Guillot did the same examination for the Lie groups G 2 and Spin(7). More recen tly in [T ot17], T otaro describ ed the computation of the Ho dge and de Rham cohomology groups of classifying stac ks. The classifying space BG 2 is deriv ed from the simplest exceptional Lie group and th us mak es for a go o d starting p oin t for testing computational metho ds. This thesis co v ers the computation of the motivic cohomology of BG 2 with in tegral co ecien ts. The primary approac h is a restriction/transfer map statemen t inspired b y the previous w orks of [Gui07], [V ez00] and [Y ag10]. Y agita conjectured the mo d- 2 motivic cohomology based on the existence of a motivic transfer map in v olving G 2 and its maximal torus SO 4 . V ezzosi’s metho d for construction of a transfer is adaptable to the motivic setting due to the fact that Blo c h’s higher Cho w groups are a mo del for motivic cohomology whic h directly extend Guillot’s w ork. This w ork also co v ers the deriv ation of a p olynomial relating the generators of the singular mo d- 2 cohomology of BG 2 through Steenro d squares. viii Chapter 1 Conventions and Summary of Results 1.1 Conventions Selected notational con v en tions used are the follo wing: All rings are assumed to b e unital and comm utativ e. Unless otherwise stated, assume that sc hemes are No etherian and nite-dimensional; Lie groups are tak en to b e in their split forms unless otherwise stated. F or example, Sch/B refers to the category of No etherian nite-dimensional sc hemes o v er a sc heme B ; there is also the sub category Sm/B of smo oth sc hemes of the same t yp e. In addition, G 2 refers to the automorphism group of the split-o ctonions and SO 4 is tak en as the indenite sp ecial orthogonal group SO (2,2) . Co v arian t functors b et w een categories S and C are written S → C while con tra v arian t functors are written S op →C . The standard pro jection map from a pro duct space on to its i th comp onen t will b e denoted as the map p i . When con v enien t, if X :∆ op →Set is a simplicial set then X n ma y b e used for X([n]). The exception to this for ob vious reasons is Λ n k . Spc(B) is tak en to b e the category of Nisnevic h shea v es on Sm/B with v alues in Set, while ∆ op Spc(B) tak es v alues in the category of simplicial sets (follo wing the con v en tion of V o ev o dsky). W e denote the sc hemes G m = Spec(C[x,x − 1 ]) and µ p = Spec(C[x,x − 1 ]/(x p − 1)). Ho w ev er, outside the con text of sc hemes G m is used to denote the m ultiplicativ e group C−{ 0}. 1 The co ecien ts for cohomology are alw a ys tak en to b e Z orZ p unless otherwise stated. Let X + =X F Spec(K) b e X with a disjoin t base-p oin t added for a sc heme X o v er a eld K. It’s w ell kno wn that H 0,1 (Spec(C);Z p ) ∼ = Z p ; in the con text of motivic cohomology , τ refers to a generator of H ∗ ,∗ ′ (Spec(C);Z p ) with deg(τ ) = (0,1) and corresp onds to a c hoice of generator of Z p . Otherwise, τ is either the transgression map from H r− 1 (F;R) to H r (B;R) in a bration F →E→B or the symmetry map of a symmetric monoidal category . In the con text of ordinary cohomology and Cho w rings, the subscript of a generator is its degree. F or motivic cohomology , a subscript of n means the degree is (2n,n). The one exception to this is H ∗ ,∗ ′ ((G m ) n ;Z)=H ∗ ,∗ ′ (Spec(C);Z)[t 1 ,...,t n ] where deg(t i )=(2,1) for all i ([Fie12],[Gui07]). Let A b e a graded R-algebra. A set {a 1 ,...,a n } of homogeneous elemen ts of A is called a simple set or system of gener ators for A i {a i1 ...a ip : 1≤ p≤ n,1≤ i 1 <... r ; denote suc h an extension as ι (α ). Consider the constan t map c xo ∈ C([0,∞),X) where c xo (t) = x 0 for all t ≥ 0. Note that c x0 is not an elemen t of PX b ecause its domain is not of the correct form. Ho w ev er, if w e dene a family of paths c x0,r :[0,r]→X where c x0,r (t) = x 0 for all t ∈ [0,r], then eac h of these c x0,r paths is in PX and in fact ι (c xo,r ) = c x0 for an y r . Next, let H :PX× [0,1]→PX where H(α,s )(t)=α (t(1− s)) is a homotop y from α to c xo,r for an y path α ∈ PX (i.e. H(α, 0) = α and H(α, 1) = c x0,r ). The pro of is completed b y the fact that eac h of the c x0,r paths is homotopic to eac h other. Sp ecically , eac h one can b e mo v ed via homotop y to c x0,0 . T o conclude, the space PX is con tractible b ecause an y path in PX can b e shifted con tin uously to the p oin t c x0,0 . The Mo ore path and lo op spaces are the to ols necessary for shifting principal bundles and creating a longer sequence. As a bit of notation, w e can inductiv ely dene Ω k+1 X to b e ΩΩ k X . The idea of shifting is that a bundle G→T →X can b e extended to the left to create ...→Ω 2 X →Ω G→Ω T →Ω X →G→T →X where an y three consecutiv e terms form a principal bundle. T o see this, rst consider the follo wing diagram: Ω X vv T × X PX // PX G 77 // T // X p:PX →X maps a path α to its endp oin t q :T →X pro jection map T × X PX ={(e,α )∈T × PX :q(e)=p(α )} ρ G :T × G→T,(e,g)7→ρ G (e,g) action of G on T e ρ G :(T × X PX)× G→T × X PX,((e,α ),g)7→(ρ G (e,g),α ) lifted action of G on T × X PX 10 µ :PX× Ω X →PX action of Ω X on PX e µ :(T × X PX)× Ω X →T × X PX,((e,α ),β )7→(e,µ (α,β )) lifted action of Ω X on T × X PX In the ab o v e diagram, b er bundle Ω X →T× X PX →T b ecomes Ω X →G→T b y homotop y equiv alence. The follo wing is a summary of the sp ecics of this equiv alence: By the homotop y lifting prop ert y of b er bundles, for an y path α :[0,r]→X and p oin t e∈q − 1 (p(α )) there is a unique lift e α (e) :[0,r]→T suc h that e α (e) (r)=e and q◦ e α (e) =α Note that (e,α )∈ T × X PX i e α (e) (r) = e (where α : [0,r]→ X), i.e. T × X PX is a a collection of pairs of p oin ts in PX and the endp oin ts their corresp onding endp oin ts W e can deformation retract T× X PX to the subspace q − 1 (x 0 )×{ c x0,0 } b y a con tin uous map H giv en b y H :(T× X PX)× [0,1]→T× X PX where H((e,α ),s)=(e α (e) (1− s),α s ) for α s :[0,r(1− s)]→X and α s (t)=α (t) As an aside, let us co v er the topic of fundamen tal classes. Recall that the Eilen b erg-Mac Lane space K(G,n) (only dened up to up to w eak homotop y equiv alence) is path-connected and has the prop ert y that π n (K(G,n)) ∼ = G and π i (K(G,n)) = 0 for all other i ≥ 1. Because π k+1 (X) ∼ = π k (Ω X) in the general case, w e ha v e that K(G,n) is w eakly homotop y equiv alen t to Ω K(G,n+1). This holds b ecause of the isomorphism π n (K(G,n)) ∼ = G ∼ = π n+1 (K(G,n+1)) = π n (Ω K(G,n+1)), and similarly for all other π i (K(G,n)) ∼ = 0. Next, recall a (rough) statemen t of the Hurewicz Theorem: if X is (n− 1)-connected then H i (X) ∼ = π i (X) for all i ∈ {0,...,n}. This is applicable b ecause, b y denition, K(G,n) is (n− 1)- connected and so H n (K(G,n);Z) ∼ =π n (K(G,n)) ∼ =G. The isomorphism H n (K(G,n);Z)→G is an elemen t u n ∈ Hom Z (H n (K(G,n);Z),G) ∼ = H n (K(G,n);G) ∼ = Hom G (G,G). W e sa y that u n is a fundamental class of K(G,n). This fundamen tal class allo ws us to describ e the bijection from [X,K(G,n)] to H n (X;G); a homotop y class [f : X → K(G,n)] is mapp ed to f ∗ (u n ) ∈ H n (X;G). Note that this is not a pro of but rather a statemen t of the end result. The concluding p oin t of this section serv es to establish some in tuition for exactly what classifying a space represen ts. F act 2.1.16. G and Ω BG have isomorphic c ohomolo gy gr oups in every de gr e e. Pro of: By the ab o v e discussion, w e ha v e that G is homotop y equiv alen t to EG× BG PBG and that the shifted principal bundle asso ciated with G→EG→BG is Ω BG→G→EG. Th us, the diagram 11 Ω BG uu EG× BG PBG // PBG G 66 // EG // BG giv es the bundle Ω BG→G→EG. The Serre sp ectral sequence H i (EG;Z)⊗ Z H j (Ω BG;Z) =⇒ H i+j (G;Z) sho ws that H j (Ω BG;Z) ∼ =H j (G;Z) for all j≥ 0. This holds b ecause the w eak con tractibilit y of EG means that H i (EG;Z)=0 for all i≥ 1. There is a stronger statemen t to b e pro v ed, namely that G and Ω BG are w eakly homotop y equiv alen t. This means that, in some sense, taking lo ops of a space and classifying a space are opp osite op erations. 2.2 Basics and Constructions of BG 2 The ob ject BG 2 is the classifying space of the automorphism group of the o ctonions. Here w e consider the split o ctonions instead of the usual Ca yley o ctonions, and care m ust also b e tak en in notation as the term G 2 can refer to the Lie algebra, the compact Lie group, or the split Lie group of these automorphisms. The goal of this section is to mak e clear the ob jects used in the remainder of this pap er and to consider v arious constructions of G 2 and BG 2 . 2.2.1 Classical background As far as this w ork is concerned, the group of in terest is the exceptional group G 2 . The exc eptional Lie gr oups are the simple, 1-connected groups that are not isomorphic to SU(n), Spin(n) or Sp(n). Recall that the Cartan sub algebr a of a Lie algebra is the largest subset of elemen ts of the algebra that all comm ute with eac h other. The r ank of a Lie group corresp onds to the rank of its Lie algebra and is tak en to b e the dimension of the Cartan subalgebra. As suc h, G 2 is a compact Lie group whose rank is 2 and dimension is 14 ([Gui07]). 12 e j e k e k e j e 0 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 0 e 0 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 1 e 1 − e 0 e 3 − e 2 − e 5 e 4 − e 7 e 6 e 2 e 2 − e 3 − e 0 e 1 − e 6 e 7 e 4 − e 5 e 3 e 3 e 2 − e 1 − e 0 − e 7 − e 6 e 5 e 4 e 4 e 4 e 5 e 6 e 7 e 0 e 1 e 2 e 3 e 5 e 5 − e 4 − e 7 e 6 − e 1 e 0 e 3 − e 2 e 6 e 6 e 7 − e 4 − e 5 − e 2 − e 3 e 0 e 1 e 7 e 7 − e 6 e 5 − e 4 − e 3 e 2 − e 1 e 0 T able 2.1: Multiplication table of the split o ctonions O ′ . Here is base eld has c haracteristic other than 2. Recall that a real Lie algebra g 0 is the r e al form of a complex Lie algebra g if g is the complexication of g 0 . Note that a corresp onding idea holds for Lie groups. In general, a complex semisimple Lie algebra will ha v e a complex form as w ell as compact and split real forms ([BH14]). It is common to tak e the the automorphism group of the o ctonions O to b e compact real form of G 2 . Ho w ev er, the more relev an t construction here will b e to tak e the automorphism group of the split o ctonions, denoted here as O ′ , to b e the split real form of G 2 . Recall that applying the Ca yley-Dic kson construction to the quaternions H has a m ultiplication rule giv en b y (a,b)(c,d) = (ac− d ¯b,¯ad+cb). A mo died Ca yley-Dic kson construction with m ultiplication giv en b y (a,b)(c,d)=(ac+d ¯b,¯ad+cb) giv es rise toO ′ . If w e tak e generators {e i } 7 i=0 , then w e can dene the conjugate, inner pro duct, length, and real part in O ′ , with e 0 b eing b oth the m ultiplicativ e iden tit y and the real generator. Also, w e ha v e that m ultiplication in O ′ (and not c haracteristic 2) is giv en in T able 2.1 ([BH14],[Y ok09]). F or an alternativ e approac h to the o ctonions, w e lo ok to the follo wing t w o denitions coming from [AHW19] 2.1.1 and 2.1.9: Denition 2.2.1. Supp ose R is a comm utativ e unital ring. An o ctonion algebr a o v er R is a quadruple (O,◦ ,1 0 ,N O ) where O is a rank 8 pro jectiv e R-mo dule, ◦ : O× O → O is a binary comp osition (t ypically suppressed in notation), 1 0 : R → O is an R-mo dule homomorphism acting as a t w o-sided unit for ◦ , and N O : O→ R is an R-mo dule homomorphism making (O,N O ) in to a non-degenerate quadratic space where N(x◦ y)=N(x)N(y) for all x,y∈O . W e call N O the asso ciate d norm form of O and denote 1 O =1 0 (1 R ). Let ⟨− ,−⟩ O :O× O→R b e the symmetric bilinear form asso ciated with the norm where ⟨x,y⟩ O = 1 2 (N O (x+y)− N O (x)− N O (y)) Let (− ) ∨ :O→O b e the c onjugation involution giv en b y x ∨ =1 0 (⟨x,1 O ⟩ O )− x 13 Let Tr O :O→R b e the R-linear tr ac e map giv en b y Tr O (x)=⟨x,1 O ⟩ O . Note that this is equiv alen t to 1 0 (Tr O (x))=x+x ∨ W e can also dene the split o ctonions O ′ in the follo wing w a y; tak e O = M 2× 2 (R)⊕ M 2× 2 (R), and if A ∈ M 2x2 (R) then dene A ∗ to b e A T conjugated b y the matrix    0 1 − 1 0    . Multiplication is giv en b y (A,B)(C,D) = (AC + DB ∗ ,A ∗ D + CB) (as in the mo died Ca yley-Dic kson construction). Dene the maps 1 0 (r) = (rI 2 ,0 2× 2 ) and N O ((A,B)) = det(A)− det(B). One can then sho w that det(A)I 2 = AA ∗ , (A,B) ∨ =(A ∗ ,− B), Tr O ((A,B))=Tr(A) and if Tr(A)=0 then A ∗ =− A. As an aside b efore con tin uing, note that it is p ossible to dene an analog of the cross pro duct on rank 3 pro jectiv e mo dules. If P is suc h a mo dule, let P ∗ = Hom R (P,R) b e the dual mo dule of P and ⟨− ,−⟩ :P ∗ × P →R b e the of the ev aluation map of P ∗ on P . With a sligh t abuse of notation, it is p ossible to dene × φ : P × P → P ∗ and × φ : P ∗ × P ∗ → P in the follo wing w a y . The highest non-zero exterior p o w er of P is denoted det(P) and an orientation φ on P is an isomorphism from det(P) to R. In this case, det(P) = V 3 P b ecause P is rank 3. W e construct × φ : P × P → P ∗ as ⟨p 1 × φ p 2 ,p⟩ = φ(p 1 ∧p 2 ∧p). It is clear that this satises the an ti-comm utativit y of the cross pro duct b ecause of the elemen ts of exterior algebras are alternating and φ is R-linear: i.e. p 2 × φ p 1 =− (p 1 × φ p 2 ) b ecause p 2 ∧p 1 ∧p 3 =− (p 1 ∧p 2 ∧p 3 ). Similarly , w e construct × φ :P ∗ × P ∗ →P using the fact that P ∗ is isomorphic to P b ecause P is nite rank; in particular, w e tak e p ∗ 1 × φ p ∗ 2 ∈P suc h that ⟨p ∗ ,p ∗ 1 × φ p ∗ 2 ⟩=φ(p∧p 1 ∧p 2 ). With these to ols in place, w e ma y no w dene Zorn ’s ve ctor matric es : Denition 2.2.2. Supp ose R is a comm utativ e and unital ring. Let (P,φ) b e an orien ted, rank 3, pro jectiv e R-mo dule. Denote Zorn((P,φ)) ={    r 1 − → p 1 − → p 2 r 2   : r 1 ,r 2 ∈ R, − → p 1 ∈ P, − → p 2 ∈ P ∗ } to b e the pro jectiv e R-mo dule equipp ed with the pro duct    r 1 − → p 1 − → p 2 r 2       s 1 − → q 1 − → q 2 s 2    =    r 1 s 1 +⟨ − → q 2 , − → p 1 ⟩ r 1 − → q 1 +s 2 − → p 1 − − → p 2 × φ − → q 2 r 2 − → q 2 +s 1 − → p 2 + − → p 1 × φ − → q 1 r 2 s 2 +⟨ − → p 2 , − → q 1 ⟩    2.2.2 Zorn’s Octonions and Its Automorphisms It is classically kno wn that if {y 1 ,...,y 7 } is an orthonormal basis for Hom R (R 7 ,R) then the subgroup of GL + 8 (R) preserving the follo wing form φ is isomorphic to G 2 ([Jo y96]): φ=y 1 ∧y 2 ∧y 7 +y 1 ∧y 3 ∧y 6 +y 1 ∧y 4 ∧y 5 +y 2 ∧y 3 ∧y 5 − y 2 ∧y 4 ∧y 6 +y 3 ∧y 4 ∧y 7 +y 5 ∧y 6 ∧y 7 14 Ho w ev er, this description is a bit to o abstract for applicabilit y here and it turns out that further in v estigation leads to useful results. Zorn’s o ctonions, denoted O Z , comprise the space {   a − → v − → w d   : a,d∈R, − → v, − → w ∈R 3 }. Let the standard basis ofR 3 b e denoted b y { − → s i } 3 i=1 . The isomorphism from O ′ toO Z is giv en b y: e 0 7→    1 − → 0 − → 0 1    e 4 7→    1 − → 0 − → 0 − 1    e 1 7→    0 − − → s 1 − → s 1 0    e 5 7→    0 − → s 1 − → s 1 0    e 2 7→    0 − − → s 2 − → s 2 0    e 6 7→    0 − → s 2 − → s 2 0    e 3 7→    0 − − → s 3 − → s 3 0    e 7 7→    0 − → s 3 − → s 3 0    One can c hec k b y hand that these matrices satisfy the relations for the split o ctonions depicted in T able 1. Ho w ev er, they are a fairly complicated basis to w ork with in the remainder of the section; w e will instead use the more standard lo oking basis:   1 − → 0 − → 0 0   ,   0 − → s 1 − → 0 0   ,   0 − → s 2 − → 0 0   ,   0 − → s 3 − → 0 0   ,   0 − → 0 − → s 1 0   ,   0 − → 0 − → s 2 0   ,   0 − → 0 − → s 3 0   ,   0 − → 0 − → 0 1   Mo ving forw ard, w e will use {⃗ s 1 ,...,⃗ s 8 } to denote this basis of O Z . Also, it is clear that this basis is similar to the standard orthonormal basis of R 8 ; as suc h, our iden tication O Z withR 8 will send our c hoice of basis to the standard one. Next, let us consider the automorphism group of O Z . As v ector spaces, O Z andR 8 are isomorphic. This means that an y automorphism of O Z will b e giv en b y a c hange of basis; in other w ords, Aut(O Z ) ma y b e though t of as a subspace of GL 8 . Thinking of elemen ts of O Z as just elemen ts of R 8 written in a dieren t w a y , the standard action of GL 8 onR 8 in the standard basis tak es the follo wing form on O Z : M·                 a 1       a 2 a 3 a 4             a 5 a 6 a 7       a 8                 =                 P 8 i=1 M 1i a i       P 8 i=1 M 2i a i P 8 i=1 M 3i a i P 8 i=1 M 4i a i             P 8 i=1 M 5i a i P 8 i=1 M 6i a i P 8 i=1 M 7i a i       P 8 i=1 M 8i a i                 15 Ho w ev er, not all matrices M ∈GL 8 will b e compatible with the needed m ultiplication condition: M·                                 a 1       a 2 a 3 a 4             a 5 a 6 a 7       a 8                                 b 1       b 2 b 3 b 4             b 5 b 6 b 7       b 8                                 =                 M·                 a 1       a 2 a 3 a 4             a 5 a 6 a 7       a 8                                                 M·                 b 1       b 2 b 3 b 4             b 5 b 6 b 7       b 8                                 This is to o abstract a requiremen t as written. Let’s b egin b y writing the equation for m ultiplication in O Z :                 a 1       a 2 a 3 a 4             a 5 a 6 a 7       a 8                                 b 1       b 2 b 3 b 4             b 5 b 6 b 7       b 8                 =                 a 1 b 1 +a 2 b 5 +a 3 b 6 +a 4 b 7       a 1 b 2 +a 2 b 8 − a 6 b 7 +a 7 b 6 a 1 b 3 +a 3 b 8 +a 5 b 7 − a 7 b 5 a 1 b 4 +a 4 b 8 − a 5 b 6 +a 6 b 5             a 3 b 4 − a 4 b 3 +a 5 b 1 +a 8 b 5 − a 2 b 4 +a 4 b 2 +a 6 b 1 +a 8 b 6 a 2 b 3 − a 3 b 2 +a 7 b 1 +a 8 b 7       a 5 b 2 +a 6 b 3 +a 7 b 4 +a 8 b 8                 Up on insp ection, it is clear that eac h en try on the righ t hand side of T able 2.2 is a homogeneous degree 2 p olynomial. In other w ords, m ultiplication in O Z is really giv en b y 8 quadratic forms and their asso ciated bilinear forms. As suggested b y the la y out of T able 2.2, there are man y relationships and symmetries b et w een these matrices. Consider the S 8 -action on M 8× 8 (R) giv en b y ρ (σ )((m ij )) = (m σ (i)σ (j) ). The in teresting part ab out these matrices and this action is that there are only a few p erm utations needed to describ e all in teractions b et w een them. The p erm utation α = (18)(27)(36)(45) sends φ i to φ 8− i for all i, and the p erm utation β =(1)(234)(567)(8) rotates φ 2 ,φ 3 ,φ 4 and φ 5 ,φ 6 ,φ 7 eac h among themselv es in that order. The τ and σ maps are represen ted visually b elo w: φ 1 OO α φ 2 OO α β && φ 4 OO α β oo φ 3 OO α β 88 φ 6 β xx φ 8 φ 7 β // φ 5 β ff 16 φ1 =            1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0            ⇔            0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1            =φ8 φ2 =            0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0            ⇔            0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0            =φ7 φ3 =            0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0            ⇔            0 0 0 0 0 0 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0            =φ6 φ4 =            0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 − 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0            ⇔            0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 − 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0            =φ5 T able 2.2: Matrices of the forms dening m ultiplication in O Z in standard basis. W e can no w rewrite the ab o v e equations in a more meaningful w a y . F or example, if w e tak e v ectors − → a =[ a 1 ... a 8 ] T and − → b =[ b 1 ... b 8 ] T then w e can write the pro duct of the corresp onding elemen ts ofO Z as the follo wing:                a 1      a 2 a 3 a 4           a 5 a 6 a 7      a 8                               b 1      b 2 b 3 b 4           b 5 b 6 b 7      b 8                =                − → a T φ 1 − → b      − → a T φ 2 − → b − → a T φ 3 − → b − → a T φ 4 − → b           − → a T φ 5 − → b − → a T φ 6 − → b − → a T φ 7 − → b      − → a T φ 8 − → b                F or shorthand, let us denote the righ t-hand side of this statemen t though t as living in R 8 as − → φ ab . If w e tak e R i to the i th ro w of our matrix M ∈GL 8 then w e can also rewrite the result of the GL 8 -action: M·                a 1      a 2 a 3 a 4           a 5 a 6 a 7      a 8                =                R 1 − → a      R 2 − → a R 3 − → a R 4 − → a           R 5 − → a R 6 − → a R 7 − → a      R 8 − → a                Keep in mind that this is just another w a y of writing the v ector M − → a . Finally , the t w o sides of our 17 m ultiplication compatibilit y condition are written in the follo wing w a ys: M·                                 a 1       a 2 a 3 a 4             a 5 a 6 a 7       a 8                                 b 1       b 2 b 3 b 4             b 5 b 6 b 7       b 8                                 =                 R 1 − → φ ab       R 2 − → φ ab R 3 − → φ ab R 4 − → φ ab             R 5 − → φ ab R 6 − → φ ab R 7 − → φ ab       R 8 − → φ ab                                 M·                 a 1       a 2 a 3 a 4             a 5 a 6 a 7       a 8                                                 M·                 b 1       b 2 b 3 b 4             b 5 b 6 b 7       b 8                                 =                 R 1 − → a       R 2 − → a R 3 − → a R 4 − → a             R 5 − → a R 6 − → a R 7 − → a       R 8 − → a                                 R 1 − → b       R 2 − → b R 3 − → b R 4 − → b             R 5 − → b R 6 − → b R 7 − → b       R 8 − → b                 =                 − → a T M T φ 1 M − → b       − → a T M T φ 2 M − → b − → a T M T φ 3 M − → b − → a T M T φ 4 M − → b             − → a T M T φ 5 M − → b − → a T M T φ 6 M − → b − → a T M T φ 7 M − → b       − → a T M T φ 8 M − → b                 Comparing en tries comp onen t-wise, w e need R i − → φ ab = − → a T M T φ i M − → b for all indices i ∈ {1,...,8} and v ectors − → a, − → b ∈R 8 . If w e tak e { − → s i } 8 i=1 to the standard basis for R 8 and C i to the b e i th column of M , w e get the all imp ortan t compact statemen t of the m ultiplication condition. Theorem 2.2.3. A matrix M ∈ GL 8 satises the multiplic ation c omp atibility c ondition r e quir e d for b eing an automorphism of O Z i R i − → φ sjs k =C T j φ i C k for al l i,j,k∈{1,...,8}. This theorem is merely an adaption of the algebra homomorphism requiremen t M( − → a − → b)=M( − → a)M( − → b). This form is more computationally friendly due to its ease to program and its direct relation to a basis. Eectiv ely the same theorem holds as a condition for a matrix in GL 4 b eing an automorphism of Zorn’s quaternions; see the next section for details. 18 ˜ φ s j s k s k s j s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 1 s 1 s 2 s 3 s 4 0 0 0 0 s 2 0 0 s 7 − s 6 s 1 0 0 s 2 s 3 0 − s 7 0 s 5 0 s 1 0 s 3 s 4 0 s 6 − s 5 0 0 0 s 1 s 4 s 5 s 5 s 8 0 0 0 − s 4 s 3 0 s 6 s 6 0 s 8 0 s 4 0 − s 2 0 s 7 s 7 0 0 s 8 − s 3 s 2 0 0 s 8 0 0 0 0 s 5 s 6 s 7 s 8 T able 2.3: Multiplication table of the split o ctonions O Z in standard basis. One can directly v erify Lemma 3.2 in [Gui07] using the ab o v e metho d, as w e will do in the follo wing lemma. Ho w ev er, an explanation of notation is needed b efore doing so. In line with common computer programming con v en tions, M a:b,c:d means the submatrix of M taking ro w indices in {a,...,b} and column indices in {c,...,d}. If only a colon is presen t, this is understo o d to range from 1 to 8, e.g. M 1,: is the rst ro w of M and M :,1 is the rst column. Lemma 2.2.4. G 2 c ontains the semi-dir e ct pr o duct SL 3 ⋊Z 2 . Sp e cic al ly, the fol lowing matric es ar e in G 2 ⊂ GL 8 for al l matric es G∈SL 3 :          1 G (G − 1 ) T 1          and          1 − (G − 1 ) T − G 1          Pro of: By expanding the 8× 8 matrices from Theorem 2.2.3 in to a system of equations and re-grouping based on shared v ariables, w e get that an equiv alen t form ulation is M 1,: − → φ sjs k =M 1,j M 1,k +M 2:4,j · M 5:7,k M 2:4,: − → φ sjs k =M 1,j M 2:4,k +M 8,k M 2:4,j − M 5:7,j × M 5:7,k M 5:7,: − → φ sjs k =M 8,j M 5:7,k +M 1,k M 5:7,j +M 2:4,j × M 2:4,k M 8,: − → φ sjs k =M 8,j M 8,k +M 5:7,j · M 2:4,k These equations indicate that there are four related t yp es of en tries in M : (1) the rst ro w, (2) ro ws 2 through 4, (3) ro ws 5 through 7, and (4) the last ro w. As suc h, from no w on w e will rewrite M as 19 M =          a 1 ... a 8 ⃗ u 1 ... ⃗ u 8 ⃗ v 1 ... ⃗ v 8 b 1 ... b 8          The details of the computations are left to the App endix Section 7.1. Most imp ortan tly: If w e assume that: ⃗ u 1 , ⃗ v 1 ,⃗ u 8 ,⃗ v 8 = ⃗ 0 a i +b i =0 for all i∈{2,...,7} Then w e get that: a 1 ,b 1 ,a 8 ,b 8 ∈{0,1} a i =0 and b i =0 for all i∈{2,...,7} ⃗ u 5 =− ⃗ v 3 × ⃗ v 4 , ⃗ u 6 =⃗ v 2 × ⃗ v 4 , ⃗ u 7 =− ⃗ v 2 × ⃗ v 3 ⃗ v 5 =⃗ u 3 × ⃗ u 4 , ⃗ v 6 =− ⃗ u 2 × ⃗ u 4 , ⃗ v 7 =⃗ u 2 × ⃗ u 3 if a 1 ,b 8 =1 then b 1 ,a 8 =0, ⃗ v 2 ,⃗ v 3 ,⃗ v 4 = ⃗ 0 and (⃗ u 3 × ⃗ u 4 )· ⃗ u 2 =1 if a 1 ,b 8 =0 then b 1 ,a 8 =1, ⃗ u 2 ,⃗ u 3 ,⃗ u 4 = ⃗ 0 and (⃗ v 3 × ⃗ v 4 )· ⃗ v 2 =− 1 Put together, these giv e the needed result. This subgroup SL 3 ⋊Z 2 stabilizes a cop y of the split-complex n um b ers con tained in the split-o ctonions. The algebra ofO Z corresp onding to the split complex n um b ers is giv en get all elemen ts of the form    a ⃗ 0 ⃗ 0 b    InR 8 this corresp onds to the v ector (a,0,...,0,b) T . T o see ho w SL 3 ⋊Z 2 stabilizes:          1 G (G − 1 ) T 1                   a ⃗ 0 ⃗ 0 b          =          a ⃗ 0 ⃗ 0 b          and          1 − (G − 1 ) T − G 1                   a ⃗ 0 ⃗ 0 b          =          b ⃗ 0 ⃗ 0 a          20 Th us, one cop y of SL 3 xes the subalgebra while the other ips it. In addition, a bit more w ork sho ws that there is a cop y of SL 2 in G 2 that xes the split-quaternions. Corollary 2.2.5. If ad− bc=1 then                       1 a b 1 c d d − c 1 − b a 1                       xes the c opy of the split-quaternions in the split-o ctonions given as a sub algebr a of O Z by                                             w      0 x 0           0 y 0      z                :w,x,y,z∈R 8                              In addition, an element of G 2 ⊂ GL 8 xes this sub algebr a i it is of this form. Pro of: The corollary follo ws easily from the matrix form in Lemma 2.2.4 com bined with the fact that a system xing an elemen t of the split-quaternions will tak e the form                       w 0 x 0 0 y 0 z                       =                       1 a 2 0 a 4 a 5 0 a 7 0 0 u 21 0 u 41 u 51 0 u 71 0 0 u 22 1 u 42 u 52 0 u 72 0 0 u 23 0 u 43 u 53 0 u 73 0 0 v 21 0 v 41 v 51 0 v 71 0 0 v 22 0 v 42 v 52 1 v 72 0 0 v 23 0 v 43 v 53 0 v 73 0 0 b 2 0 b 4 b 5 0 b 7 1                                             w 0 x 0 0 y 0 z                       This corollary is helpful in nding a cop y of SO 4 inG 2 b ecause there is a w ell kno wn 2-to-1 and surjectiv e map SL 2 × SL 2 →SO 4 ; the SL 2 xes the split-quaternions while the SO 4 stabilizes them ([nLab] G2 3.3). T aking all linear groups to b e dened o v er C: 21 Let V =M 2× 2 (C) ∼ =C 4 Ha v e a left and righ t action of SL 2 onC 4 tak en from matrix m ultiplication   a b c d     w x y z   =   aw +by ax+bz cw +dy cx+dz   =⇒ ρ L     a b c d                       w x y z                   =          aw +by ax+bz cw +dy cx+dz            w x y z     e f g h   − 1 =   hw− gx − fw +ex hy− gz − fy +ez   =⇒ ρ R     e f g h                       w x y z                   =          hw− gx − fw +ex hy− gz − fy +ez          Th us, there is an action of SL 2 × SL 2 onC 4 giv en b y ρ       a b c d   ,    e f g h                         w x y z                   =ρ R       e f g h                ρ L       a b c d                         w x y z                            Clearly ρ L (− ) = ρ ( , − ,I) and ρ R (− ) = ρ (I,− ). It is easy to sho w that ρ is equiv alen t to acting with the tensor pro duct via left-m ultiplication, i.e. ρ       a b c d   ,    e f g h       =    a b c d   ⊗    e f g h    − 1 =          ah − af bh − bf − ag ae − bg be ch − cf dh − df − cg ce − dg de          One can sho w that this matrix is an elemen t of SO 4 without m uc h dicult y , and the 2-to-1 nature of the map is clear b ecause its k ernel is generated b y (− I,− I), i.e. ρ       a b c d   ,    e f g h       =ρ   −    a b c d   ,−    e f g h       Th us, if w e can nd suitable a second em b edding of SL 2 then w e will get an em b edding of SO 4 as w ell b y this 2-to-1 map. This idea supp orted b y the claim that SO 4 ∩SL 3 is the cop y of SL 2 as found ab o v e. 22 Finally , here w e think of G 2 as a subgroup of GL 8 and it is w ell kno wn that the diagonal in v ertible matrices form maximal tori of general linear groups. Th us, natural questions to ask are: 1. What are the diagonal matrices in this form of G 2 ? 2. Do these diagonal matrices form a maximal torus of G 2 ? T urning to Theorem 2.2.3, the follo wing relations hold when considering the case of M b eing a diagonal matrix: M 11 =M 22 M 55 =M 33 M 66 =M 44 M 77 =M 88 M 22 =M 66 M 77 M 33 =M 55 M 77 M 44 =M 55 M 66 M 55 =M 33 M 44 M 66 =M 22 M 44 M 77 =M 22 M 33 With a little bit of w ork, one gets that M tak es the follo wing form for some non-zero a and b: M =                       1 a a − 1 b − 1 b a − 1 ab b − 1 1                       It’s classically kno wn that the rank of b oth SO 4 and G 2 is 2, so the collection of all matrices of the ab o v e t yp e is indeed a maximal torus for this em b edding. In addition, it’s clear that suc h matrices M are con tained in the cop y of SL 3 from Lemma 2.2.4. These facts will b e useful in the pro of of Lemma 5.4.6. 2.3 G 2 and SO 4 as Lie Groups This section rev olv es around the denition of transgressiv e and univ ersally transgressiv e elemen ts in cohomology . Consider the Serre sp ectral sequence of the bration F → E p − → B for 1-connected B and 0-connected F . The use of suc h a sp ectral sequence H i (B;H j (F;R)) =⇒ H i+j (E;R) giv es that the cohomology of the total space E can b e computed sequen tially from the cohomologies of the base space B and b er F . This sequence consists of groups E i,j r and morphisms d i,j r :E i,j r →E i+r,j− r+1 r for eac h i,j∈N and r≥ 2 where E i,j 2 =H i (B;H j (F;R)) and E i,j r+1 =Ker(d i,j r )/Im(d i− r,j+r− 1 r ) ([Hat04] 1.5.15). 23 Giv en a sp ectral sequence H i (B;H j (F;R)) =⇒ H i+j (E;R) as ab o v e, it’s clear that there is the map d 0,r− 1 r :E 0,r− 1 r →E r,0 r that go es from b et w een the v ertical and horizon tal edges of the sequence. Note that E 0,r− 1 r is con tained in E 0,r− 1 2 = H r− 1 (F;R) while E r,0 2 is a quotien t of E r,0 2 = H r (B;R). Recall the long exact sequence of cohomology giv en b y ...→H i (E,F;R)→H i (E;R)→H i (F;R)→H i+1 (E,F;R)→... induced b y the short exact sequence 0 → C i (E,F;R) → C i (E;R) → C i (F;R) → 0. W e ma y think of F as p − 1 (b) for some b∈ B , meaning that p tak es the p oin ted pair (E,F) to (B,b). T aking the induced map p ∗ i :H i (B;R)→H i (E,F;R) and the cob oundary map δ i :H i− 1 (F;R)→H i (E,F;R) is enough to giv e the needed denition. Denition 2.3.1. ([MT91] 7.2.5) An elemen t x ∈ H r− 1 (F;R) is tr ansgr essive if δ r− 1 (x) ∈ Im(p ∗ r ). Sim- ilarly , an elemen t y ∈ H r (B;R) is susp ensive if p ∗ r (y) ∈ Im(δ r− 1 ). If δ r− 1 (x) = p ∗ r (y) then w e sa y y is tr ansgr ession image of x and that x is the susp ension image of y ; in this case, w e write y = τ (x) and x=σ (y). In terms of the sp ectral sequence: x∈H r− 1 (F;R)=E 0,r− 1 2 is transgressiv e i x∈E 0,r− 1 r y∈H r (B;R)=E r,0 2 is susp ensiv e i the class of y in E r,0 r is in Im(d 0,r− 1 r ), whic h is equiv alen t to the class of y in E r,0 r+1 b eing 0 This construction leads to the follo wing v ery useful fact: Corollary 2.3.2. ([MT91] 7.2.8) L etK b e a eld and F →E p − → B b e a br ation with 1-c onne cte d B and 0- c onne cte d F . Assume H ∗ (E;K)=K and either F =Ω B orB =BF . IfH ∗ (F;K) ∼ =∆( x) forx tr ansgr essive then H ∗ (B;K) ∼ =K[τ (x)]. Similarly, if H ∗ (B;K) ∼ =K[y] for y susp ensive then H ∗ (F;K) ∼ =∆( σ (y)). When in the setting of principal bundles, there is the stronger notion of b eing univ ersally transgressiv e. As a refresher, recall that for a top ological group G, a principal G-bundle tak es the form G→E→B . Suc h a bundle is univ ersal if E is w eakly con tractible; in this case w e write E = EG and B = BG and sa y that BG is a classifying space of G. W e sa y that an elemen t x∈H n (G;R) for n> 0 is universal ly tr ansgr essive if it is transgressiv e for an y principal G-bundle. By the prop erties of classifying spaces, this is equiv alen t to sa ying that x is transgressiv e in the univ ersal bundle G→EG→BG ([MT91] 7.5.11). F act 2.3.3. ([MT91] 7.5.4) A ny c omp act, 1-c onne cte d simple Lie gr oup G is r ational ly e quivalent to a pr o duct of o dd dimensional spher es. In other wor ds, ther e exist inte gers m 1 ,...,m l such that 1=m 1 <m 2 ≤ ...≤ m l 24 andH ∗ (G;Q) ∼ =H ∗ (S 2m1+1 × ...× S 2m l +1 ;Q). These inte gers ar e unique and we say that (2m 1 +1,...,2m l +1) is the rational t yp e of G. F or example, the rational t yp e of G 2 is (3,11) and therefore l =2 with (m 1 ,m 2 )=(1,5). Putting all of the ab o v e together w e get the follo wing results: Theorem 2.3.4. ([MT91] 7.5.9-10) F or a c omp act, 1-c onne cte d simple Lie gr oup G of r ational typ e given by (2m 1 +1,...,2m l +1), if m l <min(pm 2 ,p 2 − 1) then H ∗ (G;Z p )=Λ Zp (x 2m1+1 ,...,x 2m l +1 ) In addition, these x 2mi+1 ’s c an b e chosen to b e universal ly tr ansgr essive so that y 2mi+1 =τ (x 2mi+1 ) and H ∗ (BG;Z p )=Z p [y 2m1+2 ,...,y 2m l +2 ] In p articular, the c ohomolo gy of G and BG have no p-torsion. Note: Recall that Λ Zp (x 2m1+1 ,...,x 2m l +1 ) denotes the exterior algebra of the Z p -v ector space spanned b y the elemen ts x 2m1+1 ,...,x 2m l +1 and that eac h x 2mi+1 corresp onds to an elemen t of H 2mi+1 (G;Z p ). This ab o v e theorem will b e v ery useful in the computation of motivic cohomology later in this text. In particular, this sa ys that H ∗ (BG 2 ;Z p )=Z p [y 4 ,y 12 ] and has no p-torsion for all primes p≥ 3. Finally , there is one more imp ortan t fact and t w o more computations that will b e useful in Sections 5.2 through 5.4. Recall that ([Bum10]): A group G is solvable if there are subgroups G 0 ,...,G k suc h that G 0 = {1 G }, G k = G, eac h G i is a normal subgroup of G i+1 , and eac h G i+1 /G i is ab elian A Bor el sub gr oup of an algebraic group G is a maximal, Zariski closed, connected and solv able algebraic subgroup F act: all Borel subgroups are conjugate and the in tersection of an y t w o con tains a maximal torus A p ar ab olic sub gr oup of an algebraic group G is a Zariski closed algebraic subgroup H suc h that G/H exists as a pro jectiv e v ariet y equiv alen tly , a Borel subgroup is a minimal parab olic subgroup F or A ∈ GL n (C), w e sa y that A is semisimple i it is diagonalizable and unip otent i all of its eigen v alues are 1 25 A Lie subgroup H of a Lie group G is a torus i it is connected, ab elian, and all elemen ts are semisimple unip otent i it is connected and all elemen ts are unip oten t A Lie group G is r e ductive i it has no non-trivial normal unip oten t subgroups It is w ell kno wn that if G is a compact connected Lie group and T is a torus of G then the follo wing are equiv alen t: T is maximal, C G (T) = T , G = ∪ g∈G gTg − 1 . As suc h, w e dene the W eyl gr oup of (G,T) for suc h a G with maximal T to b e W =N G (T)/T . Lemma 2.3.5. T aken over C, the W eyl gr oup of SO 4 is isomorphic to Z 2 × Z 2 . In p articular, we c an identify it with the fol lowing sub gr oup:                           1 1 1 1          ,          1 1 1 1          ,          1 1 1 1          ,          1 1 1 1                           Pro of: It’s w ell kno wn that the W eyl groups of SO 2n and SO 2n+1 are isomorphic to S n ⋉Z n− 1 2 and S n ⋉Z n 2 resp ectiv ely and that their ma y b e tak en to b e T 2n =                  A 1 . . . A n       :A i ∈SO 2            T 2n+1 =                           A 1 . . . A n 1          :A i ∈SO 2                  Here, S n acts onZ n 2 b y p erm uting the factors and so the m ultiplication rule in S n ⋉Z n 2 is giv en b y (α, (a 1 ,...,a n ))(β, (b 1 ,...,b n ))=(αβ, (a 1 +b α (1) ,...,a n +b α (n) )) Corresp ondingly , S n acts on T 2n+1 b y p erm uting the A i ’s while theZ n 2 acts b y in v ersion, i.e. 26 s:Z 2 →{− 1,1},a7→        1 − 1 ,if a=0 ,if a=1 =⇒ (α, (a 1 ,...,a n ))          A 1 . . . A n 1          =          A s(a1) α (1) . . . A s(an) α (n) 1          One can sho w that this is in fact a w ell-dened action of S n ⋉Z n 2 on T 2n+1 . Dene the map Σ n :Z n 2 →Z 2 , (a 1 ,...,a n )7→a 1 +...+a n . It is clear that Ker(Σ n ) ∼ =Z n− 1 2 and it is p ossible to sho w that the W eyl group of SO 2n (C) is isomorphic to the subgroup S n ⋉Ker(Σ n ), whic h in turn acts on T 2n via conjugation as M (α, (a1,...,an− 1))       A 1 . . . A n       M − 1 (α, (a1,...,an− 1)) Here, the matrix M (α, (a1,...,an− 1)) is the pro duct Q (a1,...,an− 1) P α where: P α is the p erm utation matrix with 2× 2 blo c ks    1 0 0 1    in indices (2i− 1 : 2i,2α (i)− 1 : 2α (i)) for eac h i Notation: F or a matrix A = (a ij ), en tries (a : b,c : d) form the submatrix where a≤ i≤ b and c≤ j≤ d Q (a1,...,an− 1) =          Q a1 . . . Q an− 1 Q Σ n(a1,...,an)          where Q a =                            1 0 0 1         0 1 1 0     ,if a=0 ,if a=1 T aking n = 2, w e can no w nd the matrices in the W eyl group W ∼ = S 2 ⋉Z 2 ∼ =Z 2 × Z 2 and complete the pro of: M ((1)(2),(0)) =          1 1 1 1                   1 1 1 1          =          1 1 1 1          27 M ((1)(2),(1)) =          1 1 1 1                   1 1 1 1          =          1 1 1 1          M ((1 2),(0)) =          1 1 1 1                   1 1 1 1          =          1 1 1 1          M ((1 2),(1)) =          1 1 1 1                   1 1 1 1          =          1 1 1 1          Borel subgroups are helpful esp ecially due to their role in the Bruhat de c omp osition . Sp ecically , supp ose G is a connected, reductiv e group o v er and algebraically closed eld. Fix a Borel subgroup B and a W eyl group W corresp onding to a maximal torus in B . Then: G=BWB = G w∈W BwB Note that W ma y not b e a subgroup of G, but that the coset wB is w ell-dened b ecause the giv en maximal torus is con tained in B . As a bit of notation, for a matrix A let T A denote the transp ose across the rev erse diagonal. Recall that for a closed subgroup G of GL n (C), dene Lie(G) = {A ∈ M n× n (C) : e tA ∈ G∀t ∈ R} to b e the corresp onding Lie algebra. As the indenite sp ecial orthogonal groups are all isomorphic o v er C (see Section 5.2), w e use the form of SO n giv en b y J =       0 ... 1 . . . . . . . . . 1 ... 0       ∈M n× n (K) =⇒ SO ′ n ={A∈SL n (K):A T JA=J} As stated in [Sin19] Chapter 8, taking G=SO ′ 2n giv es asso ciated Lie algebra and Borel subgroup as follo ws: 28 Lie(SO ′ 2n )=         P Q R S    :P,Q,R,S∈M n× n (C),Q=− T Q,R =− T R,S =− T P      B =      exp   t    P Q 0 − T P − 1       : P,Q∈M n× n (C),P upper triangular, T P − 1 =− T P,Q=− T Q,t∈R      T aking n=2: P =    a b 0 c    =⇒ T P − 1 =    1 c − b ac 0 1 a    =    − c − b 0 − a    =− T P =⇒ (a,b,c)∈{(i,0,i),(− i,0,− i)}∪{(i,α, − i):α ∈C}∪{(− i,α,i ):α ∈C} Q=    d e f g    =    − g − e − f − d    =⇒ d+g =0,e=0,f =0 This sho ws that all elemen ts of the Borel subalgebra Lie(B) are of the follo wing form for some α,β ∈C:   P Q 0 − T P − 1   =          i β i − β − i − i          ,          − i β − i − β i i          ,          i α β − i − β i − α − i          or          − i α β i − β − i − α i          Th us, the exp onen tial map sho ws that all elemen ts of the Borel group B tak e the form          e it βsin (t) e it − βsin (t) e − it e − it          ,          e − it βsin (t) e − it − βsin (t) e it e it          ,          e it αsin (t) βe it − αβsin (t) e − it − βe − it e it − αsin (t) e − it          or          e − it αsin (t) βe − it − αβsin (t) e it − βe it e − it − αsin (t) e it          W e need to mak e a small adjustmen t b efore concluding. Our W eyl group is asso ciated with SO 4 while our Borel subgroup comes from SO ′ 4 . As discussed in Sections 5.2 and 7.2, there is an isomorphism SO ′ 4 →SO 4 giv en b y A7→gAg − 1 where 29 g =          a 1 2a b 1 2b − ib i 2b − ia i 2a          for an y a,b∈G m . F or con v enience w e will tak e a and b to b e 1. All of the ab o v e information allo ws us to break do wn elemen ts of SO 4 in more meaningful w a y . T o summarize: The Bruhat decomp osition of SO 4 is SO 4 =(gBg − 1 )W(gBg − 1 ) where: the W eyl group W of SO 4 is W =                           1 1 1 1          ,          1 1 1 1          ,          1 1 1 1          ,          1 1 1 1                           the Borel subgroup gBg − 1 of SO 4 is all matrices of one of the follo wing forms:          cos(t) βsin (t) − iβsin (t) − sin(t) − βsin (t) cos(t) − sin(t) iβsin (t) iβsin (t) sin(t) cos(t) βsin (t) sin(t) − iβsin (t) − βsin (t) cos(t)          ,          cos(t) βsin (t) − iβsin (t) sin(t) − βsin (t) cos(t) sin(t) iβsin (t) iβsin (t) − sin(t) cos(t) βsin (t) − sin(t) − iβsin (t) − βsin (t) cos(t)          ,          cos(t)− αβsin (t) 1 2 αsin (t)+βe it i 2 αsin (t)− iβe it (− 1+iαβ )sin(t) − 1 2 αsin (t)− βe − it cos(t) sin(t) i 2 αsin (t)+iβe − it − i 2 αsin (t)+iβe − it − sin(t) cos(t) − 1 2 αsin (t)+βe − it (1+iαβ )sin(t) − i 2 αsin (t)− iβe it 1 2 αsin (t)− βe it cos(t)+αβsin (t)          or          cos(t)− αβsin (t) 1 2 αsin (t)+βe − it i 2 αsin (t)− iβe − it (1+iαβ )sin(t) − 1 2 αsin (t)− βe it cos(t) − sin(t) i 2 αsin (t)+iβe it − i 2 αsin (t)+iβe it sin(t) cos(t) − 1 2 αsin (t)+βe it (− 1+iαβ )sin(t) − i 2 αsin (t)− iβe − it 1 2 αsin (t)− βe − it cos(t)+αβsin (t)          30 Chapter 3 Motivic Homotopy 3.1 Homotopy Categories Denition 3.1.1. ([Qui73]) A c ate gory with we ak e quivalenc es is a pair (C,W) where C is a category with sub category W suc h that for all A,B,C ∈Ob(C), f ∈Hom C (A,B) and g∈Hom C (B,C): If f ∈Hom W (A,B) and g◦ f ∈Hom W (A,C) then g∈Hom W (B,C) If g∈Hom W (B,C) and g◦ f ∈Hom W (A,C) then f ∈Hom W (A,B) In other w ords, a c hoice of w eak equiv alences W in C amoun ts to a c hoice of ob jects and morphisms in C suc h if an y t w o out of the three sides of a comm utativ e triangle from C are in W then so is the third side. W e only need to sp ecify the t w o conditions ab o v e b ecause the natural third condition is already included in the denition of a sub category . Recall that W is a sub c ate gory ofC if the follo wing hold: Ob(W)⊂ Ob(C) If f ∈Hom W (A,B) then A,B∈Ob(W) If f ∈Hom W (A,B) and g∈Hom W (B,C) then g◦ f ∈Hom W (A,C) If A∈Ob(W) then Id A ∈Hom W (A,A) Denition 3.1.2. ([Qui73], [nLab] homotop y category) Giv en a category C with w eak equiv alences W , its homotopy c ate gory Ho(C) (if it exists) is the category with the univ ersal prop ert y that there is a functor Q : C → Ho(C) suc h that Q(f) is an isomorphism for ev ery f in W . Note that the homotop y category is also commonly denoted as W − 1 C orC[W − 1 ] and called the lo c alization of C at W . 31 There is the follo wing general construction of lo calization C[W − 1 ] of a category C with w eak equiv alences W ([Qui73], [nLab] lo calization): Let G(C,W) b e the follo wing directed m ulti-graph where v ertices giv en b y are Ob(C) and the edges b et w een A and B are Hom C (A,B) F Hom W op(A,B) Notation: f ∈Hom W (B,A) corresp onds to f ∈Hom W op(A,B) Let PG(C,W) b e the fr e e c ate gory on G(C,W), namely: Ob(PG(C,W))=Ob(C) Hom PG(C,W) (A,B)= paths in G(C,W) from A to B Notation: (A 0 ;f 1 ,...,f n ;A n ) means the path A 0 f1 − → A 1 f2 − → ... fn− 1 − −− → A n− 1 fn −→ A n W e dene an equiv alence relation ∼ on the morphisms of PG(C,W) generated b y the follo wing: for all A∈Ob(C), set (A;Id A ;A)∼ (A;;A) for all f ∈Hom C (A,B), g∈Hom C (B,C), set (A;f,g;C)∼ (A;g◦ f;C) for all f ∈Hom W (B,A), set (A;f,f;A)∼ (A;Id A ;A) and (B;f,f;B)∼ (B;Id B ;B) Dene C[W − 1 ] to b e PG(C,W) mo dulo this equiv alence relation It’s clear that the goal of this construction is to formally in v ert all morphisms in W using W op . Ho w ev er, the existence of homotop y category is not guaran teed; the ab o v e construction of PG(C,W)/∼ could fail to actually b e a small category ev en giv en that C is small to b egin with ([Qui73]). F act 3.1.3. If it exists, the homotopy c ate gory Ho(C) is unique up to e quivalenc e of c ate gories. In addition, this is an abstract and complicated construction in practice; if w e assume some additional conditions then w e can explicitly construct these homotop y categories in a more simple w a y . Let f ∈ Hom C (A,B), g ∈ Hom C (C,D) b e t w o morphisms in C . W e sa y that f has the left lifting pr op erty with resp ect to g (resp. g has the right lifting pr op erty with resp ect to f ) if for all morphisms u∈Hom C (A,C), v∈Hom C (B,D) w e ha v e that A u // f C g A u // f C g B v // D =⇒ ∃ w∈Hom C (B,C) s.t. B v // w ?? D Here, it’s clear that u=w◦ f and v =g◦ w ; this matc hes our in tuition ab out w b eing on the left of f and the righ t of g . W e ma y think of w as an extension of u and a lift of v . 32 Denition 3.1.4. ([Qui73], [nLab] w eak factorization system) A we ak factorization system on a category C is a pair (L,R) of classes of morphisms in C suc h that: L(A,B)⊂ Hom C (A,B) andR(A,B)⊂ Hom C (A,B) for all A,B∈Ob(C) ev ery morphism f ∈ Hom C (A,B) can b e factored as a comp osition h◦ g for some ob ject C ∈ Ob(C) with morphisms g∈L(A,C) and h∈R(C,B) L is precisely the class of maps ha ving the left lifting prop ert y with resp ect to ev ery morphism in R R is precisely the class of maps ha ving the righ t lifting prop ert y with resp ect to ev ery morphism in L W eak factorization systems pro vide a general structure that deriv es from brations and cobrations. Recall that a br ation is a con tin uous map p : E→B that satises the homotop y lifting prop ert y with resp ect to an y space, i.e. for an y space X , homotop y f : X× [0,1]→ B , and map e f 0 : X → E lifting f 0 = f| X×{ 0} (i.e. p◦ e f 0 = f 0 ), there exists a homotop y e f : X× [0,1]→ E lifting f suc h that e f 0 = e f| X×{ 0} . Similarly , a c obr ation is a con tin uous map i:B→E that satises the homotop y extension prop ert y with resp ect to an y space, i.e. for an y space X , homotop y f :B× [0,1]→X , and map e f 0 :E→X suc h that e f 0 ◦ i =f| B×{ 0} , there exists a homotop y e f :E× [0,1]→X extending f suc h that e f◦ (i× Id [0,1] )=f . Denition 3.1.5. ([Qui73]) A mo del structur e on a category C with w eak equiv alences W is a c hoice of distinguished classes of cobrations Cof and brations Fib suc h that: Cof(A,B)⊂ Hom C (A,B) and Fib(A,B)⊂ Hom C (A,B) for all A,B∈Ob(C) (Cof∩W,Fib) and (Cof,Fib∩W) are b oth w eak factorization systems A category equipp ed with a mo del structure is called a mo del c ate gory . Eectiv ely a mo del category is a category that allo ws for homotop y; W pla ys the role of homotop y equiv alences, and elemen ts of Cof (resp. Fib) are analogous to injectiv e maps lik e cobrations (resp. surjections lik e brations) ([Qui73]). R emark 3.1.6. In order to construct the homotop y category of a mo del category ([Qui73], [nLab] homotop y category of a mo del category 3.2): Let ∅ and∗ b e the initial and terminal ob jects of a mo del category C F or eac h A ∈ Ob(C), c ho ose factorizations ∅ i A −−−→ ∈Cof QA p A − −−−−− → ∈Fib∩W A suc h that if ∅ → A is already in Cof (i.e. A is a c obr ant ob ject) then QA=A and p A =Id A F or eac h A ∈ Ob(C), c ho ose factorizations A j A − −−−−− → ∈Cof∩W RA q A −−−→ ∈Fib ∗ suc h that if A → ∗ is already in Fib (i.e. A is a br ant ob ject) then RA=A and j A =Id A 33 W e can tak e Ho(C) to b e a sub category of C ; it is the image of the functor γ R,Q :C→Ho(C)⊂C where γ R,Q (A)=RQA and giv en some f ∈Hom C (A,B) w e tak e γ R,Q (f)=RQf ∈Hom Ho(C) (RQA,RQB) to b e constructed b y rst lifting f to Qf as ∅ i B // i A QB p B QA f◦ p A // Qf 88 B then extending Qf to RQf as QA j QB ◦ Qf // j QA RQB q QB RQA q QA // RQf 77 ∗ Note: These maps Qf and RQf exist b y the left and righ t lifting prop erties of mo del categories, and one can sho w that this v ersion of Ho(C) satises the univ ersal prop erties needed for a homotop y category of C Homotop y categories mirror m uc h of the b eha vior that homotop y do es in top ology . In particular, recall that w e can obtain cohomology from homotop y using the fact that H n (X;A) ∼ = [X,K(A,n)]. Here, K(A,n) is an Eilen b erg-Mac Lane space for an ab elian group A and [X,K(A,n)] is the collection of homotop y classes of maps from X to K(A,n). This op eration [− ,− ] is of particular in terest here and can b e generalized to other categories: Denition 3.1.7. ([Dyc18]) Let C b e a symmetric monoidal category . An internal hom in C is a functor [− ,− ]:C op ×C→C suc h that for ev ery ob ject X ∈Ob(C) w e ha v e a pair of adjoin t functors −⊗ X :C→C and [X,− ] :C →C , i.e. there is a natural isomorphism Hom C (−⊗ X,− )≃ Hom C (− ,[X,− ]). Note that if the category is not symmetric then there are notions of left and righ t in ternal homs. The construction of an in ternal hom v aries greatly dep ending on the category . F or example, when C = SimpSet there is the in ternal hom where [X,Y]([n]) = Hom SimpSet (X× ∆[ n],Y). When in Set w e often tak e [A,B] = Hom Set (A,B). The v alidit y here is due to the fact that Set is a close d c ate gory (i.e. Hom C (A,B)∈Ob(C) for all A,B∈Ob(C)) and so the equation Hom Set (A× B,C)≃Hom Set (A,Hom Set (B,C)) 34 ts a tensor-hom adjunction b ecause Hom Set (B,C) is itself a set ([Dyc18] 2.6). There is a problem for the case of Sm/B , i.e. the category of smo oth sc hemes o v er a sc heme B . Supp ose there are maps p : X → B , q : Y → B , and r : Z → B along with the pullbac k X × B Y that together satisfy the tensor-hom adjunction Hom Sm/B (X× B Y,Z)≃ Hom Sm/B (X,[Y,Z]). W e cannot set [Y,Z] to b e Hom Sm/B (Y,Z) b ecause this is not a sc heme, i.e. Sm/B is not a closed category . In fact, one can ev en sho w that there is no in ternal hom structure on Sm/B for this reason. On the other hand, it’s w ell kno wn that the inclusion functor F :Ho(C),→C for a mo del category alw a ys admits the left adjoin t lo calization functor γ R,Q :C → Ho(C), as in Remark 3.1.6, whic h has the prop ert y that Hom Ho(C) (γ R,Q (A),B)≃Hom C (A,F(B)) for all A∈Ob(C) and B∈Ob(Ho(C)) ([Qui73]). 3.2 The tale T opology The Øtale top ology is imp ortan t as it relates to the more hea vily used Nisnevic h top ology in this w ork. As suc h, a brief summary will b e presen ted here. Denition 3.2.1. ([Stac ks] 00S0) Let φ:R→S b e a ring homomorphism suc h that φ(1 R )̸=0 S . Denote b y R[S] the p olynomial ring whose v ariables are elemen ts of S . If w e denote [s] to b e the v ariable corresp onding to s∈S then R[S] is the free R-mo dule with basis elemen ts {[s]} s∈S . There is a canonical surjection giv en b y φ poly : R[S] → S,r[s 1 ] α 1 ...[s n ] α n → φ(r)s α 1 1 ...s α n n . Note that K = Ker(φ poly ) is generated b y all elemen ts of the form [s+s ′ ]− [s]− [s ′ ], [ss ′ ]− [s][s ′ ] and [r]− r . In addition, recall that the map φ:R→S mak es S in to an R-mo dule b y taking r· s=φ(r)s. Denition 3.2.2. ([Stac ks] 00RM) Let φ : R → S b e a ring homomorphism. The mo dule of Khler dier entials of S over R is the pair (Ω S/R ,d) where d : S → Ω S/R asso ciates a to [a] and Ω S/R is the cok ernel of the map (⊕ (a,b)∈S 2S[(a,b)])⊕ (⊕ (f,g)∈S 2S[(f,g)])⊕ (⊕ r∈R S[r])→⊕ a∈S S[a] [(a,b)]7→[a+b]− [a]− [b] [(f,g)]7→[fg]− [f]g− f[g] [r]7→[φ(r)] The Khler dieren tials are dieren t from t ypical dieren tials in that they satisfy an imp ortan t univ ersal prop ert y . Recall that for an S -mo dule o v er M , an R-deriv ation with v alues in M is a map D :S→M suc h that D◦ φ(r)=0, D(s+s ′ )=D(s)+D(s ′ ) and D(ss ′ )=s ′ D(s)+sD(s ′ ) for all r∈R and s,s ′ ∈S ; denote 35 the collection of all suc h deriv ations as Der R (S,M). It’s clear that Ω S/R is itself an S -mo dule b y the map d:S→Ω S/R , so with this in mind: F act 3.2.3. ([Stacks] 00RM, 00S0) (Ω S/R ,d) has the fol lowing pr op erties: 1. (Univ ersal Prop ert y) The map Hom S (Ω S/R ,M)→ Der R (S,M),α 7→ α ◦ d is an isomorphism for all S -mo dules M . 2. If φ:R→S is surjectiv e then Ω S/R =0. 3. Dene the map δ : K/K 2 → Ω R[S]/R ⊗ R[S] S,f +K 2 7→ d(f)⊗ 1 S , where d : R[S] → Ω R[S]/R . The cok ernel of this map δ is canonically isomorphic to Ω S/R . More concretely , supp ose R is a nitely generated K-algebra for char(K) = 0. Then Ω R/K is the R-mo dule generated b y the sym b ols {dr} r∈R mo dulo the relations {d(ab)− a(db)− b(da)} a,b∈R and{dr} r∈K ; th us Ω R/K is nitely generated and has deriv ation d:R→Ω R/K ,r7→dr [Ked08]. Denition 3.2.4. ([Stac ks] 07BN) The naive c otangent c omplex NL S/R of S o v er R is the c hain complex K/K 2 δ − → Ω R[S]/R ⊗ R[S] S , with K/K 2 placed in (homological) degree 1 and Ω R[S]/R ⊗ R[S] S in degree 0. Example 3.2.5. Let φ:R→S b e a ring morphism. If S is a lo calization of R then NL S/R is trivial. Denition 3.2.6. ([Stac ks] 00U1) Let φ:R→S b e a ring homomorphism. W e sa y that φ:R→S is Øtale if it is of nite presen tation and NL S/R quasi-isomorphic to zero. Recall the common denition that a morphism A→ B of c hain complexes is a quasi-isomorphism if the induced maps on (co)homology groups are all isomorphisms. One imp ortan t and related notion here is the concept of smo othness for ring and sc heme morphisms. W e sa y that a ring morphism φ :R→S is smo oth if it is of nite presen tation and NL S/R is quasi-isomorphic to a nite pro jectiv e S -mo dule M placed in degree 0 ([Stac ks] 00T2). This means that there is a c hain map K/K 2 δ // f1 Ω R[S]/R ⊗ R[S] S // f0 0 0 // M // 0 suc h that (f 0 ) ∗ : (Ω R[S]/R ⊗ R[S] S)/Im(δ )→ M and (f 1 ) ∗ : Ker(δ )→ 0 are isomorphisms. It’s clear that this implies δ :K/K 2 →Ω R[S]/R ⊗ R[S] S is injectiv e when φ is smo oth. All Øtale maps are smo oth b ecause b eing quasi-isomorphic to zeros is equiv alen t to taking M =0; recall that suc h an S -mo dule is free and th us pro jectiv e. In this case (Ω R[S]/R ⊗ R[S] S)/Im(δ ) is isomorphic to 36 0, whic h only happ ens when Im(δ ) = (Ω R[S]/R ⊗ R[S] S) (i.e. δ is surjectiv e). W e can conclude that the follo wing results hold: φ:R→S is smo oth i δ :K/K 2 →Ω R[S]/R ⊗ R[S] S is injectiv e and S ∼ =R[x 1 ,...,x n ]/(f 1 ,...,f m ) as R-algebras φ : R → S is Øtale i δ : K/K 2 → Ω R[S]/R ⊗ R[S] S is bijectiv e and S ∼ = R[x 1 ,...,x n ]/(f 1 ,...,f m ) as R-algebras Denition 3.2.7. ([Stac ks] 02GI, 01V5) Let f :X →S b e a morphism of sc hemes. W e sa y that the morphism f is Øtale (r esp. smo oth) at x∈X if there exists an ane op en neigh b orho o d Spec(A) = U ⊂ X of x and ane op en Spec(R) = V ⊂ S with f(U)⊂ V suc h that the induced ring map R→A is Øtale (resp. smo oth). W e sa y that f is Øtale (resp smo oth ) if it is Øtale (resp. smo oth) at ev ery p oin t x∈X . 3.3 Nisnevich T opology An imp ortan t v arian t on the Zariski and Øtale top ologies is the Nisnevic h top ology . It is coarser than Øtale but ner than Zariski, meaning that it has some desirable prop erties of b oth. T o construct it, recall that the r esidual eld can b e dened in the follo wing w a y . Let X and a sc heme and x ∈ X . Supp ose U =Spec(A) is an ane neigh b orho o d of x for some comm utativ e ring A. In the neigh b orho o d U , the p oin t x corresp onds to some prime ideal P of A. The lo calization A P has maximal ideal PA P , meaning that w e can tak e the residual eld of X at x to b e k(x) = A p /PA p . Note that the eld is indep enden t of c hoice of neigh b orho o d. Denition 3.3.1. ([Ho y]) F or a sc heme morphism p:U →X andx∈X , w e sa y p is c ompletely de c omp ose d at x if there is some u∈p − 1 (x) suc h that the residual eld extension k(x)→k(u) is an isomorphism. One can sho w that for a Cartesian diagram V g // q U p Y f // X with f(y)=x, if p is completely decomp osed at x then q is completely decomp osed at y . 37 Denition 3.3.2. ([Ho y]) LetB b e a sc heme and X ∈Ob(Sch/B). A family of morphisms{φ i :U i →X} i∈I is called a Nisnevich c overing of X if eac h φ i is Øtale of nite t yp e, I is nite, for ev ery x∈ X there exists some i∈I suc h that φ i is completely decomp osed at x, and X =∪ i∈I φ i (U i ). Equiv alen t Denition: ([Ho y] 1.6, [MHT]) Let B b e a sc heme and X ∈Ob(Sch/B). A family of morphisms {φ i : U i → X} i∈I is called a Nisnevich c overing of X if eac h φ i is Øtale, I is nite, and there is a ltration ∅⊂ Z n ⊂ ...⊂ Z 0 =X suc h that eac h Z j is nitely presen ted, closed, and eac h φ − 1 i (Z j − Z j+1 )→Z j − Z j+1 admits a section. F or example, letK b e a eld of c haracteristic not 2, tak e B =Spec(K) and let a b e non-zero: A 1 K −{ 0} x 2 A 1 K −{ a} //A The ab o v e diagram is a Nisnevic h co v er with ltration ∅⊂{ a}⊂ A 1 K i √ a∈K ([MHT] 1.3, [V o e99] 1.5). Prop osition 3.3.3. ([Hoy] 1.1) L et B b e a scheme and C a ful l sub c ate gory of Sch/B such that the pul lb ack diagr am U× X V p2 // p1 V ψ U ϕ // X in C in which φ is Øtale of nite typ e is in C . Then Nisnevich c overings form a b asis for a Gr othendie ck top olo gy on C . The induced top ology on C ab o v e is called the Nisnevich top olo gy on C . With sligh t abuse of notation, w e use C Nis to denote the Nisnevic h site whose underlying category is C . Note that if in addition ψ is an op en em b edding and ϕ − 1 (X− ψ (V)) is isomorphic to X− ψ (V) w e sa y that the ab o v e square is elementary distinguishe d ([V o e99] 1.4). Before ending, recall that a presheaf is merely a con tra v arian t functor from one category to another; a sheaf requires compatibilit y with a site in the form of an equalizer diagram. Th us, it mak es sense to consider the category of Nisnevic h shea v es on a category C , denoted Shv Nis (C). Denition 3.3.4. ([V o e98] 2.2, [V o e99] 1.4, [Ho y] 1.1) Let C b e a full sub category of Sch/B . A presheaf p:C op →Set is a Nisnevich she af if the image of ev ery elemen tary distinguished square in C is a Cartesian square in Set, i.e. p(∅) is a one-p oin t set and p(U× X V)=p(U)× p(X) p(V). This denition extends naturally 38 to preshea v es taking v alues in something other than Set b y requiring that the image is a square in the new category . 3.4 Construction of Unstable Motivic Homotopy 3.4.1 A dding Colimits for Homotopy Section 3.1 giv es the denition and construction of general homotop y categories, but it is y et unclear as to wh y they are imp ortan t in the con text of algebraic geometry . Categorically it is con v enien t for categories to ha v e colimits; they are generally related to gluing ob jects together through pushouts. In the ab o v e discussion w e frequen tly reference pullbac ks, but for sc hemes pullbac ks and pushouts are deeply connected from dualit y of a pulbac k of rings v ersus a pushout of sc hemes. F or example, let A,B andC b e rings with maps f :A→C andg :B→C . The b ered pro duct of A andB o v er C isA× C B ={(a,b)∈A× B :f(a)=g(b)}. Ho w ev er, the arro ws rev erse to form a pushout when passing to the corresp onding ane sc hemes, i.e. A× C B p2 // p1 B g Spec(A× C B) Spec(B) p − 1 2 oo A f // C becomes Spec(A) p − 1 1 OO Spec(C) f − 1 oo g − 1 OO Unfortunately , the category Sm/B fails to ha v e colimits. There is a standard w a y to formally add colimits to an y category C . Note that an y ob ject X ∈Ob(C) denes a represen table presheaf R X :C op →Set where Y 7→ Hom C (Y,X). Denote the category of suc h preshea v es as PreShv(C). The Y oneda lemma sho ws that PreShv(C) has colimits b ecause of the corresp ondence X ←→ R X ([V o e98]). This same topic is discussed in [V o e99] on pg. 2 under dieren t language and notation (i.e. uses h X instead of R X ). In principal, one could attempt to apply homotop y theoretic to ols to PreShv(Sm/B). Ho w ev er, the follo wing example demonstrates the dicult y in doing so: Example 3.4.1. ([V o e98] pp. 581-2) Let X ∈ Ob(Sm/B) with U and V Zariski op en subsets of X . Assuming X =U∪V , the square U∩V // V U // X is a pushout in Sm/B . Ho w ev er, the corresp onding square of preshea v es 39 R U∩V // R V R U // R X is only a pushout in PreShv(Sm/B) if U = X or V = X . As stated in [V o e99] on pg. 3, iden tit y map Id X ∈R X (X)=Hom Sm/B (X,X) cannot come from an elemen t of R U (X) unless U =X . The ab o v e example sho ws that PreShv(Sm/B) is insucien t for our needs, but the construction can still b e salv aged. The sp ecic case of Shv Nis (Sm/B) is the full sub category PreShv(Sm/B) whic h consists of shea v es compatible with the Nisnevic h top ology . This particular sub category is useful b ecause the functor Sm/B → PreShv(Sm/B) where X 7→ R X factors through the em b edding ι : Sm/B → Shv Nis (Sm/B). This allo ws us to iden tify smo oth sc hemes with their corresp onding shea v es. The construction of ι is presen ted in F act 3.5.4. The follo wing theorem sho ws that Shv Nis (Sm/B) is sucien t for studying homotop y theoretic constructions. Theorem 3.4.2. ([V o e98] 2.3) The c ate gory Shv Nis (Sm/B) has al l (smal l) limits and c olimits. The inclu- sion functor Shv Nis (Sm/B) → PreShv(Sm/B) has left adjoint, denote d a Nis , which c ommutes with b oth limits and c olimits. Thus, c olimits in Shv Nis (Sm/B) may b e c ompute d by rst doing so in PreShv(Sm/B) and then applying a Nis . More generally , w e w an t to study a category C through the lens of homotop y theory . The category itself is usually not nice enough, so w e c ho ose a category of spaces that con tains C and is nicer; this new category is exactly what is commonly denoted as Spc. In a more classical example, one migh t tak e C to b e the category of CW-complexes and Spc the category of compactly generated spaces ([V o e98]). 3.4.2 Unstable Homotop y Category The shorthand of Spc(B) means one of t w o things dep ending on the source. In [V o e98] pg. 583 and [V o e99] pg. 3, Spc(B) = Shv Nis (Sm/B) is the category of Nisnevic h shea v es on Sm/B with v alues in Set. Ho w ev er in [VR O07] 2.1, Spc(B) is giv en as ∆ op Shv Nis (Sm/B), i.e. is the category of Nisnevic h shea v es Sm/B with v alues in the category SimpSet of simplicial sets; another w a y of sa ying this is that ∆ op Shv Nis (Sm/B) is the category of simplicial ob jects in Shv Nis (Sm/B). This construction is also in [V o e99] on pg. 5 but is denoted ∆ op Spc(B). This w ork will follo w V o ev o dsky’s con v en tion of taking Spc(B) to mean Shv Nis (Sm/B). It is w ell kno wn that Spc(B) is mo del category , and so it satises the nice prop erties that V o ev o dsky refers to ([MV99] Thm 1.4): 40 Denition 3.4.3. The (unstable) homotopy c ate gory of schemes over B , denoted H(B), is the homotop y category of ∆ op Spc(B) ([MV99] 1.4, [CD19] 1.1.5). In other w ords: H(B)=Ho(∆ op Shv Nis (Sm/B)) In line with con v en tions from top ology , w e will denote the homotop y class with square brac k ets, i.e. if F :Shv Nis (Sm/B)→SimpSet then its represen tativ e in H(B) is denoted [F]. It’s w orth noting that H(B) can b e computed using the metho d in Remark 3.1.6. W eak equiv alences in Spc(B) are giv en b y Denition 1.8 of [V o e99]. W e sa y that a (co)bration is trivial if is also a w eak equiv alence. First, tak e Cof and Fib to b e the classes of sc heme cobrations and brations from Sch(B). Next, the sub categoryW ofSm/B has the smallest class of morphisms suc h that W con tains all isomorphisms and satises the follo wing axioms: 1. (homotop y) Ev ery pro jection X× A 1 K →X is inW 2. (saturation/category with w eak equiv alences) If t w o of f , g and g◦ f are in W then so is the third 3. (con tin uit y) If {f ij : X i → X j } i,j∈I is a family of trivial cobrations then eac h X i → lim −→ j∈J X j is in W 4. The pushout of a w eak equiv alence along a cobration is also a w eak equiv alence. 5. The pushout of a trivial cobration along an y map is a w eak equiv alence. T o get a corresp onding sub category of ∆ op Spc(B), one uses the em b edding ι : Sm/B → Spc(B) together with the c hains functor C ∗ : Spc(B) → ∆ op Spc(B). Th us, in tuitiv ely H(B) is the category of smo oth sc hemes o v er B tak en up to isomorphism b y A 1 -w eak homotop y equiv alence. 3.5 F rom Motivic Stable Homotopy to Cohomology There are w ell kno wn isomorphisms T ∧n ∼ = S 2n,n ∼ =A n K /(A n K −{ 0}) and S 2n− 1,n ∼ =A n K −{ 0} ([V o e99] 2.6). As in the classical case, motivic stable homotop y rev olv es around the idea of sp ectra: Denition 3.5.1. ([V o e99] 2.4) A motivic sp e ctrum or a T -sp e ctrum is a sequence (E 0 ,E 1 ,...) in Sm/B together with structure maps T ∧E i →E i+1 . The category of T -sp ectra is denoted TSpectra. There is the susp ension T -sp e ctrum functor Σ ∞ T (− ):Sm/B→TSpectra 41 where the i th space of Σ ∞ T X is E i = T ∧i X . This functor is useful but ultimately needs to b e mo died. Statemen ts ab out homotop y require base-p oin ts in man y cases but this is more complicated in the algebraic geometry setting. As suc h, one adds a disjoin t p oin t Spec(K) and denotes this as X + =X F Spec(K). The functor of adding a base-p oin t is adjoin t to the functor that forgets base-p oin ts of p oin ted spaces, i.e. colimits are preserv ed after adding base-p oin ts. W e simply dene a new functor that com bines b oth of the others: Σ ∞ T (− ) + :Sm/B→TSpectra,X 7→Σ ∞ T (X G Spec(K)) This functor preserv es w eak equiv alences and smash pro ducts, meaning that Σ ∞ T (X ∧ Y) + is equiv alen t to Σ ∞ T X + ∧Σ ∞ T Y + ([V o e99]). Using this, w e can extend Σ ∞ T (− ) + to a functor Σ ∞ T (− ) + on H(B) b y the em b edding ι :Sm/(B),→Spc(B) and the fact that H(B) can b e considered to b e a sub category of Spc(B). Denition 3.5.2. The motivic stable homotopy c ate gory of schemes over B , denotedSH(B), is the category of motivic sp ectra where the E i are ob jects in H(B). Th us, the extension of Σ ∞ T (− ) + :Sm/B→TSpectra is giv en as Σ ∞ T (− ) + :H(B)→SH(B). There is an in ternal hom on SH(B) as in Denition 3.1.7 giv en b y the mapping sp e ctrum . In particular, for t w o motivic sp ectra E = (E 0 ,E 1 ,...) and E ′ = (E ′ 0 ,E ′ 1 ,...) in SH(B) w e dene the inner hom to b e [E,E ′ ]=Hom SH(B) (E,E ′ ) ([V o e99] 3.4). Recall that for an ab elian group A the Eilen b erg-Mac Lane space K(A,n) is a space with only a single non-trivial homotop y group, namely π n (K(A,n)) = A. This space has the useful prop ert y that H n (X;A) is in bijectiv e corresp ondence with [X,K(A,n)], i.e. cohomology of X is equiv alen t to homotop y classes of paths from X to K(A,n). There is a corresp onding notion called the motivic Eilenb er g-Mac L ane sp ac e K(A(q),p) with the similar prop ert y stated b elo w in Prop osition 3.5.6. Muc h lik e in the classical setting, these motivic Eilen b erg-Mac Lane spaces are not unique and there are man y mo dels that ma y b e used to deriv e them. One suc h construction is presen ted here. Let Spc(K) denote the category of Nisnevic h shea v es on the category Sm/K of smo oth sc hemes o v er Spec(K), as in Section 3.4. Denote H(K) andSH(K) in similar w a ys. Denition 3.5.3. ([V o e99] 3.1) F or X ∈Sm/K and ab elian group A, let L(X,A):Sm/K→Ab b e the presheaf sending a connected U to the free ab elian group o v er A on the set of all closed irreducible W ⊂ U× X suc h that the pro jection W →U is nite and surjectiv e. 42 F act 3.5.4. The emb e dding ι : Sm/B → Spc(B) maps X to L(X,Z) c omp ose d with the for getful functor Ab→ Set; the natur al gener alization is to dene ι A : Sm/B → Spc(B),X 7→ L(X,A). F urther c omp osing this with the chains functor C ∗ me ans that X c an b e thought of as a simplicial Nisnevich she af. Mor e gener al ly, we c an think of L(X,A) as a p ointe d sp ac e by for getting the gr oup structur e but r ememb ering the identity ([V o e98]). Notation: F or simplicit y , w e use H A to denote the comp osition Sm/B ι A −→ Spc(B) C∗ − − → ∆ op Spc(B) [− ] − − →H (B) The susp ension sp ectrum Σ ∞ S 1 X + of a p oin ted top ological space X is the sequence (E 0 ,E 1 ,...) where E i = S i ∧X and the structure isomorphisms φ i : S 1 ∧E i → E i+1 . There is an analog in the con text of motivic sp ectra. F or that v ersion, the structure maps come from L(X,A)∧L(X ′ ,A)→L(X∧X ′ ,A) and are induced b y the bilinear L(X,A)(U)× L(X ′ ,A)(U)→L(X× X ′ ,A)(U) where (W,W ′ )7→W× U W ′ . Denition 3.5.5. ([V o e99] 3.3) The susp ension sp e ctrum of X ∈ Sm/K with c o ecients in an ab elian gr oup A is in the motivic stable homotop y category SH(K). Roughly sp eaking, the sp ectrum is giv en b y Σ ∞ T X + = (E 0 ,E 1 ,...) where E i is L(S 2i,i ∧ X + ,A) and eac h map φ i is induced b y the isomorphism T ∧S 2i,i ∧X + →S 2i+2,i+1 ∧X + . Strictly sp eaking: T ak e E i =H A (S 2i,i ∧X + )∈H(K) Ha v e a map L(T,A)∧L(S 2i,i ∧X + ,A)→L(T ∧S 2i,i ∧X + ,A) ∼ =L(S 2i+2,i+1 ∧X + ,A) Get induced equiv alences φ i :H A (T)∧E i →E i+1 The motivic Eilen b erg-Mac Lane space K(A(n),2n) and corresp onding sp ectrum HA ma y b e tak en to b e the follo wing ([V o e98], [V o e99] 3.4): T ak e K(A(n),2n)=[C ∗ (L(P 1 K ,A) ∧n )]∈H(B) with the con v en tion that K(A(0),0) is the image of the constan t sheaf A Sm/K i.e. A Sm/K (X)=A for all X ∈Sm/B and K(A(0),0)=[C ∗ (A Sm/K )] The ab o v e pro duct b ecomes K(A(m),2m)∧K(A(n),2n)→K(A(m+n),2(m+n)) Sp ecically , w e ha v e maps φ n :H A (T)∧K(A(n),2n)→K(A(n+1),2(n+1)) 43 Motivic Eilen b erg-Mac Lane sp ectrum HA is (K(A(0),0),K(A(1),2),...) with structure maps giv en b y the φ n morphisms Recall that the top ological Eilen b erg-Mac Lane spaces satisfy the prop ert y that lo oping shifts their degree, i.e. Ω K(A,n) ∼ =K(A,n− 1); adjoin tness of lo oping and susp ension sho ws that Σ K(A,n) ∼ =K(A,n+1). There is a similar construction for the motivic v ersion that allo ws for K(A(q),p) where p and q are arbitrary . The lo oping is done b y Ω T =Hom H(B) (T,− ) and satises the prop ert y Ω T (K(A(q+1),p+2))=K(A(q),p) b y V o ev o dsky’s cancellation theorem ([V o e99]3.7, [V o e10]). The corresp onding statemen t in v olving susp ension is exactly the maps φ n :H A (T)∧K(A(n),2n)→K(A(n+1),2n+2) ab o v e. A dapting the ab o v e sho ws that there is a pro duct map H A (S i,j )∧K(A(n),2n)→K(A(n+j),2n+i) The analog of Σ p,q X = S p,q ∧ X ∈ Sm/B for K(A(n),2n) ∈ H(B) is to tak e Σ p,q K(A(n),2n) to b e H A (S p,q )∧K(A(n),2n). Th us: K(A(q),p)=Σ p,q K(A(0),0) + Σ p,q HA=(Σ p,q K(A(0),0),Σ p,q K(A(1),2),...) Prop osition 3.5.6. ([V o e99] 3.4, [nL ab] motivic homotopy the ory) F or an ab elian gr oup A and every choic e p≥ q ≥ 0 and r,s∈Z, dene the (unr e duc e d) (p− r,q− s)-motivic c ohomolo gy gr oup of X ∈ Sm/B with c o ecients in A to b e H p− r,q− s (X;A)=Hom H(B) (Σ r,s H A (X) + ,K(A(q),p)) In addition, the motivic sp ac es K(A(n),2n) form a motivic sp e ctrum HA with the pr op erty that: H p,q (X;A)=Hom SH(B) (Σ ∞ T H A (X) + ,Σ p,q HA) The requiremen t for p≥ q here comes from the incorp oration of S p,q =(∆ p− q /∂∆ p− q )∧(G ∧q m )=S p− q s ∧S q t It is p ossible to consider the case of p < q with formal in v ersion of sp ecically purely T ate susp ension Σ 0,q (− ) + ([V o e98] pg. 595, [V o e99] pg. 9). Ho w ev er, this and ev en K(A(q),p) for p ̸= 2q is unnecessary b ecause H p,q (X;A)=0 when p>2q ([V o e99] pg. 10). Th us: 44 H 2q− i,q (X;A)=Hom H(B) (Σ i,0 H A (X) + ,K(A(q),2q)) Finally , there is the idea of reduced motivic cohomology as seen b elo w. Denition 3.5.7. F or p ≥ q ≥ 0, dene the reduced motivic cohomology group of X ∈ Sm/B with co ecien ts in A to b e equiv alen tly giv en b y the follo wing: ˜ H p− r,q− s (X;A)=Hom H(B) (Σ r,s H A (X),K(A(q),p)) ˜ H p,q (X;A)=Hom SH(B) (Σ ∞ T H A (X),Σ p,q HA) ˜ H 2q− i,q (X;A)=Hom H(B) (Σ i,0 H A (X),K(A(q),2q)) In short, this satises the prop ert y that the reduced motivic cohomology of X + coincides with the unreduced motivic cohomology of X , i.e. ˜ H p,q (X + ;A)=H p,q (X;A). Muc h lik e in the classical setting, there is a w a y to split unreduced motivic cohomology with the reduced motivic cohomology as a summand. Recall that if X is a v ariet y o v er a eld K then it is the set of common zeros inK n whereK is the algebraic closure of K. AK-p oint of X is a p oin t in X that b elongs toK n ⊂ K n . Prop osition 3.5.8. If X is a smo oth variety over K with aK-p oint then H ∗ ,∗ ′ (X;A)= ˜ H ∗ ,∗ ′ (X;A)⊕ H ∗ ,∗ ′ (Spec(K);A) 3.6 T ransfers and Euler Characteristics T ransfer maps are an imp ortan t to ol in algebra that come in man y forms. The relev an t v ersion here is the Bec k er-Gottlieb transfer homomorphism; this map has the cohomological prop ert y that pre-comp osing it with the restriction map of a bundle is simply m ultiplication b y the b er ([BG75]). Ho w ev er, these maps ha v e a broader con text in category theory that relates to motivic homotop y theory . Denition 3.6.1. ([Kle18] 2.1) A duality datum in a symmetric monoidal category (C,⊗ ,1 C ) is a pair X,X ∨ ∈Ob(C) with morphisms 1 C coev − −− → X⊗ X ∨ and X ∨ ⊗ X ev −→ 1 C suc h that the comp ositions X coev⊗ Id −−−−−→ X⊗ X ∨ ⊗ X Id⊗ ev − −−− → X X ∨ Id⊗ coev −−−−−→ X ∨ ⊗ X⊗ X ∨ ev⊗ Id − −−− → X ∨ 45 are iden tities. In this case, w e sa y that X ∨ is a right dual of X ; similarly , X is a left dual of X ∨ . W e sa y that X is str ongly dualizable if in addition X ∨ is a left dual of X . R emark 3.6.2. ([Kle18] 2.2) T aking X ∨ =Hom C (X,1 C ) means that X is strongly dualizable i the induced map X ∨ ⊗ X →Hom C (X,X) is an equiv alence. Denition 3.6.3. ([Kle18] 2.3) In a symmetric monoidal category (C,⊗ ,1 C ), supp ose that w e x a morphism ∆ ∈ Hom C (X,X ⊗ C) for some X,C ∈ Ob(C) with X strongly dualizable. Then the tr ansfer of X with r esp e ct to ∆ is dened to b e the comp osition tr x,∆ :1 C coev − −− → X⊗ X ∨ switch −−−−→ X ∨ ⊗ X Id⊗ ∆ − −−− → X ∨ ⊗ X⊗ C ev⊗ Id − −−− → 1⊗ C ∼ =C Denition 3.6.4. ([Ana20] 2.2) In a symmetric monoidal category (C,⊗ ,1 C ), let X b e strongly dualizable. Then the Euler char acteristic of X is dened to b e the comp osition χ C (X):1 C coev − −− → X⊗ X ∨ switch −−−−→ X ∨ ⊗ X ev −→ 1 C R emark 3.6.5. ([Ana20] 2.3) Let X b e a top ological space with the homotop y t yp e of a simplicial set with only nitely man y non-degenerate simplices (i.e. nite homotop y t yp e) and SH the category of top ological sp ectra. Then χ SH (Σ ∞ S 1X + )=χ Top (X)= X i (− 1) i dim(H i (X;Q)) Recall that the Gr othendie ck-Witt ring GW(K) is the collection of isometry classes of non-singular quadratic spaces o v er K. Note that there is a natural iden tication of GW(K) with space giv en b y Hom SH(K) (Σ ∞ T (SpecK) + ,Σ ∞ T (SpecK) + ) ([Ana20] 2.6). W e sa y that X ∈ Sm K is strongly dualizable if Σ ∞ T X + is strongly dualizable in SH(K); set χ A 1 K (X)=χ SH(K) (Σ ∞ T X + )∈GW(K) ([Ana20] 2.5, 2.6). With these to ols in mind, w e can no w turn to results regarding l-adic cohomology: Lemma 3.6.6. ([A na20] 3.1) L etK b e an algebr aic al ly close d eld, G a r e ductive gr oup overK, T a maximal torus of G, and N =N G (T) the normalizer of T in G. Then for a prime l̸=char(K) we have H i (G/N,Q l )=        Q l 0 ,if i=0 ,if i̸=0 46 Lemma 3.6.7. ([A na20] 2.11) L et K b e a eld and K b e its algebr aic closur e. Then for a given prime l̸=char(K) and X ∈Sm K we have: 1. If char(K)=0 then rankχ A 1 K (X)= P i (− 1) i dim Q l H i (X K ,Q l ) 2. If char(K)=p̸=0 then rankχ A 1 K Z[ 1 p ] (X)= P i (− 1) i dim Q l H i (X K ,Q l ) Corollary 3.6.8. ([A na20] 3.2) L et G b e a r e ductive gr oup over a eld K, T a maximal torus of G, and N =N G (T) the normalizer of T in G. Then: 1. If char(K)=0 then rankχ A 1 K (G/N)=1. 2. If char(K)=p̸=0 then rankχ A 1 K Z[ 1 p ] (G/N)=1. Theorem 3.6.9. ([A na20] 5.1) L et G b e a r e ductive gr oup over a eld K, T a maximal torus of G, and N =N G (T) the normalizer of T in G. Then: 1. If char(K)=0 then χ A 1 K (G/N) is a unit. 2. If char(K)=p̸=0 then χ A 1 K Z[ 1 p ] (G/N) is a unit. 3. If char(K)=p̸=0 and G/N is strongly dualizable then χ A 1 K (G/N) is a unit. There is another imp ortan t theorem to k eep in mind here: Theorem 3.6.10. ([Kle18] 2.9) L et B b e a scheme over S . Supp ose that E ∈SH(B) and ther e is a nite Nisnevich c overing family {j i : U i → B} and that j ∗ i E ∈SH(U i ) is str ongly dualizable. Then E is str ongly dualizable as wel l. If E → B is a Nisnevich-lo c al ly trivial b er bund le with smo oth b er F and B is smo oth over S , then E/B is str ongly dualizable in SH(B) if F/S is str ongly dualizable in SH(S). 3.7 The Six Operations Grothendiec k’s six op erations are a construction of pushforw ards and pullbac ks that generalize to man y categories. They will b e needed to pro v e Theorem 5.1.1, but this section is primarily a summary of relev an t material b y A y oub and Ho y ois in [A y o07] and [Ho y17]. As seen in [Ana20], w e frequen tly mak e asso ciations b et w een a category and its corresp onding motivic stable homotop y category . F or example, recall that w e sa y that a sc heme is str ongly dualizable in Sm K if its susp ension sp ectrum is strongly dualizable in SH(K) (see Denition 3.6.1). F or our purp oses, there is a 47 general need to construct a functor SH G :X 7→SH G (X) =SH([X/G]) from the category of G-sc hemes to the closed symmetric monoidal triangulated category of motivic G-equiv arian t sp ectra parameterized b y X , along with analogs of Grothendiec k’s six op erations. Note that [X/G] is the quotient stack of X o v er G and that SH G (X) comes equipp ed with a tensor pro duct ⊗ , in ternal mapping ob ject Hom, monoidal unit 1 S , and monoidal symmetry τ ([Ho y17] pg. 200). The ⊗ and Hom op erations are joined b y four others: Let Sm G B denote the category of smo oth G-equiv arian t sc hemes o v er B . Let H G (B) b e the homotop y category of simplicial Nisnevic h shea v es on Sm G B . A t the lev el of Nisnevic h shea v es, if f ∈Hom Sm G B (X,Y) isG-equiv arian t then the pushforw ard functor f ∗ :Shv Nis (Sm G X )→Shv Nis (Sm G Y ) is dened b y f ∗ (F)(Z)=F(Z× Y X) ([Ho y17] pg. 233). W e obtain f ∗ : H G (X) → H G (Y) b y restriction b ecause homotop y categories ma y b e tak en to b e sub categories, as in Remark 3.1.6. The pushforw ard f ∗ preserv es limits and th us it has a left adjoin t. This is the pullbac k functor and is denoted f ∗ :H G (Y)→H G (X) ([Ho y17] pg. 234). Similarly , pushforw ard f ! : H G (X) → H G (Y) with compact supp ort has corresp onding left adjoin t f ! :H G (Y)→H G (X) under suitably nicet y conditions for f . Note that the op erations (− ) ∗ ,(− ) ∗ ,(− ) ! ,(− ) ! ha v e natural extensions to the stable homotop y setting. T ogether, these op erations satisfy the follo wing adjunctions ([Ho y17] pg. 202): The tensor-hom adjunction (− )⊗ X :SH G (B)⇆ SH G (B):Hom(X,− ) for ev ery X ∈SH G (B) The pul lb ack-pushforwar d adjunction f ∗ :SH G (Y)⇆ SH G (X):f ∗ for ev ery G-equiv arian t f :T →S The exc eptional adjunction giv en b y f ! :SH G (X)⇆ SH G (Y):f ! if f is separated and of nite t yp e Supp ose B is a quasi-compact quasi-separated sc heme and G is a nitely presen ted group sc heme o v er B . The conditions for G to a tame gr oup scheme are complicated, but the follo wing are imp ortan t examples: G is nite lo cally free of order in v ertible on B , G is of m ultiplicativ e t yp e, and G is reductiv e and B has c haracteristic 0 (i.e. there exists a morphism B→Spec(Q)) ([Ho y17]). Theorem 3.7.1. ([Hoy17] 1.1) L et B b e a quasi-c omp act quasi-sep ar ate d scheme and G a tame gr oup scheme over B . Assume G is nite or B has the G-r esolution pr op erty. Then the six op er ations (− ) ∗ , (− ) ∗ ,(− ) ! , (− ) ! , ⊗ , Hom satisfy the fol lowing pr op erties on nitely pr esente d G-quasi-pr oje ctive B schemes whenever the exc eptional functors ar e dene d: 48 1. (Prop er pushforw ard) If f is a prop er G-morphism then there is an equiv alence f ! ≃f ∗ 2. (Smo oth pullbac k) If f is a smo othG-morphism then there is a self-equiv alence Tw f and an equiv alence Tw f ◦ f ! ≃f ∗ 3. (Base c hange) There are equiv alences f ∗ p ! ≃q ! g ∗ and f ! p ∗ ≃q ∗ g ! if the follo wing is a Cartesian square of G-sc hemes: • g // q • p • f // • 4. (Gluing) F or i and j complemen tary closed and op en G-immersions, there are cob er sequences j ! j ! →Id→i ∗ i ∗ i ! i ! →Id→j ∗ j ∗ 5. (Immersion pushforw ard) If i is a G-immersion then the functors i ∗ and i ! are fully faithful. 6. (Monoidalit y) If f is an y G-morphism then there is an equiv alence f ∗ (−⊗− )≃f ∗ (− )⊗ f ∗ (− ) 7. (Pro jection form ulas) If f is an y G-morphism then there is equiv alences f ! (−⊗ f ∗ (− ))≃f ! (− )⊗− Hom(f ! (− ),− )≃f ∗ Hom(− ,f ! (− )) f ! Hom(− ,− )≃Hom(f ∗ (− ),f ! (− )) 8. (Homotop y in v ariance) If f is a G-ane bundle then the functors f ∗ and f ! are fully faithful. 9. (Constructible separation) If {f i } is a co v er for the G-equiv arian t constructible top ology then the families of functors {f ∗ i } and{f ! i } are conserv ativ e. 49 There are man y v ariations of the ab o v e theorem, though they are generally called Grothendiec k’s six op era- tions. A v ersion for arbitrary triangulated categories b ered o v er sc hemes is giv en in [CD19] A.5.1 and 11.4. That v ersion is used with the category DM of stable motivic c omplexes and is closer to the corresp ondence and T ate motiv e side of motivic cohomology , but this approac h in outside the b ounds of this pap er. Next, there are a few other op erations that are not of the main six but can b e describ ed in terms of them. F act 3.7.2. ([Hoy17] p g. 234) If f : X → Y is smo oth, then f ∗ :SH G (Y)→SH G (X) has a left adjoint denote d f ♯ :SH G (X)→SH G (Y). Note that if f : V → X is a G-equiv arian t v ector bundle with zero section s then there is the adjunction giv en b y f # s ∗ :SH G (X)⇆ SH G (X):s ! f ∗ . These functors are called the V -susp ension andV -desusp ension functors receptiv ely and are denoted Σ V =f # s ∗ and Σ − V =s ! f ∗ . Supp ose M is a lo cally free sheaf of nite rank on X . Denote Σ M = Σ V(M) and Σ − M = Σ − V(M) where V(M)=Spec(Sym(M)). Note that one imp ortan t example of suc h an M is the sheaf of relativ e dieren tials Ω f giv en b y the follo wing ([Stac ks] 08RL): Let M b e a G-mo dule, i.e. M is a sheaf on X where M(U) is a G(U)-mo dule for all U ∈ Op(X) suc h that eac h G(U)-action on M(U) satises compatibilit y conditions relativ e to the restriction maps ([Stac ks] 006Q) First, let’s adapt the denition of deriv ations to shea v es ([Stac ks] 01UN, 006D): an F -derivation is a map D :G→M suc h that for all U ∈Op(X), f ∈F(U) and g,g ′ ∈G(U) D(U)◦ φ(U)(f)=0 D(U)(g+g ′ )=D(U)(g)+D(U)(g ′ ) D(U)(gg ′ )=g ′ D(U)(g)+gD(u)(g ′ ) In an analogous w a y w e con v ert the denition the mo dule Ω S/R of Khler dieren tials in to a v ersion for shea v es. Refer to Denition 3.2.2 for the denition Ω S/R and its asso ciated map d : S → Ω S/R giv en a ring morphism φ:R→S ([Stac ks] 08RM): let G[F] denote the sheacation of the presheaf U 7→ G(U)[F(U)] where this image is the p olynomial ring with co ecien ts in G(U) and where there is one v ariable [s] for eac h elemen t s∈F(U). Similar constructions apply for G[G] and G[G× G]. set Ω G/F to b e the cok ernel of the follo wing map and dene d : G → Ω G/F where if g ∈ G(U) then d(U)(g) is the image of [g] under the quotien t G[G]→Ω G/F G[G× G]⊕ G[G× G]⊕ G[F]→G[G] [(a,b)]⊕ [(f,g)]⊕ [h]7→[a+b]− [a]− [b]+[fg]− g[f]− f[g]+[φ(h)] 50 It’s clear that that O X and f − 1 O Y are b oth shea v es on X for an y sc heme morphism f : X → Y . Dene the she af of r elative dier entials induced b y the map f :X →Y to b e Ω f =Ω O X /f − 1 O Y This sheaf Ω f is so imp ortan t b ecause it allo ws us sa y something ab out this left adjoin t f # in regards to the main six op erations: F act 3.7.3. ([Hoy14] pp. 3611-12) Given smo oth f :X →Y with adjunction f ♯ :SH G (X)⇆ SH G (Y):f ∗ , ther e ar e c anonic al e quivalenc es f ! ≃f # Σ − Ω f and f # ≃f ! Σ Ω f . 51 Chapter 4 Motivic Cohomology 4.1 Motivic Cohomology in the Abstract 4.1.1 Properties of Motivic Cohomology Let A b e an ab elian group. T o clarify con v en tions for the motivic cohomology group H m,n (X;A), w e sa y that m is the de gr e e of the group and n is the weight . Another common con v en tion is to refer to the (bi)degree as (m,n) and the w eigh t as 2n− m ([Y ag10]). Both con v en tions are used in the pap er and one can distinguish them b y the con text. In addition, recall that if X and T are sc hemes o v er a sc heme S with asso ciated morphisms ϕ X : X → S and ϕ T : T → S then the set of T -p oints of X is dened to b e X(T) ={f ∈ Hom Sch/S (X,T) : ϕ X = ϕ T ◦ f}. W e use X(C) to denote the case where S and T are b oth Spec(C). Note that X(C) is in bijectiv e corresp ondence with the set of Zariski closed p oin ts of X b ecause C is algebraically closed. Motivic cohomology H ∗ ,∗ ′ (− ;A) is a bigraded cohomology theory describ ed in Section 3.5 and giv en b y H 2q− i,q (X;A)=Hom H(B) (Σ i,0 H A (X) + ,K(A(q),2q)) This theory satises the the follo wing prop erties ([Y ag10]): 1. F or an y morphism f : X → Y , the cob er sequence X f → Y → Y/Im(f) induces the long exact sequence ...→H ∗− 1,∗ ′ (X;A)→H ∗ ,∗ ′ (Y/Im(f);A)→H ∗ ,∗ ′ (Y;A)→H ∗ ,∗ ′ (X;A)→... 52 2. There are realization maps t m,n C :H m,n (X;A)→H m (X(C);A) and t C =⊕ m,n∈N t m,n C . 3. There are the Bo c kstein and reduced p o w er op erations β :H ∗ ,∗ ′ (X;Z p )→H ∗ +1,∗ ′ (X;Z p ) P i :H ∗ ,∗ ′ (X;Z p )→H ∗ +2i(p− 1),∗ ′ +i(p− 1) (X;Z p ) whic h comm ute with t C in that t C ◦ β =β ◦ t C and t C ◦ P i =P i ◦ t C (where β and P i on the left and righ t are the motivic and classical v ersions of these op erations resp ectiv ely). 4. H ∗ ,∗ ′ (X∧(P n /P n− 1 );A) ∼ =H ∗ ,∗ ′ (X;R){y ′ } where deg(y ′ )=(2n,n) and t C (y ′ )̸=0. 5. Dene the weight ofx∈H m,n (X;A) to b e w(x)=2n− m and its dier enc e de gr e e to b e d(x)=m− n. F or smo othX , ifH m,n (X;A)̸=0 thenm≤ n+dim(X),m≤ 2n andm≥ 0, i.e. ifx̸=0∈H ∗ ,∗ ′ (X;R) then w(x)≥ 0 and d(x)≤ dim(X). 6. H 0,1 (pt.;Z p ) ∼ =Z p and so w e x a generator τ ∈H 0,1 (pt.;Z p ) corresp onding to a generator of Z p . 7. F or deg(τ ) = (0,1), deg(ρ ) = (1,1) and deg(y) = (2,1), w e ha v e that H ∗ ,∗ ′ (Spec(C);Z p ) ∼ =Z p [τ ], H ∗ ,∗ ′ (Spec(R);Z 2 ) ∼ =Z 2 [ρ,τ ] and H ∗ ,∗ ′ (P ∞ ;Z p ) ∼ =H ∗ ,∗ ′ (pt.;Z p )⊗ Z p [y]. 4.1.2 Bloch’s Higher Chow Groups as an Alternate Model More concretely . one can construct motivic cohomology using the Blo c h’s higher Cho w group mo del. This construction will b e summarized in the next few paragraphs. Let R q = K[x 0 ,...,x q ] ha v e ideal I q = (x 0 +... +x q − 1). Recall that the sc heme-theoretic standar d q -simplex is ∆ q = Spec(R q /I q ). In the top ological setting, the standard q -simplex comes from the p oin ts satisfying x 0 + ... + x q = 1; in that sense, a face adds in the requiremen t that x i1 ,...,x ir = 0 for some 0≤ i 1 < ... < i r ≤ q . Th us, w e sa y that ∆ q i1,...,ir = Spec(R q /I q,i1,...,ir ) is a fac e of ∆ q where here w e tak e I q =(x 0 +...+x q − 1,x i1 ,...,x ir ). It is clear that I q ⊂ I q,i1,...,ir . Th us, if f,g∈R q suc h that f− g∈I q then f− g∈I q,i1,...,ir as w ell. This means that there is a w ell-dened map φ q,i1,...,ir :R q /I q →R q /I q,i1,...,ir ,f +I q 7→f +I q,i1,...,ir whic h in turn induces a map on sc hemes giv en b y φ ∗ q,i1,...,ir :Spec(R q /I q,i1,...,ir )→Spec(R q /I q ),P 7→φ − 1 q,i1,...,ir (P) 53 This map φ ∗ q,i1,...,ir can b e in terpreted as establishing an inclusion ∆ q i1,...ir ⊂ ∆ q ; as suc h, all future statemen ts regarding inclusion of faces in simplices will omit the φ ∗ q,i1,...,ir . T o sho w that ∆ q i1,...,ir ∼ = ∆ q− r one uses induction along with the follo wing argumen t: Let σ q,i :R q →R q− 1 where σ q,i (x j ) is x j if j i It’s clear that σ q,i (I q,i )=I q− 1 and that f− g∈I q,i implies σ q,i (f)− σ q,i (g)∈I q− 1 Th us, there is the isomorphism ϕ q,i :R q /I q,i →R q− 1 /I q− 1 ,f +I q,i 7→σ q,i (f)+I q− 1 This sho ws that the induced map ϕ ∗ q,i :∆ q− 1 →∆ q i is an isomorphism Giv en a smo oth pro jectiv e v ariet y X o v erK, w e sa y that a closed sub v ariet y of X is a prime algebr aic cycle . Let C r (X) b e the free ab elian group generated b y prime algebraic cycles of co dimension r . W e sa y that t w o prime algebraic cycles A,B interse ct pr op erly if codim(A∩B)=codim(A)+codim(B). With this in mind, let C r (X,q)⊂ C r (X× ∆ q ) b e the subgroup generated b y the prime algebraic cycles of X× ∆ q that in tersect prop erly with X× F for all faces F of ∆ q . In order to mak e the nal jump to higher Cho w groups, consider the map ∂ q,i =Id X × ϕ ∗ q,i :X× ∆ q− 1 →X× ∆ q i ⊂ X× ∆ q This is an isomorphism of sc hemes whic h induces the maps C r (∂ q,i ) and d r q where C r (∂ q,i ):C r (X,q)→C r (X,q− 1),Z7→∂ − 1 q,i ((X× ∆ q i )∩Z) d r q = P q i=0 (− 1) i C r (∂ q,i ) One can sho w that d r q : C r (X,q)→ C r (X,q− 1) is a dieren tial, i.e. d r q ◦ d r q+1 = 0. As suc h, w e can tak e the homology of the asso ciated sequence to get the higher Cho w groups: CH r (X,q)=Ker(d r q )/Im(d r q+1 ) This serv es as a mo del for motivic cohomology b ecause H m,n (X) ∼ = CH n (X,2n− m) and equiv alen tly CH r (X,q) ∼ = H 2r− q,r (X) ([V o e02]). Note that it is common in other pap ers to use indices p,q for mo- tivic cohomology with Z l co ecien ts; ho w ev er, here the con v en tion is to use indices m,n and to ha v e Z p co ecien ts. 54 Figure 4.1: Images represen ting the corresp ondence b et w een motivic cohomology groups and higher Cho w groups. Cy an regions are areas that v anishing according to the Beilinson-SoulØ v anishing conjecture; green areas v anish b ecause m>2n; y ello w areas v anish b ecause m>dim(X)+n; gra y areas v anish b ecause r <0; the gradien ts from blue to red are areas that can p oten tially b e non-trivial (assuming Beilinson-SoulØ). The region in the images on the left ll the image b ecause these are domains; the shap es in the righ t side images are the corresp onding ranges. 4.1.3 V anishing Theorems and Knneth F orm ula There are v anishing theorems for motivic cohomology . Sp ecically , they state that H m,n (X) is trivial when n<0,m>2n, orm>dim(X)+n. Because H m,n (X)=CH n (X,2n− m), it is clear wh y the rst t w o hold. The third condition is more complicated but ultimately it is most easily seen in the corresp ondence construction of motivic cohomology; this construction is outside the scop e of m y w ork and so it w on’t b e discussed further. Finally , the Beilinson-SoulØ v anishing conjecture claims that motivic cohomology also v anishes when m < 0. V o ev o dsky and Rost sho w ed that this conjecture holds when in nite c haracteristic b y pro ving the Blo c h-Kato conjecture; as suc h Blo c h-Kato is no w called the norm r esidue isomorphism the or em ([V o e11], [Ros02]). Recall the classical v ersion of the Knneth form ula in singular cohomology: if A is an ab elian group and X,Y are top ological spaces suc h that the ring H ∗ (Y;A) is a nitely generated free mo dule o v er A then the 55 cross pro duct map H ∗ (X;A)⊗ R H ∗ (Y;A)→H ∗ (X× Y;A) is an isomorphism ([Hat02]). There is a v ersion of the Knneth form ula in motivic cohomology but it is more complicated; the conditions for the map b eing an isomorphism are more subtle and the map frequen tly isn’t an isomorphism for general X and Y when in mo d p cohomology ([Y ag10] pg. 185). As suc h, w e sa y that X and Y satisfy the mo d-p Knneth formula if H ∗ ,∗ ′ (X;Z p )⊗ H ∗ ,∗ ′ (pt.;Zp) H ∗ ,∗ ′ (Y;Z p ) ∼ =H ∗ ,∗ ′ (X× Y;Z p ) Note that Y agita claims and uses the fact that the innite pro jectiv e space P ∞ and the classifying space BZ p b oth satisfy the Knneth form ula when paired with an y other space. 4.2 Classifying Spaces in Motivic Cohomology The follo wing is a summary of needed denitions of equiv arian t motivic cohomology , classifying space motivic cohomology , and their relations to restriction and transfer maps. This section b egins with a motivic cohomology adaptation of T otaro’s construction of Cho w rings of classifying spaces as presen ted in [T ot99]. This particular construction follo ws the material on pg. 189 of [Y ag10]. Supp ose G is a linear algebraic group o v er a eld K and let X b e a smo oth algebraic set with G- action. One can sho w that for ev ery m≥ 0 there exists a represen tation G→GL(V m ) withG-in v arian t closed subset S m ⊂ V m suc h that codim Vm (S m )≥ m and suc h that G acts freely on U m = V m − S m . The ob ject (X× U m )/G exists as a smo oth algebraic space and one can sho w for eac h n that the motivic cohomology groupsH m,n ((X× U m )/G) are indep enden t of c hoice of pair (V m ,S m ) up to isomorphism. W e can no w dene theG-e quivariant (m,n)-motivic c ohomolo gy gr oup ofX to b e giv en b y H m,n G (X)=H m,n ((X× U m )/G) and denote H m,n (BG)=H m,n G (pt.). Finally , these groups are joined via direct sum to create the bigraded rings denoted H ∗ ,∗ ′ G (X) and H ∗ ,∗ ′ (BG). W e sa y that this latter ring is the motivic c ohomolo gy of the classifying sp ac e of G. The ab o v e is an equiv arian t cohomology mo del and as suc h there are useful to ols that come along with it. Most imp ortan tly w e ha v e the transfer map construction. Note that if H is an algebraic subgroup of G then H ∗ ,∗ ′ G ((X× G)/H) ∼ =H ∗ ,∗ ′ H (X) 56 T o see this, supp ose 1→H ϕ →G→F →1 is an exact sequence of algebraic groups o v er K withF nite. It’s w ell kno wn that if X is a smo oth algebraic set with G-action then π :X× F →X prop er and G-equiv arian t. There is a G-equiv arian t pushforw ard map π ∗ : H ∗ ,∗ ′ G (X× F)→ H ∗ ,∗ ′ G (X). Next, supp ose w e ha v e a xed c hoice of (V m ,S m ) as ab o v e and let U m =V m − S m . Then w e ha v e the follo wing implication: (X× U m )/H ∼ =((X× F)× U m )/G =⇒ ι :H ∗ ,∗ ′ H (X) ∼= − → H ∗ ,∗ ′ G (X× F) Th us, the comp osition π ∗ ◦ ι denes the tr ansfer map tr G H : H ∗ ,∗ ′ H (X) → H ∗ .∗ ′ G (X) in motivic cohomology ([V ez00] pp. 4-5). An equally useful construction is that of the restriction map. Let G,G ′ b e a linear algebraic groups o v erK andX a smo oth algebraic set with G-action. Fix a morphism φ:G ′ →G of algebraic groups, th us endo wing X with a G ′ -action. Cho ose pairs (V m ,S m ) and (V ′ m ,S ′ m ) of represen tations of G and G ′ for eac h m≥ 0, taking U m =V m − S m andU ′ m =V ′ m − S ′ m . Note that G ′ acts on V m × V ′ m as g ′ · (v,v ′ )=(φ(g ′ )· v,g ′ · v ′ ) and that the pro jection map X× U m × U ′ m →X× U m induces a at map (X× U m × U ′ m )/G ′ →(X× U m )/G. The pullbac k of this map induces H m,n G (X)→H m,n G ′ (X), whic h when put together for all m,n is the r estriction map res G ′ G :H ∗ ,∗ ′ G (X)→H ∗ ,∗ ′ G ′ (X) ([V ez00] pp. 4-5). T o conclude this section, there are t w o imp ortan t applications of the ab o v e that will b e used later in this pap er. First, w e can create a comp osition of the restriction and transfer maps in the follo wing map. T aking G ′ = H and φ = ϕ ab o v e giv es the map res H G : H ∗ ,∗ ′ G (X)→ H ∗ ,∗ ′ H (X). The pro jection form ula states that the comp ositions tr G H ◦ res H G and res H G ◦ tr G H | H ∗ ,∗ ′ H (X) F are just m ultiplication b y |F|, where H ∗ ,∗ ′ H (X) F is the F -in v arian t subring of H ∗ ,∗ ′ H (X) ([V ez00] pp. 4-5). Second, the lo c alization long exact se quenc e is highly useful to ol for computing motivic cohomology , as demonstrated in [Y ag10]. If Y is a G-equiv arian t algebraic subset of X with inclusions giv en b y i : Y ,→ X and j : X− Y ,→ X suc h that codim X (Y) = s, there is a long exact sequence giv en b y ...→H ∗− 1,∗ ′ G (X− Y)→H ∗− 2s,∗ ′ − s G (Y) i∗ →H ∗ ,∗ ′ G (X) j ∗ →H ∗ ,∗ ′ G (X− Y)→... 4.3 Steenrod Operations Steenro d op erations mak e sev eral app earances in this w ork. Most notable are the presen tations of the motivic cohomology of BSO 4 and BG 2 withZ 2 -co ecien ts; ho w ev er, the Steenro d squaring op eration as it relates to p olynomials is relev an t for the statemen t and pro of of Theorem 5.5.2. 57 4.3.1 Classical Steenro d Square Operations The follo wing section deriv es from [WW01] and [PW02]. As a motiv ating example for the Steenro d squares, let R =Z 2 [x 1 ,...,x n ] b e the ring isomorphic to the cohomology ring of Gr n (C ∞ ). T ak e the ring homomorphism ϕ : R→ R where w e extend R-linearly from the equations ϕ (x i ) = x i +x 2 i . It’s clear that for an y p olynomial g(x 1 ,...,x n ) ∈ R, the image ϕ (p) can b e decomp osed uniquely as P ∞ k=0 ϕ (k) (g) where deg(ϕ (k) (g)) = deg(g) + k . Uniqueness allo ws us to dene the Ste enr o d homomorphisms Sq k : R → R, g7→ϕ (k) (g) with the prop ert y that if deg(p)=d then: Sq 0 (g)=g Sq d (g)=g 2 k >d =⇒ Sq k (g)=0 Example 4.3.1. ϕ (x n i )=ϕ (x i ) n =(x i +x 2 i ) n = P n j=0 n j x n+j i (mod 2) =⇒ Sq k (x n i )=        n k x n+k i (mod 2) 0 ,if 0≤ k≤ n ,if k >n In general, all equations in v olving the Steenro d homomorphisms are assumed to b e tak en mo d 2 from no w on. It is w ell kno wn that these homomorphisms satisfy the A dem r elations . Theorem 4.3.2. ([MT91] 7.4.13) The Ste enr o d homomorphisms satisfy the A dem r elations. In p articular, if 0d(q− 1) Example 4.3.3. ϕ q (x n i )=ϕ q (x i ) n =(x i +x q i ) n = P n j=0 n j x nq+j i (mod p) =⇒ ϕ (k) q (x n i )=        n j x nq+j i (mod p) 0 ,if k =j(q− 1)∈{0,q− 1,...,d(q− 1)} ,otherwise 58 Ov er R = Z p [x 1 ,...,x n ], w e ma y no w dene the r e duc e d p th -p ower homomorphisms {P k } k∈N where P k : R → R,g 7→ ϕ (2k) q (g). Note the presence of the 2k here instead of the only k . The reason for this is related to the fact that the univ ersal Chern classes for H ∗ (Gr n (C ∞ );Z) are all ev en degree, and this remains true when passing to Z p co ecien ts. When p=2 w e get that P k =Sq 2k . 4.3.2 Cohomological P ower Operations A short exact sequence 0→G→H →K→0 of ab elian groups induces a corresp onding sequence 0→C n (X;G)→C n (X;H)→C n (X;K)→0 via the co v arian t functor Hom(C n (X),− ) ([Hat02] pg. 303). This sequence is exact b ecause eac h C n (X;− ) is free. This in turn induces a long exact sequence ...→H n (X;G)→H n (X;H)→H n (X;K) β − → H n+1 (X;G)→... This map β : H ∗ (X;K)→ H ∗ +1 (X;G) is commonly referred to as the Bo ckstein homomorphism . P er- haps the most commonly used v ersion of this is the Bo ckstein op er ation whic h comes from the short exact se- quence 0→Z p →Z p 2 →Z p →0, meaning that the induced map is giv en b y β :H ∗ (X;Z p )→H ∗ +1 (X;Z p ). The Steenro d and reduced p th -p o w er op erations, denoted Sq k andP k resp ectiv ely , are b oth deriv ed from the iden tit y and Bo c kstein op erations. Sp ecically: Sq 0 =Id:H ∗ (X;Z 2 )→H ∗ (X;Z 2 ) P 0 =Id:H ∗ (X;Z p )→H ∗ (X;Z p ) Sq 1 =β :H ∗ (X;Z 2 )→H ∗ +1 (X;Z 2 ) is induced b y 0→Z 2 →Z 4 →Z 2 →0 Note: β : H ∗ (X;Z p ) → H ∗ +1 (X;Z p ) is induced b y 0 →Z p →Z p 2 →Z p → 0 and is not one of the P i ’s More generally , these op erations are designed with the p olynomial v ersion ab o v e in mind. On the univ ersal classes, whic h w e kno w can b e represen ted with elemen tary symmetric p olynomials, these cohomological op erations coincide with their p olynomial homomorphism coun terparts ab o v e. As c haracteristic classes can b e dened to b e pullbac ks of the univ ersal ones via the prop erties of classifying spaces, the Ste enr o d and r e duc e d p th -p ower op er ations can b e constructed. 59 Sq k :H ∗ (X;Z 2 )→H ∗ +k (X;Z 2 ) P k :H ∗ (X;Z p )→H ∗ +2k(p− 1) (X;Z p ) It’s imp ortan t to note that ev en though the Steenro d and reduced p th -p o w er op erations can b e though t of in terms of the corresp onding p olynomials, the Bo c kstein op eration of β : H ∗ (X;Z p )→ H ∗ +1 (X;Z p ) cannot for sev eral reasons. First, as seen in Example 4.3.3 ab o v e, ϕ (1) p is the only map of that t yp e in Z p that degree 1 shift; ho w ev er, it is clear that ϕ (1) p (x n i ) is alw a ys equal to 0 when p is not equal to 2. It’s clear that β is not the zero map in general, so w e cannot tak e β to b e the cohomological v ersion of ϕ (1) p . Second, m ultiplication of p olynomials in Z p [x 1 ,...,x n ] is comm utativ e, but the pro duct of cohomology classes only satises graded comm utativit y , e.g. if φ ∈ H p (X;Z) and ψ ∈ H q (X;Z) then ψ ⌣ φ = (− 1) pq φ ⌣ ψ for the cup pro duct in singular cohomology . It’s clear that comm utativit y holds in the cup pro duct example if p or q or ev en and a similar idea holds for pro ducts of cohomology classes in general. Th us, a cohomology op eration with o dd degree shift will inheren tly ha v e problems with comm utativit y and th us will incompatible with our p olynomial view. Despite this dicult y , the map β is a v ery useful map for sev eral reasons. P erhaps the most imp ortan t reason is the fact that all maps P k ha v e ev en degree shift. One gets o dd degree op erations b y comp osing the reduced p th -p o w er op erations with the Bo c kstein, and one can sho w that an y stable o dd degree map can b e obtained as a comp osition of these maps P k and β . This idea is seen in the follo wing theorem: Theorem 4.3.4. ([MT91] 7.4.22) The Bo ckstein and r e duc e d p th -p ower homomorphisms satisfy the A dem r elations. In p articular, if 0 0. W e turn to [Y ag10] for the extension of an equiv arian t Cho w ring of a smo oth v ariet y to Blo c h’s higher Cho w groups (see Section 4.2), but could also use the discussion in [T ot99] of Edidin and Graham’s idea instead. A k ey p oin t here is the follo wing: if a group G acts freely on a space X and H is a subgroup of G then H also acts freely on X and there is a map X/H → X/G with b er G/H . In the v ain of [T ot99], if G acts freely on some U then N G (T) do es as w ell and there is a b er sequence G/N G (T) → U/N G (T) → U/G. Note that w ork b y V o ev o dsky demonstrates that these U ’s are unnecessary , but they are used here b ecause of ties in to [V ez00]. The ab o v e maps giv en b y f : U/N G (T) → U/G are smo oth and hence at, meaning that there is a pullbac k on motivic cohomology f ∗ : H ∗ ,∗ ′ (BG;Z) → H ∗ ,∗ ′ (BN G (T);Z). This is precisely the denition of a restriction map and so w e will denote res=f ∗ from here on out. 63 Theorem 5.1.1. L et G b e an algebr aic gr oup over C, T a maximal torus of G and N G (T) its normalizer in G. Then the r estriction map res:H ∗ ,∗ ′ (BG;A)→H ∗ ,∗ ′ (BN G (T);A) is inje ctive for A=Z andZ p . Pro of: The goal of this pro of is to sho w that this restriction map, whic h is merely the usual pullbac k, is split-injectiv e b y the existence of a Bec k er-Gottlieb transfer tr :H ∗ ,∗ ′ (BN G (T))→H ∗ ,∗ ′ (BG). This is done b y sho wing that tr◦ res=Id. F or simplicit y , w e omit the co ecien ts in this pro of. If a pushforw ard f ∗ : H ∗ ,∗ ′ (BN G (T))→ H ∗ ,∗ ′ (BG) exists, the pro jection theorem sa ys that comp osing it with res w ould b e m ultiplication b y the Euler c haracteristic of G/N G (T). Unfortunately , pushforw ards are a bit more tric ky here b ecause these maps are not prop er; the maps U/N G (T) → U/G are ane but prop er ane maps m ust b e nite. These are tec hnical issues that most b e resolv ed in order to nd a suitable pushforw ard and this fact is the heart of this pro of. Supp ose f : X → Y is a smo oth prop er morphism of relativ e dimension k b et w een smo oth separated sc hemes of nite t yp e. The pushforw ard f ∗ :H ∗ ,∗ ′ (X)→H ∗− 2k,∗ ′ − k (Y) is dened as the follo wing: [Z]∈H p,q (X) =⇒ f∗ ([Z])=        deg(f|Z :Z→f(Z))[f(Z)] 0 ,if dim(f(Z))=dim(Z) ,if dim(f(Z))<dim(Z) The degree shift of f ∗ is (− 2k,− k), meaning that f ∗ can’t ll the role of the transfer morphism tr b ecause deg(f ∗ ◦ res)=(− 2k,− k) but deg(Id)=(0,0). In order to x this, w e can dene the mo died pushforw ard f # : H ∗ ,∗ ′ (X) → H ∗ ,∗ ′ (Y),[Z] 7→ f ∗ (c k (T f )· [Z]), where c k is the k th Chern class and T f is the relativ e tangen t bundle. The pr oje ction the or em giv es that f # ◦ res is just m ultiplication b y the Euler c haracteristic of the b er of f ([V ez00]). The problem with the ab o v e metho d is that the maps w e consider here are nev er prop er. W e m ust consider the case of g : U/N G (T) → U/G that is a smo oth but non-prop er morphism b et w een smo oth sc hemes. W e will turn to [Ho y14] and [Ho y17] in order to sho w the existence of a suitable alternativ e. Grothendiec k’s six op erations pro vide a go o d starting p oin t for nding suc h an alternativ e. Here w e ma y consider the case of smo oth nitely-presen ted quasi-pro jectiv e sc hemes o v er a xed quasi-compact quasi-separated sc heme B . This category is a sub category of Sm/B and as suc h it has a corresp onding sub category of SH(B). This sub category is a monoidal category with the six op erations (− ) ∗ , (− ) ∗ , (− ) ! , (− ) ! , ⊗ , and Hom. Note that the equiv arian t v ersion of this topic is co v ered in detail in Section 3.7, but w e need not concern ourselv es with equiv ariance b ecause here the group sc heme is trivial. Regardless, these 64 op erations satisfy the follo wing relev an t prop erties: F or h:X →Y there is a pullbac k-pushforw ard adjunction h ∗ :SH(Y)⇆ SH(X):h ∗ . If h is smo oth then there is also an adjunction h # :SH(X)⇆ SH(Y):h ∗ . By Prop osition 3.5.6, there are analogs of these op erations for motivic cohomology that are induced from the motivic homotop y v ersion. Most imp ortan tly , w e obtain the adjunction g # :H ∗ ,∗ ′ (U/N G (T))⇆ H ∗ ,∗ ′ (U/G):g ∗ (5.1) that xes the issue of degree shift. One do es need to b e careful to c hec k that there are no singularities in these quotien ts and therefore compactication can b e b e p erformed; this is not a concern here b ecause resolutions of singularities exist when char(K)=0 ([SV00]). As in the case when the map is prop er, one can v erify that the comp osition g # ◦ res is m ultiplication b y the Euler c haracteristic of the b er. Again b ecause w e are w orking o v er C, [Ana20] 3.2 and 5.1 imply that the Euler c haracteristic of the b er is rank 1 and a unit. The only p ossibilities for this v alue are 1 and − 1 and so together with [JP20] Theorem 1.6 w e see that the Euler c haracteristic is actually 1. Th us, these maps g # tak en o v er dieren t U dene a Bec k er-Gottlieb transfer homomorphism denoted tr : H ∗ ,∗ ′ (BN G (T)) → H ∗ ,∗ ′ (BG) where tr◦ res = Id. The pro of is concluded b y the fact that res is injectiv e b ecause it is split-injectiv e via tr . 5.2 Discussion of SO 2n Before Results It’s w ell kno wn that SO 4 is a subgroup of G 2 and con tains its maximal torus ([KV04], [Gui07]). The exact v ersion of SO 4 needs to b e sp ecied b ecause some constructions yield easier v ersions of this torus than others, ev en when tak en to b e dened o v er C. Recall that o v er a eld K of c haracteristic other than 2, w e ma y tak e the (p,q)-indenite sp e cial ortho gonal gr oup to b e SO p,q ={A∈M (p+q)× (p+q) (K):A T Q p,q A=Q p,q ,det(A)=1} where Q p,q is the diagonal matrix with p ones follo w ed b y q negativ e ones (i.e. the signatur e of Q p,q is (p,q)). The matrix Q p,q corresp onds to a quadratic form f p,q :K p+q →K, − → v 7→ − → v T Q p,q − → v and a bilinear form b p,q : K p+q × K p+q → K,( − → x, − → y) 7→ − → x T Q p,q − → y . Because char(K) ̸= 2, these forms are related b y b p,q ( − → x, − → y) = 1 2 (f p,q ( − → x + − → y)− f p,q ( − → x)− f p,q ( − → y)). W e ma y think of SO p,q as the Lie group of linear 65 transformations of a (p+q)-dimensional K-v ector space with preserv es a non-degenerate bilinear form of signature (p,q) and whic h ha v e determinan t 1. The underlying eld K is v ery imp ortan t to the structure and prop erties of SO p,q . If o v erC then these are all the same after a c hange of basis, i.e. SO p,q ∼ =SO p ′ ,q ′ whenev er p+q =p ′ +q ′ . Ov erR these are all non-isomorphic but do accoun t for all p ossible isomorphism classes of indenite sp ecial orthogonal groups o v erR b ecause the isomorphism class of a real quadratic form is determined b y its signature. Finally , things are ev en more complicated o v er Q b ecause the signature no longer determines the quadratic form up to isomorphism. All of these facts deriv e from Sylv ester’s la w of inertia ([Syl52]). In summary , the split sp e cial ortho gonal gr oup SO n,n is isomorphic to usual sp ecial orthogonal group SO 2n when o v er C but not when o v er R. Ho w ev er, when talking ab out tori of a split sp ecial orthogonal group it is often more con v enien t to instead use J =       0 ... 1 . . . . . . . . . 1 ... 0       ∈M m× m (K) =⇒ SO ′ m ={A∈SL m (K):A T JA=J,det(A)=1} (5.2) It is easy to v erify that the signature of J is(n,n) whenJ ∈M 2n× 2n (K), meaning thatSO ′ 2n is isomorphic to SO n,n . More generally , recall that if T K is an algebraic group o v er a eld K with corresp onding T K o v er the algebraic closure K then w e sa y that T K is torus if T K is isomorphic to G ⊕ n m for some n. If T K itself is isomorphic to G ⊕ n m then w e sa y it is a split torus . The imp ortan t thing to note here is that all tori o v er C are split b ecause C is its o wn algebraic closure. As suc h, w e will alw a ys tak e K =C from here on out. In this case, one can directly sho w that SO ′ n is isomorphic to SO n b y the follo wing reasoning. Supp ose A ∈ SO ′ n , meaning that A T JA = J . The goal is to transform A in to some M in SO n via conjugation, i.e. M = gAg − 1 suc h that M T IM = I for some g∈GL n (C). Letting g − T denote the in v erse-transp ose of g : J =A T JA =(g − 1 Mg) T J(g − 1 Mg) =g T M T g − T Jg − 1 Mg =⇒ g − T Jg − 1 =M T (g − T Jg − 1 )M (5.3) T aking g − T Jg − 1 = I to get I = M T IM , the problem of mapping from SO ′ 2n to SO 2n reduces to nding a factorization J =g T g for some g∈GL n (C). Note that this is dieren t than Cholesky decomp osition b ecause that v ersion uses the conjugate transp ose instead. In practice this matrix g is not unique, as demonstrated 66 b y Matlab co de that I ha v e written to searc h for suc h a matrix ([Gith ub]) and b y m y computations for the ev en size case in App endix Section 7.2. More generally , this matrix is complex-v alued and exists b ecause w e already knew that all indenite sp ecial orthogonal groups are isomorphic o v er C b y Sylv ester’s la w of inertia. T 2n =                  A 1 . . . A n       :A i ∈SO 2            T ′ 2n =                                                z 1 . . . z n z − 1 n . . . z − 1 1                 :z i ∈G m                                It’s w ell kno wn that the ab o v e T 2n and T ′ 2n are maximal tori of SO 2n and SO ′ 2n resp ectiv ely . The torus T ′ 2n can often b e easier to w ork with than T 2n b ecause its elemen ts are diagonal and so it is more clearly the n-fold pro ductG n m . Th us, the reader should k eep the more con v enien t SO ′ 2n in mind. A t this p oin t it w ould b e helpful to kno w the structure of N SO4 (T) and to compute its motivic cohomology . One can sho w that the normalizer of T ′ 4 in SO ′ 4 is giv en b y matrices of the one of the follo wing 4 forms:          a b 1 b 1 a                   a b 1 b 1 a                   a b 1 b 1 a                   a b 1 b 1 a          The v alues a and b m ust b e non-zero so eac h one of these matrices represen ts a cop y of G 2 m in N SO ′ 4 (T ′ 4 ). While these copies ofG 2 m are clearly disjoin t, only the one corresp onding to the rst matrix forms a subgroup of the normalizer and that subgroup is merely the original torus itself. The other copies exist as the parts of a semi-direct pro duct G 2 m ⋊W with the W eyl group W from Lemma 2.3.5: W =                           1 1 1 1          ,          1 1 1 1          ,          1 1 1 1          ,          1 1 1 1                           67 F rom here on out there is no distinction made b et w een SO n and SO ′ n b ecause they are isomorphic; b oth are simply written as SO n and eac h torus is denoted T . It is w ell kno wn that the order of the W eyl group of SO 2n is 2 n− 1 n! and that matrices in N SO2n (T) ha v e n free v ariables. Th us: N SO2n (T) ∼ =G n m ⋊W (5.4) Theorem 5.2.1. ([ADK08] 3.10) F or ρ : G → GL(V) a faithful K-r ational r epr esentation of a line ar algebr aic gr oup G with maximal torus T , if is n suciently lar ge then V ⊕ n has a G-e quivariant op en V n on which G acts fr e ely and wher e V n /G is smo oth and quasi-pr oje ctive. Denote X G (ρ ) = lim −→ n V n × G X . If X is a smo oth G-quasi-pr oje ctive scheme pr o c essing a K-r ational p oint then the map X T (ρ )→ X G (ρ ) induc es an isomorphism H ∗ ,∗ ′ G (X;Q)→H ∗ ,∗ ′ T (X;Q) W wher e H ∗ ,∗ ′ T (X;Q) W is the W -invariant subring of H ∗ ,∗ ′ T (X;Q). This theorem sho ws that H ∗ ,∗ ′ (BN SO2n (T);Q) ∼ = L 2 n− 1 n! i=1 H ∗ ,∗ ′ (BT;Q) W . Unfortunately the in tegral motivic more complicated due to torsion and th us this idea will not b e pursued further. 5.3 Integral Motivic Cohomology of BSO 4 The goal here is to compute the in tegral motivic cohomology of BSO 4 using the univ ersal co ecien t theorem (UCT) as presen ted on pg. 27 of Lecture Notes on Motivic Cohomology b y Carlo Mazza, Vladimir V o ev o dsky and Charles W eib el as w ell its classical analog ([MVW06], see [nLab] univ ersal co ecien t theorem for classical v ersion used here) Let’s just note t w o pieces of notation b efore pro ceeding with computations; if S is a set and R is a comm utativ e ring w e let R{S} denote the free R-mo dule with S as its basis, and if A is an ab elian group then l A={a∈A:la=0 A } is the l-torsion subgroup of A. Recall that the short exact sequence 0→Z × 2 − − → Z→Z 2 →0 induces the long exact sequence ...→H ∗ (BSO 4 ;Z) × 2 − − → H ∗ (BSO 4 ;Z) µ − → H ∗ (BSO 4 ;Z 2 ) ˜ β − → H ∗ +1 (BSO 4 ;Z)→... Here, µ is the map induced b y reducing co ecien ts mo d 2 and ˜ β is the inte gr al Bo ckstein homomorphism . The short exact sequence 0→Z 2 × 2 − − → Z 4 →Z 2 →0 induces a similar long exact sequence ...→H ∗ (BSO 4 ;Z 2 ) × 2 − − → H ∗ (BSO 4 ;Z 4 )→H ∗ (BSO 4 ;Z 2 ) β − → H ∗ +1 (BSO 4 ;Z 2 )→... where β is the Bo ckstein homomorphism . These maps are related b y β =µ ◦ ˜ β ([Hat02] 3.E). 68 The univ ersal co ecien t theorem is closely related to the ab o v e, where here w e obtain a short exact sequence on cohomology instead of a long exact one: 0→H ∗ (BSO 4 ;Z)⊗ Z 2 µ C − − → H ∗ (BSO 4 ;Z 2 ) ˜ β C − − → 2 H ∗ +1 (BSO 4 ;Z)→0 These maps µ C and ˜ β C are closely related to µ and ˜ β in that µ (x) = µ C (x⊗ 1) for all x ∈ H ∗ (BSO 4 ;Z) and ˜ β C is the same as β with the only dieren t b eing the co domain. With a sligh t abuse of notation with µ , β and ˜ β b eing reused, the motivic v ersion of these constructions can b e summarized as the follo wing ([MVW06] pg. 27): H ∗ ,∗ ′ (BSO 4 ;Z) ⊗ 1 µ ** H ∗ +1,∗ ′ (BSO 4 ;Z) 0 // H ∗ ,∗ ′ (BSO 4 ;Z)⊗ Z 2 µ M // H ∗ ,∗ ′ (BSO 4 ;Z 2 ) ˜ β M // ˜ β 44 β ** 2 H ∗ +1,∗ ′ (BSO 4 ;Z) ? OO µ // 0 H ∗ +1,∗ ′ (BSO 4 ;Z 2 ) Note that the subscript C ’s andM ’s are mean t to distinguish b et w een classical and motivic. All together, these sequences form the follo wing comm utativ e diagram: 0 // H ∗ ,∗ ′ (BSO 4 ;Z)⊗ Z 2 µ M // t1 ** H ∗ ,∗ ′ (BSO 4 ;Z 2 ) ˜ β M // t2 ** 2 H ∗ +1,∗ ′ (BSO 4 ;Z) // t3 0 0 // H ∗ (BSO 4 ;Z)⊗ Z 2 µ C // H ∗ (BSO 4 ;Z 2 ) ˜ β C // 2 H ∗ +1 (BSO 4 ;Z) // 0 The mapst i are v arian ts of the realization map t m,n :H m,n (BSO 4 ;Z)→H m (BSO 4 ;Z) wheret=⊕ m,n t m,n . There is imp ortan t conclusion to dra w here that is relev an t in the pro of of the main theorem of this section. The map µ C is injectiv e, so φ ∈ Ker(µ C ◦ t 1 ) i φ ∈ Ker(t 1 ). Similarly , µ M is injectiv e, so φ∈Ker(t 2 ◦ µ M ) i µ M (φ)∈Ker(t 2 ). But µ C ◦ t 1 =t 2 ◦ µ M b y comm utativit y and therefore φ∈Ker(t 1 ) ⇐⇒ µ M (φ)∈Ker(t 2 ) (5.5) With these abstract constructions out of the w a y , the next step is to ll in the kno wn cohomology . F or the classical cases, let ζ b e the univ ersal n-plane bundle o v er BSO n . W e tak e w m ∈ H m (BSO n ;Z 2 ) to b e the m th Stiefel-Whitney class of ζ ; similarly , let p m ∈ H 4m (BSO n ;Z) b e the m th P on try agin class of the 69 complexication ζ ⊗ C ([Bro82]). Finally , let d m ∈ H 2m (BSO n ;Z) is the m th Chern class of the standard represen tation of SO n ([Gui07]). As shorthand, if I ={i 1 ,...,i l } w e denote the pro ducts w(2I)=w 2i1 ...w 2i l and p(I)=p i1 ...p i l . With this in mind: Theorem 5.3.1. ([Br o82] 1.3) The mo d 2 c ohomolo gy ring of BSO n is given by H ∗ (BSO n ;Z 2 )=Z 2 [w 2 ,w 3 ,...,w 2n ] Theorem 5.3.2. ([Br o82] 1.5) The inte gr al c ohomolo gy ring of BSO 4 is H ∗ (BSO n ;Z)= ¯ R n / ¯I n wher e ¯ R n =Z[p 1 ,...,p ⌊ n− 1 2 ⌋ ,X n , ˜ βw (2I):1≤ i 1 <...0 } So 2 H m+1 (BSO 4 ;Z) = Z 2 {p i 1 ( √ p 2 ) j ( ˜ βw 2 ) k : i,j ∈ N,k ∈ Z >0 ,4i + 4j + 3k = m + 1} is non-trivial precisely when [ 1 4 (m+2), 1 3 (m+1)]∩Z ̸= ∅; a quic k calculation sho ws that this is equiv alen t to ha ving m+1∈{3,6,7}∪Z ≥ 9 . Con v ersely , w e no w kno w that 2 H m+1 (BSO 4 ;Z)={0} only if m+1∈{1,2,4,5,8}. Consider the v alues of 0→ H m (BSO 4 ;Z)⊗ Z 2 µ C − − → H m (BSO 4 ;Z 2 ) ˜ β C − − → 2 H m+1 (BSO 4 ;Z)→ 0 for the rst sev eral v alues of m, as giv en b y the Lemma 5.3.8: m=0: 0 //Z 2 {1⊗ 1} //Z 2 {1} // 0 // 0 m=1: 0 // 0 // 0 // 0 // 0 m=2: 0 // 0 //Z 2 {w 2 } //Z 2 { ˜ βw 2 } // 0 m=3: 0 //Z 2 { ˜ βw 2 ⊗ 1} //Z 2 {w 3 } // 0 // 0 m=4: 0 //Z 2 {p 1 ⊗ 1, √ p 2 ⊗ 1} //Z 2 {w 2 2 ,w 4 } // 0 // 0 m=5: 0 // 0 //Z 2 {w 2 w 3 } //Z 2 {( ˜ βw 2 ) 2 } // 0 m=6: 0 //Z 2 {( ˜ βw 2 ) 2 ⊗ 1} //Z 2 {w 3 2 ,w 2 w 4 ,w 2 3 } //Z 2 {p 1 ˜ βw 2 , √ p 2 ˜ βw 2 } // 0 Keeping in mind that ˜ β C ◦ t 2 =t 3 ◦ ˜ β M : The m = 2 case sho ws that ˜ β C (w 2 ) = ˜ βw 2 . Because t 2 (τ k w alg 2 ) = w 2 for an y c hoice of k , w e ha v e that t 3 ◦ ˜ β M (τ k w alg 2 ) = ˜ βw 2 . This means that ˜ β M (τ k w alg 2 ) is non-zero in 2 H 3,2+i (BSO 4 ;Z), i.e. ˜ β M (τ i w alg 2 )̸=0 but 2 ˜ β M (τ i w alg 2 )=0. In other w ords, ˜ β M (τ k w alg 2 ) is a lift of ˜ βw 2 for all k . Note that this demonstrates the distinction b et w een the algebraic lifts w alg i and the top ological liftsw top i inH ∗ ,∗ ′ (BSO 4 ;Z 2 ). Both giv e rise to lifts in H ∗ ,∗ ′ (BSO 4 ;Z) and, in fact, so do innitely man y other classes: 75 w alg 2 ˜ β M // _ t2 ˜ β M (w alg 2 ) µ M // _ t3 w alg 3 _ t2 τw alg 2 ˜ β M // _ t2 ˜ β M (τw alg 2 ) µ M // _ t3 w top 3 _ t2 w 2 ˜ β C // ˜ βw 2 µ C // w 3 w 2 ˜ β C // ˜ βw 2 µ C // w 3 Similarly , the m=5 case sho ws that ˜ β M (τ k w alg 2 w alg 3 ) is a lift of ( ˜ βw 2 ) 2 for all k . On the other hand, the Cartan form ula giv es that β (τ k w alg 3 ) = 0; it is w ell-kno wn that β (τ k ) = 0 b ecause no in tegral analog of τ exists ([MVW06] 4.2) and w e also kno w that β (w alg 3 ) = 0 ([HN18]). This sho ws that β (τ k w alg 3 ) is not a non-trivial lift of an y elemen t of H 4 (BSO 4 ;Z). The same reasoning applied to the m = 4 case sho ws that ˜ β M (τ k w 2 2 ) and ˜ β M (τ k w 4 ) are b oth 0 and th us are not considered lifts either. More generally , all 2-torsion elemen ts in H ∗ (BSO 4 ;Z) will ha v e lifts to 2-torsion elemen ts in H ∗ ,∗ ′ (BSO 4 ;Z). As ab o v e, these lifts will b e of the form ˜ β (τ k x) for classes x∈ H ∗ ,∗ ′ (BSO 4 ;Z 2 ) suc h that ˜ β C ◦ t 2 (x) is 2- torsion. Ultimately the result of the ab o v e argumen t is that the in tegral motivic cohomology of BSO 4 has innitely man y generators. F or example, eac h ˜ β M (τ k w alg 2 )∈H 3,2+k (BSO 4 ;Z) is 2-torsion lift satisfying the prop ert y τ k w alg 3 =µ ◦ ˜ β M (τ k w alg 2 ). It is k ey to determine the relations b et w een these ˜ β M (τ k w alg 2 ) for dieren t k . One immediate relation mo d 2 is giv en b y k≥ k ′ =⇒ µ ◦ ˜ β M (τ k w alg 2 )=τ k− k ′ µ ◦ ˜ β M (τ k ′ w alg 2 ) (5.10) It w ould b e con v enien t if there w ere a similar relation on the in tegral side of things, i.e. k ≥ k ′ implies ˜ β M (τ k w 2 ) = T k− k ′ ˜ β M (τ k ′ w 2 ) for some T ∈ H 0,1 (BSO 4 ;Z). Unfortunately that class w ould ha v e to liv e in H 0,1 (Spec(C);Z) and it is kno wn that no suc h class exists ([MVW06] 4.2). Clearly the in tegral case is more complicated than the mo d 2 case due to the fact that no in tegral analog of τ exists. Before pro ceeding an y further, the follo wing lemma will help to greatly simplify argumen ts mo ving forw ard: Lemma 5.3.10. Gener ators of 2-torsion in H ∗ ,∗ ′ (BSO 4 ;Z) that ar e not 2-divisible ar e e qual i their mo d 2 r e ductions in H ∗ ,∗ ′ (BSO 4 ;Z 2 ) ar e e qual. Pro of of Lemma: Supp ose x,y ∈ H ∗ ,∗ ′ (BSO 4 ;Z) are 2-torsion and are not 2-divisible. Clearly if x = y then µ (x) = µ (y), so it remains to go in the other direction. The elemen ts x and y are not 2-divisible so x⊗ 1 and y⊗ 1 are b oth non-zero in H ∗ ,∗ ′ (BSO 4 ;Z)⊗ Z 2 . 76 Assume that µ (x) = µ (y). This implies that µ M (x⊗ 1) = µ M (y⊗ 1). Injectivit y of µ M giv es that x⊗ 1=y⊗ 1. Th us, some o dd m ultiples of x and y m ust b e equal, i.e. (2i+1)x=(2j+1)y for some i and j . But x and y are 2-torsion and therefore 2ix and 2jy are zero; w e conclude that x=y . W e ha v e sho wn that there are innitely man y lifts of ˜ βw 2 ; as stated ab o v e, in principal an y 2-torsion elemen t of H ∗ (BSO 4 ;Z) will ha v e a lift in a similar form. Applying the same construction as done for ˜ βw 2 to other elemen ts of Y agita’s presen tation yields all remaining lifts (where here w e mak e a sligh t simplication to the notation of the lifts): τ k w alg 2 ˜ β M // _ t2 ˜ βτ k w 2 µ // _ t τ k w alg 3 _ t2 w 2 ˜ β C // ˜ βw 2 µ // w 3 τ k− 1 w alg 2 w alg 3 ˜ β M // _ t2 ˜ βτ k− 1 w 2 w 3 µ // _ t τ k− 1 (w alg 3 ) 2 _ t2 w 2 w 3 ˜ β C // ( ˜ βw 2 ) 2 µ // w 2 3 τ k− 1 w alg 2 w alg 4 ˜ β M // _ t2 ˜ βτ k− 1 w 2 w 4 µ // _ t τ k− 1 w alg 3 w alg 4 _ t2 w 2 w 4 ˜ β C // √ p 2 ˜ βw 2 µ // w 3 w 4 τ k− 1 w alg 2 w alg 3 w alg 4 ˜ β M // _ t2 ˜ βτ k− 1 w 2 w 3 w 4 µ // _ t τ k− 1 (w alg 3 ) 2 w alg 4 _ t2 w 2 w 3 w 4 ˜ β C // √ p 2 ( ˜ βw 2 ) 2 µ // w 2 3 w 4 τ k− 2 w alg 2 (w alg 4 ) 2 ˜ β M // _ t2 ˜ βτ k− 2 w 2 w 2 4 µ // _ t τ k− 2 w alg 3 (w alg 4 ) 2 _ t2 w 2 w 2 4 ˜ β C // p 2 ˜ βw 2 µ // w 3 w 2 4 τ k− 3 w alg 2 w alg 3 (w alg 4 ) 2 ˜ β M // _ t2 ˜ βτ k− 3 w 2 w 3 w 2 4 µ // _ t τ k− 3 (w alg 3 ) 2 (w alg 4 ) 2 _ t2 w 2 w 3 w 2 4 ˜ β C // p 2 ( ˜ βw 2 ) 2 µ // w 2 3 w 2 4 77 One imp ortan t conclusion to dra w from these diagrams is that the generator d 3 is of this form: d 3 = ˜ βτ − 1 w 2 w 3 (5.11) There are t w o other t yp es lift that w e ha v e seen, namely those in v olving d 2 , d 4 , and y 2 . Ho w ev er, these t yp es do not in v olv e τ b ecause m ultiple lifts can only exist for 2-torsion elemen ts in this construction: d k1 2 d k2 4 µ // _ t µ (d 2 ) k1 µ (d 4 ) k2 _ t2 (− p 1 ) k1 ( √ p 2 ) 2k2 µ // w 2k1 2 w 2k2 4 y 2 d k1 2 d k2 4 µ // _ t y 0,2 µ (d 2 ) k1 µ (d 4 ) k2 _ t2 2(− p 1 ) k1 ( √ p 2 ) 2k2+1 µ // 0 If w e tak e simple to mean a pro duct of only generators with no co ecien ts, w e see that almost all simple elemen ts of H ∗ (BSO 4 ;Z) ha v e lifts. There is only one t yp e of simple elemen t that remains to b e considered, namely the case of the p o w er of √ p 2 b eing o dd and there b eing no ˜ βw 2 . The Euler class falls in to this category and so the follo wing lemma is a natural con tin uation of the logic from Lemma 5.3.9: Lemma 5.3.11. A n element p j1 1 ( √ p 2 ) j2 ( ˜ βw 2 ) j3 ∈H ∗ (BSO 4 ;Z) lifts H ∗ ,∗ ′ (BSO 4 ;Z) i j 2 is even or j 3 >0. Pro of of Lemma: W e’v e seen that if j 3 is greater than zero then there is a lift. In addition, w e kno w that (− p 1 ) k1 ( √ p 2 ) 2k2 lifts to d k1 2 d k2 4 . Th us, it suces to sho w that if j 2 =2k 2 +1 and k 3 =0 then there is no lift: Supp ose a lift of l alg ∈H 4k1+8k2+4,n (BSO 4 ;Z) of p k1 1 ( √ p 2 ) 2k2+1 exists. Clearly (p k1 1 ( √ p 2 ) 2k2+1 )=p 2k1 1 ( √ p 2 ) 4k2+2 b ecause p 1 and p 2 are b oth ev en degree. But p 2k1 1 ( √ p 2 ) 4k2+2 =p 2k1 1 p 2k2+1 2 lifts to d 2k1 2 d 2k2+1 4 and so ((l alg ) 2 − d 2k1 2 d 2k2+1 4 )⊗ 1 is in Ker(t 1 ). Th us, µ M (((l alg ) 2 − d 2k1 2 d 2k2+1 4 )⊗ 1)=µ M (l alg ⊗ 1) 2 − µ (d 2 ) 2k1 µ (d 4 ) 2k2+1 is in Ker(t 2 ). This giv esµ M (l alg ⊗ 1) 2 =τ 2n− 4k1− 8k2− 4 µ (d 2 ) 2k1 µ (d 4 ) 2k2+1 and there m ust b e some x∈H 4,2 (BSO 4 ;Z 2 ) suc h that x 2 =µ (d 4 ) where µ M (l alg ⊗ 1) is equal to τ n− 2k1− 4k2− 2 µ (d 2 ) k1 µ (d 4 ) k2 x. No suc h x exists in H 4,2 (BSO 4 ;Z 2 )=Z 2 {µ (d 2 ),y 0,2 } and so l alg do es not exist. 78 F amily F orm Co ecien t P o w er of p 1 P o w er of √ p 2 P o w er of ˜ βw 2 F orm of Lifts 1 p k2 1 ( ˜ βw 2 ) 2k3+1 1 an y 0 o dd d k2 2 d k3 3 ˜ βτ k1 w 2 2 p k2 1 ( ˜ βw 2 ) 2k3+2 1 an y 0 ev en >0 d k2 2 d k3 3 ˜ βτ k1− 1 w 2 w 3 3 p k2 1 ( √ p 2 ) 2k4+1 ( ˜ βw 2 ) 2k3+1 1 an y o dd o dd d k2 2 d k3 3 d k4 4 ˜ βτ k1− 1 w 2 w 4 4 p k2 1 ( √ p 2 ) 2k4+1 ( ˜ βw 2 ) 2k3+2 1 an y o dd ev en >0 d k2 2 d k3 3 d k4 4 ˜ βτ k1− 1 w 2 w 3 w 4 5 p k2 1 ( √ p 2 ) 2k4+2 ( ˜ βw 2 ) 2k3+1 1 an y ev en >0 o dd d k2 2 d k3 3 d k4 4 ˜ βτ k1− 2 w 2 w 2 4 6 p k2 1 ( √ p 2 ) 2k4+2 ( ˜ βw 2 ) 2k3+2 1 an y ev en >0 ev en >0 d k2 2 d k3 3 d k4 4 ˜ βτ k1− 3 w 2 w 3 w 2 4 7 λp k2 1 ( √ p 2 ) 2k4 an y an y ev en 0 λ (− d 2 ) k2 d k4 4 8 2λp k2 1 ( √ p 2 ) 2k4+1 ev en an y o dd 0 λy 2 (− d 2 ) k2 d k4 4 9 (2λ +1)p k1 1 ( √ p 2 ) 2k2+1 o dd an y o dd 0 N/A T able 5.1: The elemen ts of H ∗ ,∗ ′ (BSO 4 ;Z) fall in to nine families. Eigh t of these ha v e lifts as ab o v e but the nin th do es not. 79 The rst ob jectiv e is to remo v e an y generators that can b e rewritten completely in terms of other generators. All the generators from the rst six families in T able 5.1 are 2-torsion but not 2-divisible and so Lemma 5.3.10 applies for nding in tegral relations. The driving idea for these relations comes from Prop osition 7.3.1; in particular, if t w o classes in H ∗ ,∗ ′ (BSO 4 ;Z 2 )− Ker(t 2 ) ha v e the same realizations and degrees then they m ust b e equal. This fact allo ws us to rewrite families 2, 4, 5 and 6: ˜ βτ − 1 w 2 w 3 =d 3 ˜ βτ k1+k2 w 2 w 3 =( ˜ βτ k1 w 2 )( ˜ βτ k2 w 2 ) ˜ βτ k1+k2− 1 w 2 w 3 w 4 =( ˜ βτ k1 w 2 )( ˜ βτ k2− 1 w 2 w 4 ) ˜ βτ k1− 2 w 2 w 2 4 =d 4 ˜ βτ k1 w 2 ˜ βτ − 3 w 2 w 3 w 2 4 =d 3 d 4 ˜ βτ k1+k2− 2 w 2 w 3 w 2 4 =d 4 ( ˜ βτ k1 w 2 )( ˜ βτ k2 w 2 ) (5.12) As a more detailed example, w e can consider wh y ˜ βτ − 1 w 2 w 3 is equal to d 3 . W e kno w that t( ˜ βτ − 1 w 2 w 3 ) and t(d 3 ) are b oth equal to ( ˜ βw 2 ) − 1 and b oth ha v e degree (6,3). Our c hoice of notation giv es µ (d 3 )=τ − 1 (w alg 3 ) 2 , so Lemma 5.3.10 and Prop osition 7.3.1 giv e that µ ( ˜ βτ − 1 w 2 w 3 )=τ − 1 (w alg 3 ) 2 and therefore ˜ βτ − 1 w 2 w 3 =d 3 . In short, the only in tegral generators that app ear are y 2 , d 2 , d 3 , d 4 , ˜ βτ k w 2 and ˜ βτ k− 1 w 2 w 4 . It remains to determine ho w these six t yp es of generators in teract with eac h other. First, recall the original relations from Field: 2d 3 =0 y 2 d 3 =0 y 2 2 − 4d 4 =0 (5.13) Next, the comm utativ e diagram construction sho ws that the new classes are all 2-torsion: 2 ˜ βτ k w 2 =0 2 ˜ βτ k− 1 w 2 w 4 =0 (5.14) Lemma 5.3.10 along with µ ( ˜ βτ k w 2 )=τ k w alg 3 and µ ( ˜ βτ k w 2 w 4 )=τ k w alg 3 w alg 4 giv e the follo wing: ( ˜ βτ k1 w 2 )( ˜ βτ k2 w 2 )=( ˜ βτ k3 w 2 )( ˜ βτ k4 w 2 ) ⇐⇒ k 1 +k 2 =k 3 +k 4 ( ˜ βτ k1− 1 w 2 w 4 )( ˜ βτ k2− 1 w 2 w 4 )=( ˜ βτ k3− 1 w 2 w 4 )( ˜ βτ k4− 1 w 2 w 4 ) ⇐⇒ k 1 +k 2 =k 3 +k 4 ( ˜ βτ k1 w 2 )( ˜ βτ k2− 1 w 2 w 4 )=( ˜ βτ k3 w 2 )( ˜ βτ k4− 1 w 2 w 4 ) ⇐⇒ k 1 +k 2 =k 3 +k 4 (5.15) Finally , d 2 do es not in teract with an y other generators and in teractions of new generators with y 2 are similar to those b et w een y 2 and d 3 : y 2 ˜ βτ k w 2 =0 y 2 ˜ βτ k− 1 w 2 w 4 =0 (5.16) These nal equations are sho wn in the same w a y one sho ws that y 2 d 3 = 0. Y agita’s presen tation sho ws that y 0,2 ∈H ∗ ,∗ ′ (BSO 4 ;Z 2 ) only app ears as part ofZ 2 [µ (d 2 ),µ (d 4 )]{y 0,2 }. This implies that y 0,2 m ultiplied b y an y generator other than µ (d 2 ) or µ (d 4 ) is 0 ( [HN18] Theorem 0.1). Keeping in mind that 2 ˜ βw 2 = 0, t(y 2 )=2 √ p 2 , t(d 3 )=( ˜ βw 2 ) 2 , t( ˜ βτ k w 2 )= ˜ βw 2 and t( ˜ βτ k w 2 )= √ p 2 ˜ βw 2 : 80 y 2 d 3 µ // _ t y 0,2 µ (d 3 ) _ t2 y 2 ˜ βτ k w 2 µ // _ t τ k y 0,2 w alg 3 _ t2 y 2 ˜ βτ k− 1 w 2 w 4 µ // _ t τ k− 1 y 0,2 w alg 2 w alg 4 _ t2 0 µ // 0 0 µ // 0 0 µ // 0 Th us, the upp er righ t en try of eac h of these four diagrams m ust b e equal to 0. W e can demonstrate that the upp er left corners m ust all b e 0 as w ell using y 2 d 3 as an example: µ M (y 2 d 3 ⊗ 1)=y 0,2 µ (d 3 ) and µ M is injectiv e, so y 2 d 3 ⊗ 1=0 y 2 and d 3 are b oth not 2-divisible, so therefore y 2 d 3 =0 The claim that y 2 ˜ βτ k w 2 = 0 and y 2 ˜ βτ k− 1 w 2 w 4 = 0 follo ws from the same logic b ecause ˜ βτ k w 2 and ˜ βτ k− 1 w 2 w 4 are not 2-divisible. 5.4 Integral Motivic Cohomology of BG 2 It is w ell kno wn that H ∗ (BG 2 ;Z 2 ) =Z 2 [w 4 ,w 6 ,w 7 ] and H ∗ (BG 2 ;Z p ) =Z p [y 4 ,y 12 ] for p ≥ 3 ([Bor54] 22.6, [MT91] 7.5.9-10 and 7.6.3). Subscripts here alw a ys indicate degree, w i ’s b e the Stiefel-Whitney classes of the 7-dimensional irreducible represen tation of G 2 ([Gui07] pg. 4) where the y i ’s are transgression images of univ ersally transgressiv e x ′ i s where H ∗ (G 2 ;Z p )=Λ Zp (x 3 ,x 11 ) (see Section 2.3). The rst goal of this section is to compute the in tegral cohomology of BG 2 . The result will b e similar to that of BSO 4 in [Bro82], but w e start o with the follo wing lemma: Lemma 5.4.1. The r estriction map H ∗ (BG 2 ;Z)→H ∗ (BSO 4 ;Z) is inje ctive. This is merely the classical v ersion of Lemmas 5.4.6 and 5.4.7, and as suc h the ma jorit y of this simpler pro of will b e omitted. The idea is to use the facts that H ∗ (BG;Z) ,→ H ∗ (BN G (T);Z) for G = G 2 and SO 4 , N SO4 (T) is index 3 in N G2 (T), and all torsion elemen ts x ∈ H ∗ (BG 2 ;Z) are 2-torsion b y a transfer argumen t with the pro jection form ula. F or these transfers, note that the Euler c haracteristic of the b er is 1 b ecause G 2 is connected ([BG75] 6). F rom here w e can state the result where capital Y i ’s are used to reect the y i ’s from transgression but as to not b e confused with y 2 from [Fie12]: Theorem 5.4.2. Ther e ar e elements Y 4 ,Y 7 ,Y 12 ∈H ∗ (BG 2 ;Z) such that H ∗ (BG 2 ;Z)=Z[Y 4 ,Y 7 ,Y 12 ]/(2Y 7 ) 81 Pro of of Theorem 5.4.2: First let’s consider the case of p ≥ 3. F or eac h m w e ha v e the short exact sequence 0→H m (BG 2 ;Z)⊗ Z p µ − → H m (BG 2 ;Z p ) ˜ β − → p H m+1 (BG 2 ;Z)→0 W e can determine the follo wing v alues a priori; w e can ll in the rest of the v alues using exactness and injectivit y of BG 2 cohomology in to BSO 4 cohomology: m=0 1 2 3 4 5 6 7 8 9 10 11 12 H m (BG 2 ;Z)⊗ Z p Z p {1} 0 0 0 0 0 0 0 0 0 H m (BG 2 ;Z p ) Z p {1} 0 0 0 Z p {y 4 } 0 0 0 Z p {y 2 4 } 0 0 0 Z p {y 3 4 ,y 12 } p H m+1 (BG 2 ;Z) 0 0 0 0 0 0 0 0 0 W e kno w that p H m+1 (BG 2 ;Z) = 0 for all m b ecause p H m+1 (BG 2 ;Z) injects in to p H m+1 (BSO 4 ;Z) and BSO 4 has no torsion other than 2-torsion Th us, w e kno w that H 4 (BG 2 ;Z)⊗ Z p is rank 1, i.e. H 4 (BG 2 ;Z)⊗ Z p =Z p {Y 4 ⊗ 1} This tells us H 8 (BG 2 ;Z)⊗ Z p =Z p {Y 2 4 ⊗ 1} and H 12 (BG 2 ;Z)⊗ Z p =Z p {Y 3 4 ⊗ 1,Y 12 ⊗ 1} for some elemen t Y 12 T o summarize, w e kno w that H ∗ (BG 2 ;Z) has only 2-torsion and also has non-torsion generators Y 4 and Y 12 in degrees 4 and 12 resp ectiv ely . Next, let’s turn to the case of p=2 and p erform the same analysis: m=0 1 2 3 4 5 6 7 8 9 10 H m (BG 2 ;Z)⊗ Z 2 Z 2 {1} 0 0 0 0 0 H m (BG 2 ;Z 2 ) Z 2 {1} 0 0 0 Z 2 {w 4 } 0 Z 2 {w 6 } Z 2 {w 7 } Z 2 {w 2 4 } 0 Z 2 {w 4 w 6 } 2 H m+1 (BG 2 ;Z) 0 0 0 0 0 2 H 1 (BG 2 ;Z)=0 b y exactness, and 2 H 5 (BG 2 ;Z)=0 b ecause 2 H 5 (BSO 4 ;Z)=0 Th us b y exactness w e kno w that H 4 (BG 2 ;Z)⊗ Z 2 =Z 2 {Y 4 ⊗ 1} (i.e. Y 4 is not 2-divisible) and that H 8 (BG 2 ;Z)⊗ Z 2 =Z 2 {Y 2 4 ⊗ 1} In addition 2 H 8 (BG 2 ;Z)=0 b ecause 2 H 8 (BSO 4 ;Z)=0 and therefore H 7 (BG 2 ;Z)⊗ Z 2 =Z 2 {Y 7 ⊗ 1} for some Y 7 b y exactness If Y 7 w ere a non-torsion class then it w ould ha v e app eared in the p≥ 3 case; therefore w e kno w that 2 H 7 (BG 2 ;Z)=Z 2 {Y 7 } and H 6 (BG 2 ;Z)⊗ Z 2 =0 82 In short, w e ha v e no w sho wn the existence of the degree 7 generator Y 7 . It is imp ortan t to note that Y 7 is 2-torsion, unlik e the generators Y 4 and Y 12 . It remains to determine the relations b et w een these generators Y 4 , Y 7 and Y 12 . Considering that mo d p reduction is a ring homomorphism and there are no relations on H ∗ (BG 2 ;Z p ) generators, w e kno w that Y 4 and Y 12 ha v e no relations on themselv es or b et w een eac h other. Th us, the only p ossible relations that can exist m ust in v olv e Y 7 . One immediate relation is that 2Y 7 =0, seeing as w e already knew that it is 2-torsion. More generally , supp ose f = P a ijk Y i 4 Y j 7 Y k 12 is an elemen t represen ting a relation. The mo d 2 reduction of f is µ (f)= P a ijk w i 4 w 2k 6 w j 7 (mod 2). W e kno w that µ (f)=0 i a ijk is alw a ys ev en, and therefore all terms where j >1 disapp ear b ecause Y 7 is 2-torsion. This means that f is really of the form 2 P b ik Y i 4 Y k 12 but, as w e’v e already argued, no suc h relations exist b et w een Y 4 and Y 12 . R emark 5.4.3. It is w orth emphasizing that the class Y 7 is really giv en b y ˜ βw 6 . T o see this fact, recall that H ∗ (BG 2 ;Z 2 )=Z 2 [w 4 ,w 6 ,w 7 ] where the w i ’s are the Stiefel-Whitney classes of the inclusion of G 2 in SO 7 . W e kno w that H 6 (BG 2 ;Z 2 )=Z 2 {w 6 } and H 7 (BG 2 ;Z)=Z 2 {Y 7 }; clearly this implies that Y 7 = ˜ βw 6 . This will b e a useful p ersp ectiv e when considering motivic lifts later on. The next step is to mo v e in to the motivic relm. Just as with the BSO 4 case, it is imp ortan t to kno w the details of the morphisms µ C and ˜ β C : Lemma 5.4.4. In the exact se quenc e 0→H ∗ (BG 2 ;Z)⊗ Z 2 µ C − − → H ∗ (BG 2 ;Z 2 ) ˜ β C − − → 2 H ∗ +1 (BG 2 ;Z)→0: µ C (Y k1 4 Y k2 7 Y k3 12 ⊗ 1)=w k1 4 w 2k3 6 w k2 7 ˜ β C (w k1 4 w 2k3 6 w k2 7 )=0 ˜ β C (w k1 4 w 2k3+1 6 w k2 7 )=Y k1 4 Y k2+1 7 Y k3 12 Pro of of Lemma: The ab o v e demonstrates that µ (Y 4 )=w 4 , µ (Y 7 )=w 7 and µ (Y 12 )=w 2 6 . Th us, the rst line holds b y exactness of the sequence and the fact that µ C is a ring homomorphism. The second line holds b y the Cartan form ula and the fact that w 5 =0: β (w 2k3+1 6 )=w 6 β (w 2k3 6 )+β (w 6 )w 2k3 6 =w 2k3 6 w 7 =µ (Y 7 Y k3 12 ) β (w 2k3+1 6 w k2 7 )=w 2k3+1 6 β (w k2 7 )+β (w 2k3+1 6 )w k2 7 =w 2k3 6 w k2+1 7 =µ (Y k2+1 7 Y y3 12 ) β (w k1 4 w 2k3+1 6 w k2 7 )=w k1 4 β (w 2k3+1 6 w k2 7 )+β (w k1 4 )w 2k3+1 6 w k2 7 =w k1 4 w 2k3 6 w k2+1 7 =µ (Y k1 4 Y k2+1 7 Y y3 12 ) Finally , w e can state the main result. The pro of is brok en do wn in to a few parts. W e rst closely follo w the argumen ts of Guillot to sho w that the restriction map ˜ H ∗ ,∗ ′ (BG 2 ;Z) → ˜ H ∗ ,∗ ′ (BSO 4 ;Z) is injectiv e 83 thanks to Theorem 5.1.1. W e then mirror the univ ersal co ecien t theorem argumen ts made for BSO 4 to nd lifts of classical generators. Theorem 5.4.5. L et the c i ’s b e the Chern classes of the 7-dimensional irr e ducible r epr esentation of G 2 , and let the w 2i ’s b e the even Stiefel-Whitney classes of the inclusion G 2 ⊂ SO 7 wher e deg(w 2i )=(2i,i+1) ([Y ag10]). T ake ˜ β :H ∗ ,∗ ′ (BG 2 ;Z 2 )→H ∗ +1,∗ ′ (BG 2 ;Z) to b e the inte gr al Bo ckstein homomorphism. Final ly, let τ b e a gener ator of H 0,1 (Spec(C);Z 2 ) ∼ =Z 2 . R e c al ling that H ∗ (BG 2 ;Z)=Z[Y 4 ,Y 7 ,Y 12 ]/(2Y 7 ), the motivic c ohomolo gy ring of BG 2 with inte gr al c o ecients is H ∗ ,∗ ′ (BG 2 ;Z)=H ∗ ,∗ ′ (Spec(C);Z)[c 2 ,c 4 ,c 6 ,c 7 , ˜ βτ k w 6 , ˜ βτ k− 1 w 4 w 6 :k≥ 0]/I wher e I is the ide al gener ate d by the fol lowing r elations: 2c 7 =0 c 2 c 7 =0 c 2 2 − 4c 4 =0 2 ˜ βτ k w 6 =0 2 ˜ βτ k− 1 w 4 w 6 =0 c 2 ˜ βτ k w 6 =0 c 2 ˜ βτ k− 1 w 3 w 6 =0 ( ˜ βτ k1 w 6 )( ˜ βτ k2 w 6 )=( ˜ βτ k3 w 6 )( ˜ βτ k4 w 6 ) ⇐⇒ k 1 +k 2 =k 3 +k 4 ( ˜ βτ k1− 1 w 4 w 6 )( ˜ βτ k2− 1 w 4 w 6 )=( ˜ βτ k3− 1 w 4 w 6 )( ˜ βτ k4− 1 w 4 w 6 ) ⇐⇒ k 1 +k 2 =k 3 +k 4 ( ˜ βτ k1 w 6 )( ˜ βτ k2− 1 w 4 w 6 )=( ˜ βτ k3 w 6 )( ˜ βτ k4− 1 w 4 w 6 ) ⇐⇒ k 1 +k 2 =k 3 +k 4 ( ˜ βτ k− 1 w 4 w 6 ) 2 =c 4 ( ˜ βτ k w 6 ) 2 ( ˜ βτ k w 6 ) 3 =c 7 ˜ βτ 3k+1 w 6 The de gr e es of the gener ators ar e as fol lows: deg(c i )=(2i,i) deg( ˜ βτ k w 6 )=(7,4+k) deg( ˜ βτ k− 1 w 4 w 6 )=(11,6+k) F urthermor e, the images of the gener ators under the r e alization map t : H ∗ ,∗ ′ (BG 2 ;Z)→ H ∗ (BG 2 ;Z) ar e t(c 2 )=2Y 4 , t(c 4 )=Y 2 4 , t(c 6 )=− Y 12 , t(c 7 )=Y 2 7 , t( ˜ βτ k w 6 )=Y 7 , and t( ˜ βτ k− 1 w 4 w 6 )=Y 4 Y 7 . Pro of: By Lemma 2.2.4, w e kno w that G 2 con tains the semi-direct pro duct K = SL 3 ⋊Z 2 and so w e can iden tify SL 3 with SL 3 ×{ 0}⊂ K . W e also kno w that G 2 con tains SO 4 b y [Gui07] 3.4. Let T b e a maximal torus of G 2 ; w e are allo w ed to b e imprecise here b ecause all maximal tori are conjugate to eac h other. The outline of the pro of is giv en here and closely follo ws the metho d used in [Gui07]: Argue that there is only 2-torsion in ˜ H ∗ ,∗ ′ (BG 2 ;Z) using Lemmas 2.2.4 and 2.3.5 Sho w that the restriction map ˜ H ∗ ,∗ ′ (BG 2 ;Z)→ ˜ H ∗ ,∗ ′ (BSO 4 ;Z) is injectiv e b y Theorem 5.1.1 Sho w that the Cho w ring relations Guillot nds in the pro of of [Gui07] Theorem 4.4 are in fact the only relations in motivic cohomology b ecause of the lac k of other torsion 84 Lemma 5.4.6. If x∈ ˜ H ∗ ,∗ ′ (BG 2 ;Z) is torsion then 2x=0. Pro of of Lemma: First, w e w an t to sho w that if x∈ ˜ H ∗ ,∗ ′ (BG 2 ;Z) is torsion then 2x=0. It is w ell kno wn that the W eyl groups of G 2 and SL 3 are the dihedral group D 6 and symmetric group S 3 resp ectiv ely: W(G 2 ,T)=N G2 (T)/T ∼ =D 6 =⇒ |W(G 2 ,T)|=12 W(SL 3 ,T)=N SL3 (T)/T ∼ =S 3 =⇒ |W(SL 3 ,T)|=6 By the third isomorphism theorem w e ha v e N G2 (T)/N SL3 (T) ∼ =D 6 /S 3 =Z 2 Th us, there is an elemen t of G 2 − SL 3 normalizing T . Because N SL3 (T)⊂ N K (T)⊂ N G2 (T) w e ha v e 2=[N G2 (T):N SL3 (T)]=[N G2 (T):N K (T)][N K (T):N SL3 (T)] =⇒ [N G2 (T):N K (T)]=2 or [N K (T):N SL3 (T)]=2 (5.17) The claim here is that [N K (T):N SL3 (T)]=2 b ecause N K (T)− N SL3 (T) is non-empt y . T o see this, note that Lemma 2.2.4 sho ws that K is a subgroup of G 2 whic h preserv es a cop y of the split-complex n um b ers con tained in the split-o ctonions. In particular, SL 3 ×{ 0}⊂ K acts as the iden tit y x+yj 7→ x+yj while SL 3 ×{ 1} corresp onds to conjugation x+yj 7→ x− yj . As seen at the end of Section 7.1, elemen ts of SL 3 ×{ 0} and SL 3 ×{ 1} tak e the follo wing forms resp ectiv ely for G∈SL 3 :          1 G (G − 1 ) T 1          and          1 − (G − 1 ) T − G 1          (5.18) In addition, the end of Section 2.2.2 sho ws that elemen ts of T ma y b e tak en to b e diagonal matrices M with en tries 1, a, a − 1 b − 1 , b, a − 1 , ab, b − 1 and 1 for non-zero a and b. All suc h matrices are con tained in SL 3 ×{ 0} and conjugating M b y an elemen t of SL 3 ×{ 1} will pro vide an elemen t of SL 3 ×{ 0}; all that remains is to do is force this elemen t to actually b e in T . Note that if G,H ∈SL 3 :          1 − (G − 1 ) T − G 1                   1 H (H − 1 ) T 1                   1 − (G − 1 ) T − G 1          − 1 =          1 (GH T G T ) − 1 GHG T 1          Th us, all w e need to do is nd some G∈SL 3 suc h that GHG T is diagonal whenev er H is diagonal. It’s clear that an y diagonal G of determinan t 1 w orks, th us pro ving the claim that N K (T)− N SL3 (T) is non-empt y . 85 The consequence of the claim is that [N G2 (T) : N K (T)] = 1, meaning that N G2 (T) = N K (T) and that K con tains N G2 (T) b ecause N G2 (T)=N K (T)⊂ K . Theorem 5.1.1 sa ys that the restriction maps giv en b y ˜ H ∗ ,∗ ′ (BG 2 ;Z)→ ˜ H ∗ ,∗ ′ (BN G2 (T);Z) and ˜ H ∗ ,∗ ′ (BK;Z)→ ˜ H ∗ ,∗ ′ (BN K (T);Z) are injectiv e. The dotted line in the follo wing diagram represen ts an injectiv e map ˜ H ∗ ,∗ ′ (BG 2 ;Z)→ ˜ H ∗ ,∗ ′ (BK;Z) b ecause, in general, if f =g◦ h with f and g injectiv e then h m ust b e injectiv e as w ell: ˜ H ∗ ,∗ ′ (BG 2 ;Z) // ˜ H ∗ ,∗ ′ (BN G2 (T);Z) ˜ H ∗ ,∗ ′ (BK;Z) // ˜ H ∗ ,∗ ′ (BN K (T);Z) Supp ose x ∈ ˜ H ∗ ,∗ ′ (BG 2 ;Z) is torsion, i.e. nx = 0 for some n. Injectivit y of the homomorphism h : ˜ H ∗ ,∗ ′ (BG 2 ;Z)→ ˜ H ∗ ,∗ ′ (BK;Z) sho ws that nh(x) = 0 in ˜ H ∗ ,∗ ′ (BK;Z) as w ell. But SL 3 has index 2 in K and it’s w ell kno wn that ˜ H ∗ ,∗ ′ (BSL n ;Z) is torsion-free. A transfer argumen t sho ws that the only p ossible torsion in ˜ H ∗ ,∗ ′ (BK;Z) is 2-torsion and th us n b e m ust equal to 2. This rst lemma can no w b e used to pro of injectivit y of the restriction from BG 2 to BSO 4 in a w a y analogous to the metho d in [Gui07]: Lemma 5.4.7. The r estriction map ˜ H ∗ ,∗ ′ (BG 2 ;Z)→ ˜ H ∗ ,∗ ′ (BSO 4 ;Z) is inje ctive. Pro of of Lemma: Next, let N SO4 (T) and N G2 (T) b e the normalizers of T in SO 4 and G 2 resp ectiv ely . It is w ell kno wn that the index [N G2 (T):N SO4 (T)] is equal to 3 ([Gui07]). Denote the restriction maps ˜ H ∗ ,∗ ′ (BG 2 ;Z) r1 // _ r2 )) ˜ H ∗ ,∗ ′ (BSO 4 ;Z) _ r3 ˜ H ∗ ,∗ ′ (BN G2 (T);Z) r4 // ˜ H ∗ ,∗ ′ (BN SO4 (T);Z) Th us, this index of 3 along with an appropriate transfer argumen t implies that if r 4 (y) = 0 then 3y = 0 for an y y∈ ˜ H ∗ ,∗ ′ (BN G2 (T);Z). T o see this, use the constructions r 4 =res N SO 4 (T) N G 2 (T) : ˜ H ∗ ,∗ ′ (BN G2 (T);Z)→ ˜ H ∗ ,∗ ′ (BN SO4 (T);Z) tr =tr N G 2 (T) N SO 4 (T) : ˜ H ∗ ,∗ ′ (BN SO4 (T);Z)→ ˜ H ∗ ,∗ ′ (BN G2 (T);Z) in Section 4.2; the pro jection theorem sa ys that tr◦ r 4 is just m ultiplication b y 3 ([V ez00] pp. 4-5): r 4 (y)=0 implies 3y = tr◦ r 4 (y) = tr(0) = 0. By injectivit y of r 2 from Theorem 5.1.1, if r 4 ◦ r 2 (x) = 0 then 3x = 0 86 for x∈ ˜ H ∗ ,∗ ′ (BG 2 ;Z). But b y the ab o v e, 3x = 0 implies 2x = 0 as w ell and therefore x m ust b e 0. This directly sho ws that r 4 ◦ r 2 = r 3 ◦ r 1 is injectiv e and th us indirectly sho ws that r 1 is injectiv e b y injectivit y of r 3 from Theorem 5.1.1. The remainder of this pro of relies on the injectivit y of the restriction map ˜ H ∗ ,∗ ′ (BG 2 ;Z)→ ˜ H ∗ ,∗ ′ (BSO 4 ;Z) to nd lifts of generators using the univ ersal co ecien t theorem: 0 // H ∗ ,∗ ′ (BG 2 ;Z)⊗ Z 2 µ M // t1 ** H ∗ ,∗ ′ (BG 2 ;Z 2 ) ˜ β M // t2 )) 2 H ∗ +1,∗ ′ (BG 2 ;Z) // t3 0 0 // H ∗ (BG 2 ;Z)⊗ Z 2 µ C // H ∗ (BG 2 ;Z 2 ) ˜ β C // 2 H ∗ +1 (BG 2 ;Z) // 0 W e kno w that H ∗ (BG 2 ;Z) =Z[Y 4 ,Y 7 ,Y 12 ]/(2Y 7 ) and H ∗ (BG 2 ;Z 2 ) =Z 2 [w 4 ,w 6 ,w 7 ]. Y agita’s conjecture in Theorem 9.6 is v eried b y using results from [Ana20] and [JP20]; ho w ev er, there is a v arian t on Y agita’s presen tation that is more appropriate for our purp oses here (see Section 7.4 for deriv ation): Theorem 5.4.8. The motivic c ohomolo gy of BG 2 withZ 2 -c o ecients is given by Z 2 [τ,y 0,2 ,w alg 4 ,w alg 6 ,w alg 7 ,τ − 2 (w alg 4 ) 2 ,τ − 2 (w alg 6 ) 2 ,τ − 1 (w alg 7 ) 2 ,τ − 1 w alg 4 w alg 6 ,τ − 1 w alg 4 w alg 7 ,τ − 1 w alg 6 w alg 7 ]/I wher e de gr e es ar e given by deg(τ )=(0,1) deg(y 0,2 )=(4,2) deg(w alg 4 )=(4,3) deg(w alg 6 )=(6,4) deg(w alg 7 )=(7,4) and I is the ide al gener ate d by the fol lowing r elations: τy 0,2 =0 y 0,2 w alg 4 =0 y 0,2 w alg 6 =0 y 0,2 w alg 7 =0 y 0,2 τ − 1 (w alg 7 ) 2 =0 y 0,2 τ − 1 w alg 4 w alg 6 =0 y 0,2 τ − 1 w alg 4 w alg 7 =0 y 0,2 τ − 1 w alg 6 w alg 7 =0 The realization map t 2 is completely eviden t b y this presen tation and Lemma 7.4.2 (whic h states that Ker(t)=Z 2 [τ − 2 (w alg 4 ) 2 ,τ − 2 (w alg 6 ) 2 ]{y 0,2 }). Finally , the maps µ C and ˜ β C are giv en in Lemma 5.4.4. With this diagram in mind, w e can nd all the lifts of needed to describ e ˜ H ∗ ,∗ ′ (BG 2 ;Z). Guillot tells us that the CH ∗ (BG 2 )=Z[c 2 ,c 4 ,c 6 ,c 7 ](2c 7 ,c 2 c 7 ,c 2 2 − 4c 4 ) where degree argumen ts sho w that µ (c 2 )=y 0,2 , µ (c 4 )=τ − 2 (w alg 4 ) 2 , µ (c 6 )=τ − 2 (w alg 6 ) 2 , and µ (c 7 )=τ − 1 (w alg 7 ) 2 . This in turn giv es that t◦ µ (c 2 )=0 and t◦ µ (c i )=w 2 i for i∈{4,6,7}. 87 As b efore, the next step is to consider the elemen ts of CH ∗ (BG 2 ) as lifts of classes in H ∗ (BG 2 ;Z). Lemma 5.4.9 b elo w giv es that c 2 , c 4 , c 6 and c 7 are lifts of the follo wing: c 2 µ // _ t y 0,2 _ t2 c 4 µ // _ t τ − 2 (w alg 4 ) 2 _ t2 2Y 4 µ // 0 Y 2 4 µ // w 2 4 c 6 µ // _ t τ − 2 (w alg 6 ) 2 _ t2 c 7 µ // _ t τ − 1 (w alg 7 ) 2 _ t2 − Y 12 µ // w 2 6 Y 2 7 µ // w 2 7 It is clear that the same line of reasoning used for √ p 2 in H ∗ (BSO 4 ;Z) sho ws that there is no lift of Y 4 from H ∗ (BG 2 ;Z) to H ∗ ,∗ ′ (BG 2 ;Z). W e kno w this b ecause c 2 is a lift of 2Y 4 but c 2 is not 2-divisble. The univ ersal co ecien t theorem is no w used to nd all remaining lifts of elemen ts of H ∗ (BG 2 ;Z): τ k w alg 6 ˜ β M // _ t2 ˜ βτ k w 6 µ // _ t τ k w alg 7 _ t2 w 6 ˜ β C // Y 7 µ // w 7 τ k− 1 w alg 4 w alg 6 ˜ β M // _ t2 ˜ βτ k− 1 w 4 w 6 µ // _ t τ k− 1 w alg 4 w alg 7 _ t2 w 4 w 6 ˜ β C // Y 4 Y 7 µ // w 4 w 7 τ k− 1 w alg 6 w alg 7 ˜ β M // _ t2 ˜ βτ k− 1 w 6 w 7 µ // _ t τ k− 1 (w alg 7 ) 2 _ t2 w 6 w 7 ˜ β C // Y 2 7 µ // w 2 7 τ k− 2 (w alg 4 ) 2 w alg 6 ˜ β M // _ t2 ˜ βτ k− 2 w 2 4 w 6 µ // _ t τ k− 2 (w alg 4 ) 2 w alg 7 _ t2 w 2 4 w 6 ˜ β C // Y 2 4 Y 7 µ // w 2 4 w 7 88 τ k− 1 w alg 4 w alg 6 w alg 7 ˜ β M // _ t2 ˜ βτ k− 1 w 4 w 6 w 7 µ // _ t τ k− 1 w alg 4 (w alg 7 ) 2 _ t2 w 4 w 6 w 7 ˜ β C // Y 4 Y 2 7 µ // w 4 w 2 7 τ k− 3 (w alg 4 ) 2 w alg 6 w alg 7 ˜ β M // _ t2 ˜ βτ k− 3 w 2 4 w 6 w 7 µ // _ t τ k− 3 (w alg 4 ) 2 (w alg 7 ) 2 _ t2 w 2 4 w 6 w 7 ˜ β C // Y 2 4 Y 2 7 µ // w 2 4 w 2 7 Just as with the BSO 4 case, these six families of Bo c kstein-t yp e lifts in T able 5.2 collapse to just t w o families when com bined with the Chern classes. In particular, the analog of Prop osition 7.3.1 means that t w o simple non-2-divisble elemen ts not in the k ernel of the realization map are equal i their realizations and degrees agree. As suc h: ˜ βτ − 1 w 6 w 7 =c 7 ˜ βτ k1+k2 w 6 w 7 =( ˜ βτ k1 w 6 )( ˜ βτ k2 w 6 ) ˜ βτ k1+k2− 1 w 4 w 6 w 7 =( ˜ βτ k1 w 6 )( ˜ βτ k2− 1 w 4 w 6 ) ˜ βτ k1− 2 w 2 4 w 6 =c 4 ˜ βτ k1 w 6 ˜ βτ − 3 w 2 4 w 6 w 7 =c 4 c 7 ˜ βτ k1+k2− 2 w 2 4 w 6 w 7 =c 4 ( ˜ βτ k1 w 6 )( ˜ βτ k2 w 6 ) All the necessary generators and relations ha v e b een found except for one remaining set; in particular, it remains to sho w that c 2 m ultiplied b y the new families ˜ βτ k w 6 and ˜ βτ k− 1 w 4 w 6 is trivial. As b efore, this tak es the same approac h as sho wing the kno wn relation c 2 c 7 =0: c 2 c 7 µ // _ t y 0,2 µ (c 7 ) _ t2 c 2 ˜ βτ k w 6 µ // _ t τ k y 0,2 w alg 7 _ t2 c 2 ˜ βτ k− 1 w 4 w 6 µ // _ t τ k− 1 y 0,2 w alg 4 w alg 7 _ t2 0 µ // 0 0 µ // 0 0 µ // 0 Keeping in mind that µ (c 7 ) = τ − 2 (w alg 7 ) 2 , w e kno w b y the presen tation in Theorem 5.4.8 that y 0,2 µ (c 7 ), τ k y 0,2 w alg 7 , τ k− 1 y 0,2 w alg 4 w alg 7 are all zero. Th us, c 2 c 7 ⊗ 1, c 2 ˜ βτ k w 6 ⊗ 1, and c 2 ˜ βτ k− 1 w 4 w 6 ⊗ 1 are all zero as w ell. But these elemen ts are not 2-divisible so the result follo ws. The pro of concludes no w b ecause, b y injectivit y of the map ˜ H ∗ ,∗ ′ (BG 2 ;Z) → ˜ H ∗ ,∗ ′ (BSO 4 ;Z) as in Theorem 5.1.1 as w ell as Lemma 5.4.9, there are no other p ossible generators or relations. 89 F amily F orm Co ecien t P o w er of Y 4 P o w er of Y 7 P o w er of Y 12 F orm of Lifts 1 Y 2k3+1 7 Y k4 12 1 0 o dd an y c k4 6 c k3 7 ˜ βτ k1 w 6 2 Y 2k3+2 7 Y k4 12 1 0 ev en >0 an y c k4 6 c k3 7 ˜ βτ k1− 1 w 6 w 7 3 Y 2k2+1 4 Y 2k3+1 7 Y k4 12 1 o dd o dd an y c k2 4 c k4 6 c k3 7 ˜ βτ k1− 1 w 4 w 6 4 Y 2k2+1 4 Y 2k3+2 7 Y k4 12 1 o dd ev en >0 an y c k2 4 c k4 6 c k3 7 ˜ βτ k1− 1 w 4 w 6 w 7 5 Y 2k2+2 4 Y 2k3+1 7 Y k4 12 1 ev en >0 o dd an y c k2 4 c k4 6 c k3 7 ˜ βτ k1− 2 w 2 4 w 6 6 Y 2k2+2 4 Y 2k3+2 7 Y k4 12 1 ev en >0 ev en >0 an y c k2 4 c k4 6 c k3 7 ˜ βτ k1− 3 w 2 4 w 6 w 7 7 λY 2k2 4 Y k4 12 an y ev en 0 an y λc k2 4 (− c 6 ) k4 8 2λY 2k2+1 4 Y k4 12 ev en o dd 0 an y λc 2 c k2 4 (− c 6 ) k4 9 (2λ +1)Y 2k2+1 4 Y k4 12 o dd o dd 0 an y N/A T able 5.2: The elemen ts of H ∗ ,∗ ′ (BG 2 ;Z) fall in to nine families. Eigh t of these ha v e lifts as ab o v e but the nin th do es not. 90 Lemma 5.4.9. The elements Y 4 ,Y 7 ,Y 12 ∈ H ∗ (BG 2 ;Z) c an b e chosen such that fol lowing c ommutative diagr am holds: c 2 // y 2 +2d 2 c 4 // y 2 d 2 +d 2 2 +d 4 c 6 // y 2 d 4 +d 2 d 4 +d 2 3 c 7 // d 3 d 4 c 2 _ c 4 _ c 6 _ c 7 _ CH ∗ (BG 2 ) ι 1 // τ 1 CH ∗ (BSO 4 ) τ 2 y 2 _ d 2 _ d 3 _ d 4 _ 2Y 4 Y 2 4 − Y 12 Y 2 7 H ∗ (BG 2 ;Z) ι 2 // H ∗ (BSO 4 ;Z) 2 √ p 2 p 1 ( ˜ βw 2 ) 2 p 2 Y 4 // p 1 + √ p 2 Y 7 // √ p 2 ˜ βw 2 Y 12 // − p 1 p 2 − 2( √ p 2 ) 3 − ( ˜ βw 2 ) 4 Pro of of Lemma: The map CH ∗ (BG 2 )→ CH ∗ (BSO 4 ) is kno wn due to [Gui07] and Lemma 5.4.7, and the same is true for CH ∗ (BSO 4 )→H ∗ (BSO 4 ;Z) from [Fie12]. Th us, w e kno w that τ 2 ◦ ι 1 (c 2 )=2(p 1 + √ p 2 ) τ 2 ◦ ι 1 (c 4 )=(p 1 + √ p 2 ) 2 τ 2 ◦ ι 1 (c 6 )=p 1 p 2 +2( √ p 2 ) 3 +( ˜ βw 2 ) 4 τ 2 ◦ ι 1 (c 7 )=p 2 ( ˜ βw 2 ) 2 F rom here w e use the fact that τ 2 ◦ ι 1 = ι 2 ◦ τ 1 b y comm utativit y of the diagram. First let’s consider c 2 and c 4 ; purely from the comm utativ e diagram w e can sho w that τ 1 (c 2 ) =± 2Y 4 and ι 2 (Y 4 ) =± (p 1 + √ p 2 ). Ho w ev er, b ecause the second Chern class is represen ted b y an actual cycle it mak es the most sense to tak e its image to b e p ositiv e. This, com bined with the relation c 2 2 =4c 4 , tells us the follo wing τ 1 (c 2 )=2Y 4 ι 2 (Y 4 )=p 1 + √ p 2 τ 1 (c 4 )=Y 2 4 Next, b ecause CH 7 (BG 2 ;Z)=Z 2 {c 7 } and H 14 (BG 2 ;Z)=Z 2 {Y 2 7 } w e kno w that τ 1 (c 7 )=Y 2 7 ι 2 (Y 7 )= √ p 2 ˜ βw 2 It remains to determine the realization of c 6 . Recall that H 12 (BG 2 ;Z) =Z{Y 3 4 ,Y 12 } and consider the follo wing general images: τ 1 (c 6 )=aY 3 4 +bY 12 ι 2 (Y 12 )=cp 3 1 +dp 2 1 √ p 2 +ep 1 p 2 +f( √ p 2 ) 3 +g( ˜ βw 2 ) 4 W e kno w that ι 2 ◦ τ 1 (c 6 )=p 1 p 2 +2( √ p 2 ) 3 +( ˜ βw 2 ) 4 and so 91 p 1 p 2 +2( √ p 2 ) 3 +( ˜ βw 2 ) 4 = a(p 1 + √ p 2 ) 3 +bcp 3 1 +bdp 2 1 √ p 2 +bep 1 p 2 +bf( √ p 2 ) 3 +bg( ˜ βw 2 ) 4 = (a+bc)p 3 1 +(3a+bd)p 2 1 √ p 2 +(3a+be)p 1 p 2 +(a+bf)( √ p 2 ) 3 +bg( ˜ βw 2 ) 4 This in turn giv es us a system of equations: a+bc=0 3a+bd=0 3a+be=1 a+bf =2 bg =1 The v ariables are all in tegers here and therefore w e ha v e that b = g = ± 1. Th us, the remaining v ariables simplify to a=− bc, d=3c, e=3c+b and f =c+2b. Th us: τ 1 (c 6 )=b(Y 12 − cY 3 4 ) ι 2 (Y 12 )=c(p 1 + √ p 2 ) 3 +b(p 1 p 2 +2( √ p 2 ) 2 +( ˜ βw 2 ) 4 ) The nal step is to determine the v alues of b and c while k eeping in mind that b =± 1 and c∈Z. F or this, w e turn to the mo d 2 reductions of these elemen ts. Denote the maps µ 1 :H ∗ (BG 2 ;Z)→H ∗ (BG 2 ;Z 2 ), µ 2 : H ∗ (BSO 4 ;Z)→ H ∗ (BSO 4 ;Z 2 ), and ι 3 : H ∗ (BG 2 ;Z 2 )→ H ∗ (BSO 4 ;Z 2 ). W e kno w that µ 1 (Y 12 ) = w 2 6 and that µ 2 ◦ ι 2 (Y 12 )=c(w 2 2 +w 4 ) 3 +w 2 2 w 2 4 +w 4 3 . Clearly w 2 2 w 2 4 +w 4 3 =(w 2 w 4 +w 2 3 ) 2 and so ι 3 (w 6 ) 2 =c(w 2 2 +w 4 ) 3 +(w 2 w 4 +w 2 3 ) 2 The righ t hand side cannot b e written as a square unless c is ev en (due to the implied mo d 2). This is the only requiremen t giv en the these equations, so in order to closely matc h the BSO 4 case w e can tak e b=− 1 and c=0: τ 1 (c 6 )=− Y 12 ι 2 (Y 12 )=− p 1 p 2 − 2( √ p 2 ) 3 − ( ˜ βw 2 ) 4 5.5 Minor Result About Steenrod Operations It is w ell kno wn that H ∗ (BG 2 ;Z 2 ) = Z 2 [w 4 ,w 6 ,w 7 ] where w 6 = Sq 2 w 4 and w 7 = Sq 3 w 4 = Sq 1 w 6 ([Bor54] 22.6, [MT91] 7.6.3). Note that w 7 = Sq 1 w 6 is automatic from the other t w o relations b ecause one of the A dem relations is Sq 2n+1 =Sq 1 Sq 2n . Regardless, w e kno w from the univ ersal cases that c haracteristic classes can at least sometimes b e though t of as p olynomials. The question is whether or not that holds true 92 here. In particular, supp ose p is a p olynomial with Z 2 co ecien ts. Thinking of w 4 , w 6 , and w 7 as Sq 4 p, Sq 6 p, and Sq 7 p resp ectiv ely , is there a c hoice of p suc h that w 6 =Sq 2 w 4 and w 7 =Sq 3 w 4 ? Ev en more than this, ho w ev er, w e w an t Sq k p to b e non-zero if and only if k∈{0,4,6,7}. In short: Is there a p olynomial p∈Z 2 [x 1 ,...,x n ] suc h that Sq 6 p=Sq 2 Sq 4 p, Sq 7 p=Sq 3 Sq 4 p, and Sq k p̸=0 ⇐⇒ k∈{0,4,6,7}? Lo oking at the A dem relations, w e see that: Sq 2 Sq 4 = 1 X i=0 3− i 2− 2i Sq 6− i Sq i =Sq 6 Sq 0 +Sq 5 Sq 1 =Sq 6 +Sq 5 Sq 1 Sq 3 Sq 4 = 1 X i=0 3− i 3− 2i Sq 7− i Sq i =Sq 7 Sq 0 =Sq 7 Th us, Sq 7 p = Sq 3 Sq 4 p is satised for an y c hoice of p olynomial. On the other hand, the A dem relations result of Sq 2 Sq 4 = Sq 6 +Sq 5 Sq 1 do es not alw a ys matc h our condition of Sq 6 p = Sq 2 Sq 4 p, but it will for our case b ecause w e require that Sq 1 is trivial. Th us, our our question can b e simply rephrased as follo ws: Is there a non-zero p olynomial p∈Z 2 [x 1 ,...,x n ] suc h that Sq k p̸=0 ⇐⇒ k∈{0,4,6,7}? Let’s try to determine if suc h a p olynomial exists. First, p m ust b e a degree 7 p olynomial b ecause Sq 7 p̸= 0 and k > 7 =⇒ Sq k p = 0. Second, p m ust b e a homogeneous p olynomial b ecause Sq k p m ust b e canceled out when k∈{1,2,3,5}. In particular, terms m ust cancel at the lev el of these Sq k , not at lev el of the ring homomorphism ϕ :Z 2 [x 1 ,...,x n ]→Z 2 [x 1 ,...,x n ],x i 7→x i +x 2 i . Example 5.5.1. Supp ose w e w an ted to nd a p olynomial p suc h that Sq k p̸=0 ⇐⇒ k∈{0,2}. Let’s test p(x)=x+x 2 to see if it satises this prop ert y . It’s clear that ϕ (x+x 2 )=ϕ (x)+ϕ (x) 2 =(x+x 2 )+(x+x 2 ) 2 =x+x 4 W e also kno w that ϕ (x+x 2 ) = Sq 0 (x+x 2 )+Sq 1 (x+x 2 )+Sq 2 (x+x 2 ). Because there are only t w o terms in ϕ (x+x 2 ) one migh t b e lead to b eliev e that Sq 1 (x+x 2 )=0 and th us this is a solution to the p osed question. Ho w ev er: It’s clear that ϕ (x) = x+x 2 and ϕ (x 2 ) = (x+x 2 ) 2 = x 2 +x 4 . Th us, Sq 0 (x) = x,Sq 1 (x) = x 2 and Sq 0 (x 2 )=x 2 ,Sq 1 (x 2 )=0,Sq 2 (x 2 )=x 4 : Sq 0 (x)=x,Sq 1 (x)=x 2 ,Sq 0 (x 2 )=x 2 ,Sq 1 (x 2 )=0,Sq 2 (x 2 )=x 4 =⇒ Sq 0 (x+x 2 )=x+x 2 ,Sq 1 (x+x 2 )=x 2 ,Sq 2 (x+x 2 )=x 4 93 Th us, Sq 1 (x+x 2 ) is non-zero ev en though that term didn’t sho w up in ϕ (x+x 2 ), i.e. this p olynomial is not a solution. Ultimately , the problem that o ccurred in Example 5.5.1 o ccurred b ecause the original p olynomial w as not homogeneous. There w ere t w o copies of x 2 : one from Sq 0 (x 2 ) and the other from Sq 1 (x). This undesirable and deceptiv e cancellation o ccurs only when the p olynomial is non-homogeneous. In addition, recall that univ ersal Chern classes are elemen tary symmetric p olynomials, all of whic h are homogeneous. It’s w ell kno wn that y 4 ,y 6 ,y 7 are Chern classes whic h are themselv es pullbac ks of the univ ersal ones, so it mak es sense to conne our atten tion to homogeneous p olynomials. Another imp ortan t lter for searc hing for our p olynomial is simplicit y , sp ecically the n um b er of v ariables. Supp ose p(x 1 ,...,x n )∈Z 2 [x 1 ,...,x n ] is a solution; it’s clear b y linearit y that p(x 1 ,...,x n )+p(x n+1 ,...,x 2n )∈Z 2 [x 1 ,...,x 2n ] is also a solution but in t wice as man y v ariables. Th us, our goal should b e to nd a solution in the minim um n um b er of v ariables p ossible. With all of this in mind, our goal is to nd homogeneous degree 7 p olynomials that satisfy the needed prop erties. In particular, our searc h requires that Sq k is non-trivial i k ∈ {0,4,6,7}. As shorthand, let c 1 ,...,c n represen t the term x c1 1 ...x c7 n . Because our p olynomials are degree 7, there is no need for more than 7 v ariables for eac h term. T erms in dieren t v ariables can’t p ossibly cancel with eac h other, so w e will only need at most 7 v ariables total. Let’s b egin b y trying to cancel out Sq 1 with linear com binations of the dieren t em b eddings of Sq 0 p olynomials. Lo oking at T able 5.3, the only generators of linear com binations that kill o Sq 1 (and their n um b er of em b eddings in 7 v ariables) are the follo wing: f 1 =6,1+5,2 with 42 em b eddings f 2 =4,3+3,4 with 42 em b eddings f 3 =4,2,1+3,2,2 with 210 em b eddings f 4 =4,2,1+4,1,2 with 105 em b eddings f 5 =2,2,2,1+2,2,1,2 with 140 em b eddings f 6 =4,1,1,1+3,2,1,1+3,1,2,1+3,1,1,2 with 140 em b eddings f 7 =2,2,1,1,1+2,1,2,1,1+2,1,1,2,1+2,1,1,1,2 with 105 em b eddings f 8 =2,1,1,1,1,1+...+1,1,1,1,1,2 with 7 em b eddings 94 Sq 0 Sq 1 # em b eddings in 7 v ars. 7 8 7 4,3 4,4 42 5,2 6,2 42 3,2,2 4,2,2 105 6,1 6,2 42 3,3,1 4,3,1+3,4,1+3,3,2 105 4,2,1 4,2,2 210 2,2,2,1 2,2,2,2 140 5,1,1 6,1,1+5,2,1+5,1,2 105 3,2,1,1 4,2,1,1+3,2,2,1+3,2,1,2 420 4,1,1,1 4,2,1,1+4,1,2,1+4,1,1,2 140 2,2,1,1,1 2,2,2,1,1+2,2,1,2,1+2,2,1,1,2 210 3,1,1,1,1 3,2,1,1,1+...+3,1,1,1,2+4,1,1,1,1 105 2,1,1,1,1,1 2,2,1,1,1,1+2,1,1,1,1,2 42 1,1,1,1,1,1,1 2,1,1,1,1,1,1+...+1,1,1,1,1,1,2 1 T able 5.3: A list of the p olynomial terms of degree 7 in no more than 7 v ariables along with the results of applying the Steenro d homomorphism Sq 1 . Coun ts the n um b er of em b eddings of the terms in to a xed c hoice of 7 v ariables. Next, let’s tak e linear com binations of these f i ’s to try to cancel out Sq 2 . After lo oking through T able 5.4, one sees that the only linear com bination of em b edded f i ’s that cancel Sq 2 correctly will b e when f 4 cancels with itself. In particular: Let g 1 =4,2,1+4,1,2 and g 2 =2,4,1+2,1,4 and g 3 =1,4,2+1,2,4 Sq 2 (g 1 )=4,4,1+4,1,4=Sq 2 (g 2 ) note that Sq 1 (g 1 )=0 and Sq 1 (g 2 )=2,4,2+2,2,4 Sq 2 (g 3 )=1,4,4+1,4,4=0 (sinceZ 2 co ecien ts) note that Sq 1 (g 3 )=2,4,2+2,2,4 Th us, g 1 ,g 2 ha v e non-trivial but iden tical Sq 2 that cancel eac h other out, and the same is true for g 2 ,g 3 with Sq 1 . In short, it app ears that the only w a y to cancel b oth Sq 1 and Sq 2 is b y summing all three: g 1 +g 2 +g 3 =4,4,1+4,1,4+2,4,1+2,1,4+1,4,2+1,2,4 It remains to v erify that Sq 3 and Sq 5 of g 1 +g 2 +g 3 are b oth trivial, and this is easy c hec k ed. Th us, w e can nally state our main conclusion for this section: 95 Sq 0 Sq 2 Notes f 1 8,1+5,4 f 1 and f 2 partially cancel but f 2 5,4+4,5 not enough to remo v e 8,1 or 1,8 f 3 5,2,2+4,4,1+3,4,2+3,2,4 partial cancellation b et w een f 3 and f 4 , but f 4 4,4,1+4,1,4 killing o Sq 2 also kills o Sq 0 f 5 4,2,2,1+4,2,1,2+2,4,2,1+ 2,4,1,2+2,2,4,1+2,2,1,4 f 5 with f 6 has similar problem to f 1 with f 2 f 6 5,2,1,1+5,1,2,1+5,1,1,2+ 4,2,2,1+4,2,1,2+4,1,2,2+ 3,4,1,1+3,1,4,1+3,1,1,4+ 3,2,2,2 mo ving 4 here prev en ts cancellation f 7 4,2,1,1,1+...+4,1,1,1,2+ 2,4,1,1,1+...+2,1,1,1,4+ 2,2,2,2,1+...+2,1,2,2,2 mo ving 4 here prev en ts cancellation f 8 4,1,1,1,1,1+...+1,1,1,1,1,4+ 2,2,2,1,1,1+...+1,1,1,2,2,2 mo ving 4 here prev en ts cancellation T able 5.4: A list of the results of applying the Steenro d homomorphism Sq 2 to thef i ’s determined previously . Applying Sq 1 to these f i mak es them 0 and the notes describ e wh y or wh y not it is p ossible to do the same with Sq 2 . Theorem 5.5.2. A ny homo gene ous p olynomial p such that Sq k p̸= 0 ⇐⇒ k ∈{0,4,6,7} is b ase d on the form x 4 y 2 z+x 4 yz 2 +x 2 y 4 z+x 2 yz 4 +xy 4 z 2 +xy 2 z 4 In other wor ds, any such p is a line ar c ombination of various emb e ddings of the ab ove into a higher numb er of variables. It’s truly in teresting that there is precisely one w a y of doing this. The fact that canceling Sq 1 and Sq 2 actually ended up canceling Sq 3 and Sq 5 as w ell is quite un usual. In addition, up to reordering v ariables and higher em b eddings there is precisely one solution. Just giv en some testing of the ab o v e pro cess, it b ecomes clear just ho w rare this t yp e of result is; the v ast ma jorit y of the time these tests there w as either no p olynomial solution or the p olynomial w as not unique. Example 5.5.3. It’s kno wn that the rational cohomology of BG 2 is H ∗ (BG 2 ;Q) = Q[c 2 ,c 6 ] ([Gui07]). Supp ose w e w an ted to nd a p olynomial p suc h that Sq k p̸= 0 ⇐⇒ k ∈{0,2,6} in a similar v ein as the ab o v e for Z 2 -cohomology; it turns out that no suc h p olynomial exists. If w e try relaxing the condition to nd something that w orks, i.e. Sq k p̸= 0 ⇐⇒ k∈{0,2,6,c} for some c∈{1,3,4,5}, it turns out that the only v alue of c that do es so is 4. Ho w ev er, here w e don’t ev en ha v e a unique solution: Solution 1: Sq 0 =6 Sq 2 =8, Sq 4 =10, Sq 6 =12 96 Solution 2: Sq 0 =2,4 Sq 2 =4,4, Sq 4 =2,8, Sq 6 =4,8 Solution 3: Sq 0 =2,2,2 Sq 2 =4,2,2+2,4,2+2,2,4, Sq 4 =4,4,2+4,2,4+2,4,4, Sq 6 =4,4,4 There seems to b e something sp ecial going on with this p olynomial for H ∗ (BG 2 ;Z 2 ). The details of wh y are not clear at presen t, but further in v estigation migh t pro v e fruitful in the future. 97 Chapter 6 Conclusions and F uture Directions In summary , this w ork giv es results for the motivic cohomology of BSO 4 and BG 2 withZ-co ecien ts. This w ork with BG 2 has three main directions for future study: 1. As G 2 is the automorphism group of the o ctonions, there is p oten tial for computing cohomological in v arian ts of o ctonion algebras. 2. In addition, the metho ds used to compute for G 2 w ould ideally b e applicable to other exceptional Lie groups. The natural starting p oin t w ould b e F 4 as it is the isometry group of the o ctonic pro jectiv e planeOP 2 ([Y ok09]). 3. Metho ds for computations for sp ecial linear groups migh t b e extended to similar conclusions for the symplectic groups. The other main result of this w ork is a deriv ation of a p olynomial describing the Steenro d op erations and their relations to the generators of the ordinary cohomology ring of BG 2 withZ 2 co ecien ts. Again, there are sev eral directions future study can go: 1. What cohomological information can one gain from suc h a p olynomial? 2. Is the existence and uniqueness of this p olynomial signican t or merely a uk e? 3. 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[Y ok09] Y ok ota, Ic hiro: Exceptional Lie Groups. arXiv:0902.043 (2009). 103 Chapter 7 Appendix 7.1 SL 3 ⋊Z 2 Computations for Proof of Lemma 2.2.4 Recall the claim that G 2 ⊂ GL 8 con tains the semi-direct pro duct SL 3 ⋊Z 2 whic h is explicitly giv en as all matrices of the form          1 G (G − 1 ) T 1          and          1 − (G − 1 ) T − G 1          Recall that w e write an elemen t M of GL 8 in the follo wing w a y: M =          a 1 ... a 8 ⃗ u 1 ... ⃗ u 8 ⃗ v 1 ... ⃗ v 8 b 1 ... b 8          As suc h, the condition for M to b e in G 2 coming from Theorem 2.2.3 can b e rewritten as follo ws a 1 ... a 8 − → φ sjs k =a j a k +⃗ u j · ⃗ v k ⃗ u 1 ... ⃗ u 8 − → φ sjs k =a j ⃗ u k +b k ⃗ u j − ⃗ v j × ⃗ v k ⃗ v 1 ... ⃗ v 8 − → φ sjs k =b j ⃗ v k +a k ⃗ v j +⃗ u j × ⃗ u k b 1 ... b 8 − → φ sjs k =b j b k +⃗ v j · ⃗ u k 104 where the v ectors − → φ sjs k are presen ted in T able 2.3 and are the results of m ultiplication of split-o ctonions as giv en our c hoice of basis. Let’s run through the equations for the ⃗ u’s and ⃗ v ’s for dieren t v alues of j and k : 1 and 8 (j,k)=(1,1) =⇒ ⃗ u 1 =a 1 ⃗ u 1 +b 1 ⃗ u 1 ,⃗ v 1 =a 1 ⃗ v 1 +b 1 ⃗ v 1 =⇒ (1− a 1 − b 1 )⃗ u 1 = ⃗ 0,(1− a 1 − b 1 )⃗ v 1 = ⃗ 0 (j,k)=(1,8) =⇒ ⃗ u 1 × ⃗ u 8 =− b 1 ⃗ v 8 − a 8 ⃗ v 1 , ⃗ v 1 × ⃗ v 8 =a 1 ⃗ u 8 +b 8 ⃗ u 1 (j,k)=(8,1) =⇒ ⃗ u 8 × ⃗ u 1 =− b 8 ⃗ v 1 − a 1 ⃗ v 8 , ⃗ v 8 × ⃗ v 1 =a 8 ⃗ u 1 +b 1 ⃗ u 8 =⇒ (a 1 +b 1 )⃗ u 8 +(a 8 +b 8 )⃗ u 1 = ⃗ 0,(a 1 +b 1 )⃗ v 8 +(a 8 +b 8 )⃗ v 1 = ⃗ 0 (j,k)=(8,8) =⇒ ⃗ u 8 =a 8 ⃗ u 8 +b 8 ⃗ u 8 ,⃗ v 8 =a 8 ⃗ v 8 +b 8 ⃗ v 8 =⇒ (1− a 8 − b 8 )⃗ u 8 = ⃗ 0,(1− a 8 − b 8 )⃗ v 1 = ⃗ 0 =⇒ (a 1 +b 1 )(a 8 +b 8 )(⃗ u 1 +⃗ u 8 )= ⃗ 0,(a 1 +b 1 )(a 8 +b 8 )(⃗ v 1 +⃗ v 8 )= ⃗ 0 2 and 5 (j,k)=(2,2) =⇒ ⃗ u 1 =(a 2 +b 2 )⃗ u 2 ,⃗ v 1 =(a 2 +b 2 )⃗ v 2 =⇒ ⃗ u 1 ∥⃗ u 2 ,⃗ v 1 ∥⃗ v 2 (j,k)=(2,5) =⇒ ⃗ u 2 × ⃗ u 5 =− b 2 ⃗ v 5 − a 5 ⃗ v 2 +⃗ v 1 , ⃗ v 2 × ⃗ v 5 =a 2 ⃗ u 5 +b 5 ⃗ u 2 − ⃗ u 1 (j,k)=(5,2) =⇒ ⃗ u 5 × ⃗ u 2 =− b 5 ⃗ v 2 − a 2 ⃗ v 5 +⃗ v 8 , ⃗ v 5 × ⃗ v 2 =a 5 ⃗ u 2 +b 2 ⃗ u 5 − ⃗ u 8 =⇒ (a 2 +b 2 )⃗ u 5 +(a 5 +b 5 )⃗ u 2 =⃗ u 1 +⃗ u 8 ,(a 2 +b 2 )⃗ v 5 +(a 5 +b 5 )⃗ v 2 =⃗ u 1 +⃗ u 8 (j,k)=(5,5) =⇒ ⃗ u 8 =(a 5 +b 5 )⃗ u 5 ,⃗ v 8 =(a 5 +b 5 )⃗ v 5 =⇒ ⃗ u 8 ∥⃗ u 5 ,⃗ v 8 ∥⃗ v 5 3 and 6 (j,k)=(3,3) =⇒ ⃗ u 1 =(a 3 +b 3 )⃗ u 3 ,⃗ v 1 =(a 3 +b 3 )⃗ v 3 =⇒ ⃗ u 1 ∥⃗ u 3 ,⃗ v 1 ∥⃗ v 3 105 (j,k)=(3,6) =⇒ ⃗ u 3 × ⃗ u 6 =− b 3 ⃗ v 6 − a 6 ⃗ v 3 +⃗ v 1 , ⃗ v 3 × ⃗ v 6 =a 3 ⃗ u 6 +b 6 ⃗ u 3 − ⃗ u 1 (j,k)=(6,3) =⇒ ⃗ u 6 × ⃗ u 3 =− b 6 ⃗ v 3 − a 3 ⃗ v 6 +⃗ v 8 , ⃗ v 6 × ⃗ v 3 =a 6 ⃗ u 3 +b 3 ⃗ u 6 − ⃗ u 8 =⇒ (a 3 +b 3 )⃗ u 6 +(a 6 +b 6 )⃗ u 3 =⃗ u 1 +⃗ u 8 ,(a 3 +b 3 )⃗ v 6 +(a 6 +b 6 )⃗ v 3 =⃗ u 1 +⃗ u 8 (j,k)=(6,6) =⇒ ⃗ u 8 =(a 6 +b 6 )⃗ u 6 ,⃗ v 8 =(a 6 +b 6 )⃗ v 6 =⇒ ⃗ u 8 ∥⃗ u 6 ,⃗ v 8 ∥⃗ v 6 4 and 7 (j,k)=(4,4) =⇒ ⃗ u 1 =(a 4 +b 4 )⃗ u 4 ,⃗ v 1 =(a 4 +b 4 )⃗ v 4 =⇒ ⃗ u 1 ∥⃗ u 4 ,⃗ v 1 ∥⃗ v 4 (j,k)=(4,7) =⇒ ⃗ u 4 × ⃗ u 7 =− b 4 ⃗ v 7 − a 7 ⃗ v 4 +⃗ v 1 , ⃗ v 4 × ⃗ v 7 =a 4 ⃗ u 7 +b 7 ⃗ u 4 − ⃗ u 1 (j,k)=(7,4) =⇒ ⃗ u 7 × ⃗ u 4 =− b 7 ⃗ v 4 − a 4 ⃗ v 7 +⃗ v 8 , ⃗ v 7 × ⃗ v 4 =a 7 ⃗ u 4 +b 4 ⃗ u 7 − ⃗ u 8 =⇒ (a 4 +b 4 )⃗ u 7 +(a 7 +b 7 )⃗ u 4 =⃗ u 1 +⃗ u 8 ,(a 4 +b 4 )⃗ v 7 +(a 7 +b 7 )⃗ v 4 =⃗ u 1 +⃗ u 8 (j,k)=(7,7) =⇒ ⃗ u 8 =(a 5 +b 5 )⃗ u 5 ,⃗ v 8 =(a 5 +b 5 )⃗ v 5 =⇒ ⃗ u 8 ∥⃗ u 5 ,⃗ v 8 ∥⃗ v 5 All together, these equations sa y that: If ⃗ u 1 = ⃗ 0 then (a 2 +b 2 )⃗ u 2 = ⃗ 0, (a 3 +b 3 )⃗ u 3 = ⃗ 0 and (a 4 +b 4 )⃗ u 4 = ⃗ 0 If ⃗ u 1 ̸= ⃗ 0 then a 2 +b 2 ,a 3 +b 4 ,a 4 +b 4 ̸= 0, ⃗ u 2 ,⃗ u 3 ,⃗ u 4 ̸= ⃗ 0 and the v ectors ⃗ u 1 , ⃗ u 2 , ⃗ u 3 and ⃗ u 4 are all parallel Similar statemen ts hold for ⃗ v 1 , ⃗ u 8 and ⃗ v 8 . There are t w o v ery imp ortan t assumptions that w e mak e from this p oin t forw ard: ASSUMPTION 1: W e will assume that the v ectors ⃗ u 1 , ⃗ v 1 , ⃗ u 8 and ⃗ v 8 are all zero. ASSUMPTION 2: W e will assume that a i +b i =0 for all i∈{2,...,7} F or simplicit y , it is b est to assume that these four v ectors are all ⃗ 0; if not, there are p oten tial problems getting the rank of the matrix to b e the required v alue of 8 due to duplicate information from parallel v ectors. Similarly , the rst assumption without the second really complicates things. That’s not to sa y that this isn’t p ossible to get solutions, but w e will see that these assumptions allo w us to nd a cop y of SL 3 ⋊Z 2 . As w e will see, these assumptions greatly simplify our w ork from here on out. The consequence of these assumptions is that our matrix tak es the follo wing form: 106          a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 ⃗ 0 ⃗ u 2 ⃗ u 3 ⃗ u 4 ⃗ u 5 ⃗ u 6 ⃗ u 7 ⃗ 0 ⃗ 0 ⃗ v 2 ⃗ v 3 ⃗ v 4 ⃗ v 5 ⃗ v 6 ⃗ v 7 ⃗ 0 b 1 − a 2 − a 3 − a 4 − a 5 − a 6 − a 7 b 8          Next, let’s examine the equations for the a’s and b ′ s for dieren t j and k : 1 and 8 (j,k)=(1,1) =⇒ a 1 =a 2 1 +⃗ u 1 · ⃗ v 1 ,b 1 =b 2 1 +⃗ v 1 · ⃗ u 1 assumptions =⇒ a 1 ,b 1 ∈{0,1} (j,k)=(1,8) =⇒ 0=a 1 a 8 +⃗ u 1 · ⃗ v 8 ,0=b 1 b 8 +⃗ v 1 · ⃗ u 8 (j,k)=(8,1) =⇒ 0=a 8 a 1 +⃗ u 8 · ⃗ v 1 ,0=b 8 b 1 +⃗ v 8 · ⃗ u 1 assumptions =⇒ a 1 a 8 =0,b 1 b 8 =0 (j,k)=(8,8) =⇒ a 8 =a 2 8 +⃗ u 8 · ⃗ v 8 ,b 8 =b 2 8 +⃗ v 8 · ⃗ u 8 assumptions =⇒ a 8 ,b 8 ∈{0,1} 2 and 5 (j,k)=(2,2) =⇒ 0=a 2 2 +⃗ u 2 · ⃗ v 2 ,0=b 2 2 +⃗ v 2 · ⃗ u 2 =⇒ a 2 2 =b 2 2 (j,k)=(2,5) =⇒ a 1 =a 2 a 5 +⃗ u 2 · ⃗ v 5 ,b 1 =b 2 b 5 +⃗ v 2 · ⃗ u 5 (j,k)=(5,2) =⇒ a 8 =a 5 a 2 +⃗ u 5 · ⃗ v 2 ,b 8 =b 5 b 2 +⃗ v 5 · ⃗ u 2 (j,k)=(5,5) =⇒ 0=a 2 5 +⃗ u 5 · ⃗ v 5 ,0=b 2 5 +⃗ v 5 · ⃗ u 5 =⇒ a 2 5 =b 2 5 3 and 6 (j,k)=(3,3) =⇒ 0=a 2 3 +⃗ u 3 · ⃗ v 3 ,0=b 2 3 +⃗ v 3 · ⃗ u 3 =⇒ a 2 3 =b 2 3 107 (j,k)=(3,5) =⇒ a 1 =a 3 a 6 +⃗ u 3 · ⃗ v 6 ,b 1 =b 3 b 6 +⃗ v 3 · ⃗ u 6 (j,k)=(6,3) =⇒ a 8 =a 6 a 3 +⃗ u 6 · ⃗ v 3 ,b 8 =b 6 b 3 +⃗ v 6 · ⃗ u 3 (j,k)=(6,6) =⇒ 0=a 2 6 +⃗ u 6 · ⃗ v 6 ,0=b 2 6 +⃗ v 6 · ⃗ u 6 =⇒ a 2 6 =b 2 6 4 and 7 (j,k)=(4,4) =⇒ 0=a 2 4 +⃗ u 4 · ⃗ v 4 ,0=b 2 4 +⃗ v 4 · ⃗ u 4 =⇒ a 2 4 =b 2 4 (j,k)=(4,7) =⇒ a 1 =a 4 a 7 +⃗ u 4 · ⃗ v 7 ,b 1 =b 4 b 7 +⃗ v 4 · ⃗ u 7 (j,k)=(7,4) =⇒ a 8 =a 7 a 4 +⃗ u 7 · ⃗ v 4 ,b 8 =b 7 b 4 +⃗ v 7 · ⃗ u 4 (j,k)=(7,7) =⇒ 0=a 2 7 +⃗ u 7 · ⃗ v 7 ,0=b 2 7 +⃗ v 7 · ⃗ u 7 =⇒ a 2 7 =b 2 7 Th us, w e ha v e that b i = − a i and a 2 i = b 2 i for all i ∈ {2,...,7}, implying that a 2 ,...,a 7 = 0. Returning to equations for ⃗ u’s and ⃗ v ’s: 2 and 3 (j,k)=(2,3) =⇒ ⃗ u 7 =a 2 ⃗ u 3 +b 3 ⃗ u 2 − ⃗ v 2 × ⃗ v 3 ,⃗ v 7 =b 2 ⃗ v 3 +a 3 ⃗ v 2 + ⃗ u 2 × ⃗ u 3 (j,k)=(3,2) =⇒ − ⃗ u 7 =a 3 ⃗ u 2 +b 2 ⃗ u 3 − ⃗ v 3 × ⃗ v 2 ,− ⃗ v 7 =b 3 ⃗ v 2 +a 2 ⃗ v 3 + ⃗ u 3 × ⃗ u 2 assumptions =⇒ ⃗ u 7 =− ⃗ v 2 × ⃗ v 3 ,⃗ v 7 = ⃗ u 2 × ⃗ u 3 2 and 4 (j,k)=(2,4) =⇒ − ⃗ u 6 =a 2 ⃗ u 4 +b 4 ⃗ u 2 − ⃗ v 2 × ⃗ v 4 ,− ⃗ v 6 =b 2 ⃗ v 4 +a 4 ⃗ v 2 + ⃗ u 2 × ⃗ u 4 (j,k)=(4,2) =⇒ ⃗ u 6 =a 4 ⃗ u 2 +b 2 ⃗ u 4 − ⃗ v 4 × ⃗ v 2 ,⃗ v 6 =b 4 ⃗ v 2 +a 2 ⃗ v 4 + ⃗ u 4 × ⃗ u 2 assumptions =⇒ ⃗ u 6 =⃗ v 2 × ⃗ v 4 ,⃗ v 6 =− ⃗ u 2 × ⃗ u 4 3 and 4 (j,k)=(3,4) =⇒ ⃗ u 5 =a 3 ⃗ u 4 +b 4 ⃗ u 3 − ⃗ v 3 × ⃗ v 4 ,⃗ v 5 =b 3 ⃗ v 4 +a 4 ⃗ v 3 + ⃗ u 3 × ⃗ u 4 (j,k)=(4,3) =⇒ − ⃗ u 5 =a 4 ⃗ u 3 +b 3 ⃗ u 4 − ⃗ v 4 × ⃗ v 3 ,− ⃗ v 5 =b 4 ⃗ v 3 +a 3 ⃗ v 4 + ⃗ u 4 × ⃗ u 3 assumptions =⇒ ⃗ u 5 =− ⃗ v 3 × ⃗ v 4 ,⃗ v 5 = ⃗ u 3 × ⃗ u 4 Giv en the ab o v e, our new matrix form is the follo wing: 108          a 1 0 0 0 0 0 0 a 8 ⃗ 0 ⃗ u 2 ⃗ u 3 ⃗ u 4 − ⃗ v 3 × ⃗ v 4 ⃗ v 2 × ⃗ v 4 − ⃗ v 2 × ⃗ v 3 ⃗ 0 ⃗ 0 ⃗ v 2 ⃗ v 3 ⃗ v 4 ⃗ u 3 × ⃗ u 4 − ⃗ u 2 × ⃗ u 4 ⃗ u 2 × ⃗ u 3 ⃗ 0 b 1 0 0 0 0 0 0 b 8          Recall that a 1 ,b 1 ,a 8 ,b 8 ∈{0,1} and that a 1 a 8 = 0,b 1 b 8 = 0. The only w a y to get a rank 8 matrix with these requiremen ts is if (a 1 ,b 1 ,a 8 ,b 8 ) is (1,0,0,1) or (0,1,1,0). F rom no w on w e will refer to these are Case I and Case I I . There are man y equations that no w b ecome degenerate giv en our main assumptions. F or example, the equation of a’s when (j,k) = (1,2) is just a 2 = a 1 a 2 +⃗ u 1 · ⃗ v 2 ; this is automatically true due to a 2 and ⃗ u 1 b eing zero. Ho w ev er, w e can still get information here from the ⃗ u and ⃗ v equations: 1 and 2 (j,k)=(1,2) =⇒ ⃗ u 2 =a 1 ⃗ u 2 +b 2 ⃗ u 1 − ⃗ v 1 × ⃗ v 2 ,⃗ v 2 =b 1 ⃗ v 2 +a 2 ⃗ v 1 +⃗ u 1 × ⃗ u 2 (j,k)=(2,1) =⇒ ⃗ 0=a 2 ⃗ u 1 +b 1 ⃗ u 2 − ⃗ v 2 × ⃗ v 1 , ⃗ 0=b 2 ⃗ v 1 +a 1 ⃗ v 2 +⃗ u 2 × ⃗ u 1 assumptions =⇒ (1− a 1 )⃗ u 2 = ⃗ 0,(1− b 1 )⃗ v 2 = ⃗ 0,b 1 ⃗ u 2 = ⃗ 0,a 1 ⃗ v 2 = ⃗ 0 1 and 3 (j,k)=(1,3) =⇒ ⃗ u 3 =a 1 ⃗ u 3 +b 3 ⃗ u 1 − ⃗ v 1 × ⃗ v 3 ,⃗ v 3 =b 1 ⃗ v 3 +a 3 ⃗ v 1 +⃗ u 1 × ⃗ u 3 (j,k)=(3,1) =⇒ ⃗ 0=a 3 ⃗ u 1 +b 1 ⃗ u 3 − ⃗ v 3 × ⃗ v 1 , ⃗ 0=b 3 ⃗ v 1 +a 1 ⃗ v 3 +⃗ u 3 × ⃗ u 1 assumptions =⇒ (1− a 1 )⃗ u 3 = ⃗ 0,(1− b 1 )⃗ v 3 = ⃗ 0,b 1 ⃗ u 3 = ⃗ 0,a 1 ⃗ v 3 = ⃗ 0 1 and 4 (j,k)=(1,4) =⇒ ⃗ u 4 =a 1 ⃗ u 4 +b 4 ⃗ u 1 − ⃗ v 1 × ⃗ v 4 ,⃗ v 4 =b 1 ⃗ v 4 +a 4 ⃗ v 1 +⃗ u 1 × ⃗ u 4 (j,k)=(4,1) =⇒ ⃗ 0=a 4 ⃗ u 1 +b 1 ⃗ u 4 − ⃗ v 4 × ⃗ v 1 , ⃗ 0=b 4 ⃗ v 1 +a 1 ⃗ v 4 +⃗ u 4 × ⃗ u 1 assumptions =⇒ (1− a 1 )⃗ u 4 = ⃗ 0,(1− b 1 )⃗ v 4 = ⃗ 0,b 1 ⃗ u 4 = ⃗ 0,a 1 ⃗ v 4 = ⃗ 0 5 and 8 (j,k)=(5,8) =⇒ ⃗ u 5 =a 5 ⃗ u 8 +b 8 ⃗ u 5 − ⃗ v 5 × ⃗ v 8 ,⃗ v 5 =b 5 ⃗ v 8 +a 8 ⃗ v 5 +⃗ u 5 × ⃗ u 8 (j,k)=(8,5) =⇒ ⃗ 0=a 8 ⃗ u 5 +b 5 ⃗ u 8 − ⃗ v 8 × ⃗ v 5 , ⃗ 0=b 8 ⃗ v 5 +a 5 ⃗ v 8 +⃗ u 8 × ⃗ u 5 assumptions =⇒ (1− b 8 )⃗ u 5 = ⃗ 0,(1− a 8 )⃗ v 5 = ⃗ 0,a 8 ⃗ u 5 = ⃗ 0,b 8 ⃗ v 5 = ⃗ 0 6 and 8 (j,k)=(6,8) =⇒ ⃗ u 6 =a 6 ⃗ u 8 +b 8 ⃗ u 6 − ⃗ v 6 × ⃗ v 8 ,⃗ v 6 =b 6 ⃗ v 8 +a 8 ⃗ v 6 +⃗ u 6 × ⃗ u 8 (j,k)=(8,6) =⇒ ⃗ 0=a 8 ⃗ u 6 +b 6 ⃗ u 8 − ⃗ v 8 × ⃗ v 6 , ⃗ 0=b 8 ⃗ v 6 +a 6 ⃗ v 8 +⃗ u 8 × ⃗ u 6 assumptions =⇒ (1− b 8 )⃗ u 6 = ⃗ 0,(1− a 8 )⃗ v 6 = ⃗ 0,a 8 ⃗ u 6 = ⃗ 0,b 8 ⃗ v 6 = ⃗ 0 109 7 and 8 (j,k)=(7,8) =⇒ ⃗ u 7 =a 7 ⃗ u 8 +b 8 ⃗ u 7 − ⃗ v 7 × ⃗ v 8 ,⃗ v 7 =b 7 ⃗ v 8 +a 8 ⃗ v 7 +⃗ u 7 × ⃗ u 8 (j,k)=(8,7) =⇒ ⃗ 0=a 8 ⃗ u 7 +b 7 ⃗ u 8 − ⃗ v 8 × ⃗ v 7 , ⃗ 0=b 8 ⃗ v 7 +a 7 ⃗ v 8 +⃗ u 8 × ⃗ u 7 assumptions =⇒ (1− b 8 )⃗ u 7 = ⃗ 0,(1− a 8 )⃗ v 7 = ⃗ 0,a 8 ⃗ u 7 = ⃗ 0,b 8 ⃗ v 7 = ⃗ 0 These equations giv e a lot of information when put together with Case I and Case I I. Sp ecically , w e ha v e obtained the follo wing information: Case I =⇒ ⃗ v 2 = ⃗ 0, ⃗ v 3 = ⃗ 0 and ⃗ v 4 = ⃗ 0 (and that the v ectors ⃗ u 5 , ⃗ u 6 and ⃗ u 7 are zero as w ell giv en our matrix form in this case) Case I I =⇒ ⃗ u 2 = ⃗ 0, ⃗ u 3 = ⃗ 0 and ⃗ u 4 = ⃗ 0 (and that the v ectors ⃗ v 5 ,⃗ v 6 and ⃗ v 7 are zero as w ell giv en our matrix form in this case) Our new matrix forms for Case I and Case I I resp ectiv ely are: Case I          1 0 0 0 0 0 0 0 ⃗ 0 ⃗ u 2 ⃗ u 3 ⃗ u 4 ⃗ 0 ⃗ 0 ⃗ 0 ⃗ 0 ⃗ 0 ⃗ 0 ⃗ 0 ⃗ 0 ⃗ u 3 × ⃗ u 4 − ⃗ u 2 × ⃗ u 4 ⃗ u 2 × ⃗ u 3 ⃗ 0 0 0 0 0 0 0 0 1          Case I I          0 0 0 0 0 0 0 1 ⃗ 0 ⃗ 0 ⃗ 0 ⃗ 0 − ⃗ v 3 × ⃗ v 4 ⃗ v 2 × ⃗ v 4 − ⃗ v 2 × ⃗ v 3 ⃗ 0 ⃗ 0 ⃗ v 2 ⃗ v 3 ⃗ v 4 ⃗ 0 ⃗ 0 ⃗ 0 ⃗ 0 1 0 0 0 0 0 0 0          There is just one nal set of ⃗ u and⃗ v equations that needs to b e c hec k ed and expanded. These nal lines require the use of the v ector quadruple pro duct iden tit y giv en b y (⃗ a× ⃗ b)× (⃗ c× ⃗ d)=((⃗ a× ⃗ b)· ⃗ d)⃗ c− ((⃗ a× ⃗ b)· ⃗ c) ⃗ d. T o see this: 5 and 6 (j,k)=(5,6) =⇒ − ⃗ u 4 =a 5 ⃗ u 6 +b 6 ⃗ u 5 − ⃗ v 5 × ⃗ v 6 ,− ⃗ v 4 =b 5 ⃗ v 6 +a 6 ⃗ v 5 +⃗ u 5 × ⃗ u 6 (j,k)=(6,5) =⇒ ⃗ u 4 =a 6 ⃗ u 5 +b 5 ⃗ u 6 − ⃗ v 6 × ⃗ v 5 ,⃗ v 4 =b 6 ⃗ v 5 +a 5 ⃗ v 6 +⃗ u 6 × ⃗ u 5 Case I =⇒ ⃗ u 4 =(⃗ u 3 × ⃗ u 4 )× (− ⃗ u 2 × ⃗ u 4 )=− ((⃗ u 3 × ⃗ u 4 )· ⃗ u 4 )⃗ u 2 +((⃗ u 3 × ⃗ u 4 )· ⃗ u 2 )⃗ u 4 =⇒ ((⃗ u 3 × ⃗ u 4 )· ⃗ u 2 )=1 Case II =⇒ ⃗ v 4 =(⃗ v 2 × ⃗ v 4 )× (− ⃗ v 3 × ⃗ v 4 )=− ((⃗ v 2 × ⃗ v 4 )· ⃗ v 4 )⃗ v 3 +((⃗ v 2 × ⃗ v 4 )· ⃗ v 3 )⃗ v 4 =⇒ ((⃗ v 2 × ⃗ v 4 )· ⃗ v 3 )=1 110 5 and 7 (j,k)=(5,7) =⇒ ⃗ u 3 =a 5 ⃗ u 7 +b 7 ⃗ u 5 − ⃗ v 5 × ⃗ v 7 ,⃗ v 3 =b 5 ⃗ v 7 +a 7 ⃗ v 5 +⃗ u 5 × ⃗ u 7 (j,k)=(7,5) =⇒ − ⃗ u 3 =a 7 ⃗ u 5 +b 5 ⃗ u 7 − ⃗ v 7 × ⃗ v 5 ,− ⃗ v 3 =b 7 ⃗ v 5 +a 5 ⃗ v 7 +⃗ u 7 × ⃗ u 5 Case I =⇒ ⃗ u 3 =(⃗ u 2 × ⃗ u 3 )× (⃗ u 3 × ⃗ u 4 )=((⃗ u 2 × ⃗ u 3 )· ⃗ u 4 )⃗ u 3 − ((⃗ u 2 × ⃗ u 3 )· ⃗ u 3 )⃗ u 4 =⇒ ((⃗ u 2 × ⃗ u 3 )· ⃗ u 4 )=1 Case II =⇒ ⃗ v 3 =(− ⃗ v 3 × ⃗ v 4 )× (− ⃗ v 2 × ⃗ v 3 )=((⃗ v 3 × ⃗ v 4 )· ⃗ v 3 )⃗ v 2 − ((⃗ v 3 × ⃗ v 4 )· ⃗ v 2 )⃗ v 3 =⇒ ((⃗ v 3 × ⃗ v 4 )· ⃗ v 2 )=− 1 6 and 7 (j,k)=(6,7) =⇒ − ⃗ u 2 =a 6 ⃗ u 7 +b 7 ⃗ u 6 − ⃗ v 6 × ⃗ v 7 ,− ⃗ v 2 =b 6 ⃗ v 7 +a 7 ⃗ v 6 +⃗ u 6 × ⃗ u 7 (j,k)=(7,6) =⇒ ⃗ u 2 =a 7 ⃗ u 6 +b 6 ⃗ u 7 − ⃗ v 7 × ⃗ v 6 ,⃗ v 2 =b 7 ⃗ v 6 +a 6 ⃗ v 7 +⃗ u 7 × ⃗ u 6 Case I =⇒ ⃗ u 2 =(− ⃗ u 2 × ⃗ u 4 )× (⃗ u 2 × ⃗ u 3 )=− ((⃗ u 2 × ⃗ u 4 )· ⃗ u 3 )⃗ u 2 +((⃗ u 2 × ⃗ u 4 )· ⃗ u 2 )⃗ u 3 =⇒ ((⃗ u 2 × ⃗ u 4 )· ⃗ u 3 )=− 1 Case II =⇒ ⃗ v 2 =(− ⃗ v 2 × ⃗ v 3 )× (⃗ v 2 × ⃗ v 4 )=− ((⃗ v 2 × ⃗ v 3 )· ⃗ v 4 )⃗ v 2 +((⃗ v 2 × ⃗ v 3 )· ⃗ v 2 )⃗ v 4 =⇒ ((⃗ v 2 × ⃗ v 3 )· ⃗ v 4 )=− 1 With these nal equations it is clear that w e get our cop y of SL 3 ⋊Z 2 . The scalar triple pro duct form ula sa ys that (⃗ a× ⃗ b)· ⃗ c=det ⃗ a ⃗ b ⃗ c . Th us, all the ab o v e equations are sa ying is the follo wing: Case I =⇒ det ⃗ u 2 ⃗ u 3 ⃗ u 4 =1, i.e. ⃗ u 2 ⃗ u 3 ⃗ u 4 ∈SL 3 Case I I =⇒ det ⃗ v 2 ⃗ v 3 ⃗ v 4 =− 1, i.e. − ⃗ v 2 ⃗ v 3 ⃗ v 4 ∈SL 3 G= ⃗ a ⃗ b ⃗ c ∈SL 3 =⇒ (G − 1 ) T = ⃗ b× ⃗ c − ⃗ a× ⃗ c ⃗ a× ⃗ b The claims for b oth matrix form and group structure no w follo w b ecause if G,H ∈SL 3 then: Case I =⇒          1 G (G − 1 ) T 1          Case II =⇒          1 − (G − 1 ) T − G 1          111          1 G (G − 1 ) T 1                   1 H (H − 1 ) T 1          =          1 GH ((GH) − 1 ) T 1                   1 G (G − 1 ) T 1                   1 − (H − 1 ) T − H 1          =          1 − G(H − 1 ) T − (G − 1 ) T H 1                   1 − (G − 1 ) T − G 1                   1 H (H − 1 ) T 1          =          1 − ((GH) − 1 ) T − GH 1                   1 − (G − 1 ) T − G 1                   1 − (H − 1 ) T − H 1          =          1 (G − 1 ) T H G(H − 1 ) T 1          7.2 Isomorphisms Between F orms of SO n Recall that the goal is to nd a map o v er C from SO ′ n to SO n via conjugation in GL n where A7→gAg 1 suc h that J =g T g and J =       0 ... 1 . . . . . . . . . 1 ... 0       ∈M n× n (C) =⇒ SO ′ n ={A∈SL n (C):A T JA=J} SO n ={M ∈SL n (C):M T IM =I} Solutions to J = g T g are not unique and brute forcing a solution using computational metho ds do es not pro vide m uc h insigh t. T o get a meaningful family of solutions, let’s b egin with the case of n=2. 112 g =    a b c d    =⇒    0 1 1 0    =    a c b d       a b c d    =    a 2 +c 2 ab+cd ab+cd b 2 +d 2    =⇒ a 2 +c 2 =0,ab+cd=1,b 2 +d 2 =0 =⇒ c=± ia,d=± ib (c,d)=(ia,ib) =⇒ ab+(ia)(ib)=ab− ab=0̸=1 =⇒ fails (c,d)=(ia,− ib) =⇒ ab+(ia)(− ib)=ab+ab=2ab=1 =⇒ works (c,d)=(− ia,ib) =⇒ ab+(− ia)(ib)=ab+ab=2ab=1 =⇒ works (c,d)=(− ia,− ib) =⇒ ab+(− ia)(− ib)=ab− ab=0̸=1 =⇒ fails An y solution here is v alid but p erhaps the simplest tak es the follo wing form: a∈G m ,b= 1 2a ,c=− ia,d= i 2a g =    a 1 2a − ia i 2a    This matrix is particularly useful when considering maximal tori T ′ 2 andT 2 ofSO ′ 2 andSO 2 . It’s w ell kno wn that T ′ 2 =         z 0 0 z − 1    :z∈G m      T 2 =         1 2 (z+z − 1 ) − 1 2i (z− z − 1 ) 1 2i (z− z − 1 ) 1 2 (z+z − 1 )    :z∈G m      and that more generally the tori T ′ 2n and T 2n of SO ′ 2n and SO 2n are T ′ 2n =                                                z 1 . . . z n z − 1 n . . . z − 1 1                 :z i ∈G m                                113 T 2n =                  A 1 . . . A n       :A i ∈T 2            It is easy to sho w that conjugation maps b et w een these initial tori v ery neatly , i.e. for an y a∈G m    1 2 (z+z − 1 ) − 1 2i (z− z − 1 ) 1 2i (z− z − 1 ) 1 2 (z+z − 1 )    =    a 1 2a − ia i 2a       z 0 0 z − 1       a 1 2a − ia i 2a    − 1 It’s not hard to see what pattern giv es a family of matrices transforming T ′ 2n in to T 2n . F or example, the 2n=4 and 2n=6 cases are: 4× 4 =⇒ g =          a 1 2a b 1 2b − ib i 2b − ia i 2a          6× 6 =⇒ g =                 a 1 2a b 1 2b c 1 2c − ic i 2c − ib i 2b − ia i 2a                 7.3 Supporting W ork for H ∗ ,∗ ′ (BSO 4 ;Z) Lo oking at Y agita’s presen tation of H ∗ ,∗ ′ (BSO 4 ;Z 2 ) ([Y ag10] 9.4), w e see that there are the follo wing 16 generators: τ y 0,2 µ (d 2 ) µ (d 4 ) w alg 2 Q 0 w alg 2 Q 1 w alg 2 Q 0 Q 1 w alg 2 µ (d 3 )a Q 0 a Q 1 a Q 2 a Q 0 Q 1 a Q 0 Q 2 a Q 1 Q 2 a Q 0 Q 1 Q 2 a In particular, the ring breaks do wn in to the direct sum of four pieces: Z 2 [µ (d 2 ),µ (d 4 )]{y 0,2 } Z 2 [τ,µ (d 2 ),µ (d 4 )] Z 2 [τ,µ (d 2 ),µ (d 3 )]⊗ Q(1){w alg 2 } Z 2 [τ,µ (d 2 ),µ (d 3 ),µ (d 4 )]⊗ Q(2){a}− Z 2 [τ,µ (d 2 ),µ (d 4 )]{a} 114 It can b e dicult to understand the relations b et w een these generators b ecause a is a virtual elemen t that actually b elongs to H 0 (BSO 4 ;H 3 (BZ 2 ;Z 2 )). The follo wing prop osition helps in simplifying the presen tation: Prop osition 7.3.1. L et x,y ∈ H ∗ ,∗ ′ (BSO 4 ;Z 2 ) b e pr o ducts of Y agita’s 16 gener ators not in the kernel of the r e alization map. The de gr e es and r e alizations of x and y b oth agr e e i x=y . Pro of of Prop osition: Recall that t 2 :H ∗ ,∗ ′ (BSO 4 ;Z 2 )→H ∗ (BSO 4 ;Z 2 ) is the realization map. Clearly if x = y then deg(x) = deg(y) and t 2 (x) = t 2 (y), so it remains to pro v e the other direction. It is clas- sically kno wn that there are no relations b et w een the generators of H ∗ (BSO 4 ;Z 2 ) =Z 2 [w 2 ,w 3 ,w 4 ]; th us, w i1 2 w j1 3 w k1 4 =w i2 2 w j2 3 w k4 4 i i 1 =i 2 , j 1 =j 2 and k 1 =k 2 . Assume deg(x) = deg(y) and t 2 (x) = t 2 (y). The map t 2 is a ring homomorphism so t 2 (x− y) = 0, i.e. x− y ∈ Ker(t 2 ) =Z 2 [µ (d 2 ),µ (d 4 )]{y 0,2 } ([HN18] 0.1). All elemen ts of Ker(t 2 ) are of degree (4c,2c) for v arious v alues of c, and w e kno w that the degrees of the 16 generators are giv en b y deg(τ )=(0,1) deg(y 0,2 )=(4,2) deg(µ (d 2 ))=(4,2) deg(µ (d 4 ))=(8,4) deg(w alg 2 )=(2,2) deg(Q 0 w alg 2 )=(3,2) deg(Q 1 w alg 2 )=(5,3) deg(Q 0 Q 1 w alg 2 )=(6,3) deg(µ (d 3 )a)=(9,6) deg(Q 0 a)=(4,3) deg(Q 1 a)=(6,4) deg(Q 2 a)=(10,6) deg(Q 0 Q 1 a)=(7,4) deg(Q 0 Q 2 a)=(11,6) deg(Q 1 Q 2 a)=(13,7) deg(Q 0 Q 1 Q 2 a)=(14,7) If x− y = 0 then no further w ork is needed, so assume x− y̸= 0. This implies that deg(x) and deg(y) are b oth equal to (4c,2c) for some c; b ecause x and y are tak en to not b e in Ker(t 2 ) but are pro ducts of the 16 generators, it is clear that x and y m ust b e pro ducts of only µ (d 2 ), (Q 0 Q 1 w alg 2 ) 2 and µ (d 4 ): x=(µ (d 2 )) i1 (Q 0 Q 1 w alg 2 ) 2j1 (µ (d 4 )) k1 =⇒ t 2 (x)=w 2i1 2 w 4j1 3 w 2k1 4 y =(µ (d 2 )) i2 (Q 0 Q 1 w alg 2 ) 2j2 (µ (d 4 )) k2 =⇒ t 2 (y)=w 2i2 2 w 4j2 3 w 2k2 4 But t 2 (x)=t 2 (y) implies that i 1 =i 2 , j 1 =j 2 and k 1 =k 2 ; this turn means that x=y and con tradicts our assumption that x− y̸=0. With Prop osition 7.3.1 in place, w e can analyze the 16 generators more closely . The realizations of the 16 generators are giv en b y the follo wing: t 2 (τ )=1 t 2 (y 0,2 )=0 t 2 (µ (d 2 ))=w 2 2 t 2 (µ (d 4 ))=w 2 4 t 2 (w alg 2 )=w 2 t 2 (Q 0 w alg 2 )=w 3 t 2 (Q 1 w alg 2 )=w 2 w 3 t 2 (Q 0 Q 1 w alg 2 )=w 2 3 t 2 (µ (d 3 )a)=w 2 w 3 w 4 t 2 (Q 0 a)=w 4 t 2 (Q 1 a)=w 2 w 4 t 2 (Q 2 a)=w 2 w 2 4 t 2 (Q 0 Q 1 a)=w 3 w 4 t 2 (Q 0 Q 2 a)=w 3 w 2 4 t 2 (Q 1 Q 2 a)=w 2 w 3 w 2 4 t 2 (Q 0 Q 1 Q 2 a)=w 2 3 w 2 4 115 As w e can see, these 16 images are all distinct but ha v e relations b et w een them. F or example, w e see that t 2 (µ (d 2 )) = t 2 ((w alg 2 ) 2 ) and so p erhaps µ (d 2 ) and (w alg 2 ) 2 could b e equal. Ho w ev er, this is not the case b ecause deg(µ (d 2 )) = (4,2) but deg((w alg 2 ) 2 ) = (4,4). W e can systematically consider relations b et w een realizations and lo ok at the corresp onding degrees. First, for the realizations that are pro ducts of t w o w i ’s: t 2 (µ (d 2 ))=t 2 ((w alg 2 ) 2 )=w 2 2 but deg(µ (d 2 ))=(4,2), deg((w alg 2 ) 2 )=(4,4) t 2 (µ (d 4 ))=t 2 ((Q 0 a) 2 )=w 2 4 but deg(µ (d 4 ))=(8,4), deg((Q 0 a) 2 )=(8,6) t 2 (Q 1 w alg 2 )=t 2 (w alg 2 Q 0 w alg 2 )=w 2 w 3 but deg(Q 1 w alg 2 )=(5,3), deg(w alg 2 Q 0 w alg 2 )=(5,4) t 2 (Q 0 Q 1 w alg 2 )=t 2 ((Q 0 w alg 2 ) 2 )=w 2 3 but deg(Q 0 Q 1 w alg 2 )=(6,3), deg((Q 0 w alg 2 ) 2 )=(6,4) t 2 (Q 1 a)=t 2 (w alg 2 Q 0 a)=w 2 w 4 but deg(Q 1 a)=(6,4), deg(w alg 2 Q 0 a)=(6,5) t 2 (Q 0 Q 1 a)=t 2 (Q 0 w alg 2 Q 0 a)=w 3 w 4 but deg(Q 0 Q 1 a)=(7,4), deg(Q 0 w alg 2 Q 0 a)=(7,5) Next, for pro ducts of three or more w i ’s: t 2 (µ (d 3 )a)=t 2 (w alg 2 Q 0 Q 1 a)=w 2 w 3 w 4 and deg(µ (d 3 )a)=deg(w alg 2 Q 0 Q 1 a)=(9,6) µ (d 3 )a=w alg 2 Q 0 Q 1 a t 2 (µ (d 3 )a)=t 2 (Q 0 w alg 2 Q 1 a)=w 2 w 3 w 4 and deg(µ (d 3 )a)=deg(Q 0 w alg 2 Q 1 a)=(9,6) µ (d 3 )a=Q 0 w alg 2 Q 1 a t 2 (µ (d 3 )a)=t 2 (Q 1 w alg 2 Q 0 a)=w 2 w 3 w 4 and deg(µ (d 3 )a)=deg(Q 1 w alg 2 Q 0 a)=(9,6) µ (d 3 )a=Q 1 w alg 2 Q 0 a t 2 (µ (d 3 )a)=t 2 (w alg 2 Q 0 w alg 2 Q 0 a)=w 2 w 3 w 4 but deg(µ (d 3 )a)=(9,6), deg(w alg 2 Q 0 w alg 2 Q 0 a)=(9,7) t 2 (Q 2 a)=t 2 (µ (d 4 )w alg 2 )=w 2 w 2 4 and deg(Q 2 a)=deg(µ (d 4 )w alg 2 )=(10,6) Q 2 a=µ (d 4 )w alg 2 t 2 (Q 2 a)=t 2 (Q 0 aQ 1 a)=w 2 w 2 4 but deg(Q 2 a)=(10,6), deg(Q 0 aQ 1 a)=(10,7) t 2 (Q 0 Q 2 a)=t 2 (µ (d 4 )Q 0 w alg 2 )=w 3 w 2 4 and deg(Q 0 Q 2 a)=deg(µ (d 4 )Q 0 w alg 2 )=(11,6) Q 0 Q 2 a=µ (d 4 )Q 0 w alg 2 t 2 (Q 0 Q 2 a)=t 2 (Q 0 aQ 0 Q 1 a)=w 3 w 2 4 but deg(Q 0 Q 2 a)=(11,6), deg(Q 0 aQ 0 Q 1 a)=(11,7) 116 t 2 (Q 1 Q 2 a)=t 2 (µ (d 4 )Q 1 w alg 2 )=w 2 w 3 w 2 4 and deg(Q 1 Q 2 a)=deg(µ (d 4 )Q 1 w alg 2 )=(13,7) Q 1 Q 2 a=µ (d 4 )Q 1 w alg 2 t 2 (Q 1 Q 2 a)=t 2 (Q 0 w alg 2 Q 2 a)=w 2 w 3 w 2 4 but deg(Q 1 Q 2 a)=(13,7), deg(Q 0 w alg 2 Q 2 a)=(13,8) t 2 (Q 1 Q 2 a)=t 2 (w alg 2 Q 0 Q 2 a)=w 2 w 3 w 2 4 but deg(Q 1 Q 2 a)=(13,7), deg(w alg 2 Q 0 Q 2 a)=(13,8) t 2 (Q 1 Q 2 a)=t 2 (Q 1 aQ 0 Q 1 a)=w 2 w 3 w 2 4 but deg(Q 1 Q 2 a)=(13,7), deg(Q 1 aQ 0 Q 1 a)=(13,8) t 2 (Q 0 Q 1 Q 2 a)=t 2 (µ (d 4 )Q 0 Q 1 w alg 2 )=w 2 3 w 2 4 and deg(Q 0 Q 1 Q 2 a)=deg(µ (d 4 )Q 0 Q 1 w alg 2 )=(14,7) Q 0 Q 1 Q 2 a=µ (d 4 )Q 0 Q 1 w alg 2 t 2 (Q 0 Q 1 Q 2 a)=t 2 ((Q 0 Q 1 a) 2 )=w 2 3 w 2 4 but deg(Q 0 Q 1 Q 2 a)=(14,7), deg((Q 0 Q 1 a) 2 )=(14,8) t 2 (Q 0 Q 1 Q 2 a)=t 2 (Q 0 w alg 2 Q 0 Q 2 a)=w 2 3 w 2 4 but deg(Q 0 Q 1 Q 2 a)=(14,7), deg(Q 0 w alg 2 Q 0 Q 2 a)=(14,8) The ab o v e computations greatly simplify matters mo ving forw ard. First of all, w e ha v e lifts of missing Chern class µ (d 3 ) and the Stiefel-Whitney classes w alg 3 and w alg 4 : w alg 3 =Q 0 w alg 2 µ (d 3 )=Q 0 Q 1 w alg 2 w alg 4 =Q 0 a Prop osition 7.3.1 also allo ws us to rewrite 5 of the 16 generators in terms of the remaining 11: µ (d 3 )a=w alg 2 Q 0 Q 1 a=w alg 3 Q 1 a=w alg 4 Q 1 w alg 2 Q 2 a=µ (d 4 )w alg 2 Q 0 Q 2 a=µ (d 4 )w alg 3 Q 1 Q 2 a=µ (d 4 )Q 1 w alg 2 Q 0 Q 1 Q 2 a=µ (d 3 )c alg 4 Finally , w e no w ha v e an alternativ e presen tation of the mo d 2 motivic cohomology of BSO 4 . T able 7.1 serv es to summarize this corollary and demonstrates that the comm utativ e diagrams for lifts from Section 5.3 satisfy the needed prop erties. Corollary 7.3.2. The motivic c ohomolo gy of BSO 4 withZ 2 -c o ecients is given by H ∗ ,∗ ′ (BSO 4 ;Z 2 )=Z 2 [τ,y 0,2 ,µ (d 2 ),µ (d 3 ),µ (d 4 ),w alg 2 ,w alg 3 ,w alg 4 ,Q 1 w alg 2 ,Q 1 a,Q 0 Q 1 a]/I wher e I is the ide al gener ate d by the fol lowing r elations: 117 τy 0,2 =0 y 0,2 µ (d 3 )=0 y 0,2 w alg 2 =0 y 0,2 w alg 3 =0 y 0,2 w alg 4 =0 y 0,2 Q 1 w alg 2 =0 y 0,2 Q 1 a=0 y 0,2 Q 0 Q 1 a=0 τQ 1 w alg 2 =w alg 2 w alg 3 τQ 1 a=w alg 2 w alg 4 τQ 0 Q 1 a=w alg 3 w alg 4 (w alg 2 ) 2 =τ 2 µ (d 2 ) (w alg 3 ) 2 =τµ (d 3 ) (w alg 4 ) 2 =τ 2 µ (d 4 ) (Q 1 w alg 2 ) 2 =τµ (d 2 )µ (d 3 ) (Q 1 a) 2 =τ 2 µ (d 2 )µ (d 4 ) (Q 0 Q 1 a) 2 =τµ (d 3 )µ (d 4 ) w alg 2 Q 1 w alg 2 =τµ (d 2 )w alg 3 w alg 2 Q 1 a=τµ (d 2 )w alg 4 w alg 2 Q 0 Q 1 a=w alg 4 Q 1 w alg 2 w alg 3 Q 1 w alg 2 =µ (d 3 )w alg 2 w alg 3 Q 1 a=w alg 4 Q 1 w alg 2 w alg 3 Q 0 Q 1 a=µ (d 3 )w alg 4 w alg 4 Q 1 w alg 2 =w alg 4 Q 1 w alg 2 w alg 4 Q 1 a=τµ (d 4 )w alg 2 w alg 4 Q 0 Q 1 a=τµ (d 4 )w alg 3 Q 1 w alg 2 Q 1 a=µ (d 2 )w alg 3 w alg 4 Q 1 w alg 2 Q 0 Q 1 a=µ (d 3 )Q 1 a Q 1 aQ 0 Q 1 a=µ (d 4 )w alg 2 w alg 3 The presen tation from Corollary 7.3.2 is simplied in that it explicitly con tains all three Chern and Stiefel-Whitney classes and it only has 11 generators instead of 16; ho w ev er, it can still b e simplied further as it only partially remo v ed the virtual elemen t a and the Milnor op erations Q i . W e can complete this remo v al with a sligh t abuse of notation that rev olv es around in v ersion of τ . Note that a formal in v erse of τ is not added, merely used as notation: (w alg 2 ) 2 =τ 2 µ (d 2 ) =⇒ denote µ (d 2 )=τ − 2 (w alg 2 ) 2 (w alg 3 ) 2 =τµ (d 3 ) =⇒ denote µ (d 3 )=τ − 1 (w alg 3 ) 2 (w alg 4 ) 2 =τ 2 µ (d 4 ) =⇒ denote µ (d 4 )=τ − 2 (w alg 4 ) 2 w alg 2 w alg 3 =τQ 1 w alg 2 =⇒ denote Q 1 w alg 2 =τ − 1 w alg 2 w alg 3 w alg 2 w alg 4 =τQ 1 a =⇒ denote Q 1 a=τ − 1 w alg 2 w alg 4 w alg 3 w alg 4 =τQ 0 Q 1 a =⇒ denote Q 0 Q 1 a=τ − 1 w alg 3 w alg 4 This c hange in notation has man y adv an tages. First, w e can manipulate these sym b ols in the most naiv e w a y and still ha v e v alid expressions. F or example: µ (d 3 )a=w alg 2 Q 0 Q 1 a=w alg 3 Q 1 a=w alg 4 Q 1 w alg 2 ⇓ µ (d 3 )a=w alg 2 (τ − 1 w alg 3 w alg 4 )=w alg 3 (τ − 1 w alg 2 w alg 4 )=w alg 4 (τ − 1 w alg 2 w alg 3 ) Clearly all three expressions are simply dieren t factorizations of τ − 1 w alg 2 w alg 3 w alg 4 ; this matc hes the state- men t that w alg 2 w alg 3 w alg 4 = τµ (d 3 )a coming from Prop osition 7.3.1. A second adv an tage is that this naiv e treatmen t means that far few er relations m ust b e explicitly dened: an y factorization of the same set of 118 sym b ols will represen t the same class. Finally , it is no w clear what the realization of eac h class m ust b e: t 2 (τ )=1 t 2 (w alg 4 )=w 4 t 2 (y 0,2 )=0 t 2 (w alg 2 )=w 2 t 2 (w alg 3 )=w 3 t 2 (τ − 2 (w alg 2 ) 2 )=w 2 2 t 2 (τ − 1 (w alg 3 ) 2 )=w 2 3 t 2 (τ − 2 (w alg 4 ) 2 )=w 2 4 t 2 (τ − 1 w alg 2 w alg 3 )=w 2 w 3 t 2 (τ − 1 w alg 2 w alg 4 )=w 2 w 4 t 2 (τ − 1 w alg 3 w alg 4 )=w 3 w 4 All together, the ab o v e w ork sho ws that the follo wing theorem giv es an alternate and greatly simplied statemen t of the mo d 2 motivic cohomology of BSO 4 : Theorem 7.3.3. The motivic c ohomolo gy ring H ∗ ,∗ ′ (BSO 4 ;Z 2 ) is given by Z 2 [τ,y 0,2 ,w alg 2 ,w alg 3 ,w alg 4 ,τ − 2 (w alg 2 ) 2 ,τ − 1 (w alg 3 ) 2 ,τ − 2 (w alg 4 ) 2 ,τ − 1 w alg 2 w alg 3 ,τ − 1 w alg 2 w alg 4 ,τ − 1 w alg 3 w alg 4 ]/I wher e de gr e es ar e given by deg(τ )=(0,1) deg(y 0,2 )=(4,2) deg(w alg 2 )=(2,2) deg(w alg 3 )=(3,2) deg(w alg 4 )=(4,3) and I is the ide al gener ate d by the fol lowing r elations: τy 0,2 =0 y 2 0,2 =0 y 0,2 w alg 2 =0 y 0,2 w alg 3 =0 y 0,2 w alg 4 =0 y 0,2 τ − 1 (w alg 3 ) 2 =0 y 0,2 τ − 1 w alg 2 w alg 3 =0 y 0,2 τ − 1 w alg 2 w alg 4 =0 y 0,2 τ − 1 w alg 3 w alg 4 =0 119 MUL TIPLICA TION T ABLE (symmetric but only half displa y ed for clarit y): τ y 0,2 µ (d 2 ) µ (d 3 ) µ (d 4 ) w alg 2 w alg 3 w alg 4 Q 1 w alg 2 Q 1 a Q 0 Q 1 a τ τ 2 0 τµ (d 2 ) τµ (d 3 ) τµ (d 4 ) τw alg 2 τw alg 3 τw alg 4 w alg 2 w alg 3 w alg 2 w alg 4 w alg 3 w alg 4 y 0,2 0 y 0,2 µ (d 2 ) 0 y 0,2 µ (d 4 ) 0 0 0 0 0 0 µ (d 2 ) µ (d 2 ) 2 µ (d 2 )µ (d 3 ) µ (d 2 )µ (d 4 ) µ (d 2 )w alg 2 µ (d 2 )w alg 3 µ (d 2 )w alg 4 µ (d 2 )Q 1 w alg 2 µ (d 2 )Q 1 a µ (d 2 )Q 0 Q 1 a µ (d 3 ) µ (d 3 ) 2 µ (d 3 )µ (d 4 ) µ (d 3 )w alg 2 µ (d 3 )w alg 3 µ (d 3 )w alg 4 µ (d 3 )Q 1 w alg 2 µ (d 3 )Q 1 a µ (d 3 )Q 0 Q 1 a µ (d 4 ) µ (d 4 ) 2 µ (d 4 )w alg 2 µ (d 4 )w alg 3 µ (d 4 )w alg 4 µ (d 4 )Q 1 w alg 2 µ (d 4 )Q 1 a µ (d 4 )Q 0 Q 1 a w alg 2 τ 2 µ (d 2 ) τQ 1 w alg 2 w alg 2 w alg 4 τµ (d 2 )w alg 3 τµ (d 2 )w alg 4 w alg 4 Q 1 w alg 2 w alg 3 τµ (d 3 ) w alg 3 w alg 4 µ (d 3 )w alg 2 w alg 4 Q 1 w alg 2 µ (d 3 )w alg 4 w alg 4 τ 2 µ (d 4 ) w alg 4 Q 1 w alg 2 τµ (d 4 )w alg 2 τµ (d 4 )w alg 3 Q 1 w alg 2 τµ (d 2 )µ (d 3 ) µ (d 2 )w alg 3 w alg 4 µ (d 3 )Q 1 a Q 1 a τ 2 µ (d 2 )µ (d 4 ) µ (d 4 )w alg 2 w alg 3 Q 0 Q 1 a τµ (d 3 )µ (d 4 ) REALIZA TION T ABLE: τ y 0,2 µ (d 2 ) µ (d 3 ) µ (d 4 ) w alg 2 w alg 3 w alg 4 Q 1 w alg 2 Q 1 a Q 0 Q 1 a 1 0 w 2 2 w 2 3 w 2 4 w 2 w 3 w 4 w 2 w 3 w 2 w 4 w 3 w 4 DEGREE T ABLE: τ y 0,2 µ (d 2 ) µ (d 3 ) µ (d 4 ) w alg 2 w alg 3 w alg 4 Q 1 w alg 2 Q 1 a Q 0 Q 1 a (0,1) (4,2) (4,2) (6,3) (8,4) (2,2) (3,2) (4,3) (5,3) (6,4) (7,4) T able 7.1: Simplied m ultiplication, realization, and degree tables for generators of H ∗ ,∗ ′ (BSO 4 ;Z 2 ) as presen ted b y Corollary 7.3.2. Here the realization map is the canonical one H ∗ ,∗ ′ (BSO 4 ;Z 2 )→H ∗ (BSO 4 ;Z 2 ). 120 7.4 Supporting W ork for H ∗ ,∗ ′ (BG 2 ;Z) Theorem 7.4.1. ([Y ag10] 9.6) The mo d 2 motivic c ohomolo gy of BG 2 is given by H ∗ ,∗ ′ (BG 2 ;Z 2 )=Z 2 [µ (c 4 ),µ (c 6 )]⊗ (Z 2 {y 0,2 }⊕ Z 2 [τ ]⊗ (Z 2 [µ (c 7 )]Q(2)− Z 2 {1}){a}) wher e t(µ (c 7 )a)=w 4 w 6 w 7 , t(Q 0 a)=w 6 , t(Q 1 a)=w 6 and t(Q 2 a)=w 4 w 6 . There are man y insigh ts to gain from this that are similar to those for BSO 4 . Breaking do wn this theorem, w e see that H ∗ ,∗ ′ (BG 2 ;Z 2 ) decomp oses in to three pieces: Z 2 [µ (c 4 ),µ (c 6 )]{y 0,2 } Z 2 [τ,µ (c 4 ),µ (c 6 )] Z 2 [τ,µ (c 4 ),µ (c 6 ),µ (c 7 )]⊗ Q(2){a}− Z 2 [τ,µ (c 4 ),µ (c 6 )]{a} Because the restriction maps H ∗ ,∗ ′ (BG 2 ;Z 2 ) → H ∗ ,∗ ′ (BSO 4 ;Z 2 ) and H ∗ (BG 2 ;Z 2 ) → H ∗ (BSO 4 ;Z 2 ) are injectiv e, a class in H ∗ ,∗ ′ (BG 2 ;Z 2 ) is in the k ernel of the realization map H ∗ ,∗ ′ (BG 2 ;Z 2 )→ H ∗ (BG 2 ;Z 2 ) i its restriction image is in the k ernel Z 2 [µ (d 2 ),µ (d 4 )]{y 0,2 } of H ∗ ,∗ ′ (BSO 4 ;Z 2 ) → H ∗ (BSO 4 ;Z 2 ). The univ ersal co ecien t theorem along with argumen ts in [Gui07] sho w that: τ 7→τ y 0,2 7→y 0,2 a7→a µ (c 4 )7→y 0,2 µ (d 2 )+µ (d 2 ) 2 +µ (d 4 ) µ (c 6 )7→y 0,2 µ (d 4 )+µ (d 2 )µ (d 4 )+µ (d 3 ) 2 µ (c 7 )7→µ (d 3 )µ (d 4 ) In short, the only p ossible w a y for an elemen t of H ∗ ,∗ ′ (BG 2 ;Z 2 ) to b e in the k ernel is for it to include y 0,2 : Lemma 7.4.2. The kernel of the r e alization map H ∗ ,∗ ′ (BG 2 ;Z 2 )→H ∗ (BG 2 ;Z 2 ) isZ 2 [µ (c 4 ),µ (c 6 )]{y 0,2 }. W e can rewrite H ∗ ,∗ ′ (BG 2 ;Z 2 ) in a dieren t presen tation, just as w e did for H ∗ ,∗ ′ (BSO 4 ;Z 2 ). An analog of Prop osition 7.3.1 holds here, meaning that t w o classes in H ∗ ,∗ ′ (BG 2 ;Z 2 )− Z 2 [µ (c 4 ),µ (c 6 )]{y 0,2 } are equal i their degrees and realizations agree. As suc h, w e can mak e the follo wing adjustmen ts to notation: w alg 4 =Q 0 a w alg 6 =Q 1 a w alg 7 =Q 0 Q 1 a τ − 2 (w alg 4 ) 2 =µ (c 4 ) τ − 2 (w alg 6 ) 2 =µ (c 6 ) τ − 1 (w alg 7 ) 2 =µ (c 7 ) τ − 1 w alg 4 w alg 6 =Q 2 a τ − 1 w alg 4 w alg 7 =Q 0 Q 2 a τ − 1 w alg 6 w alg 7 =Q 1 Q 2 a 121 Theorem 7.4.3. The motivic c ohomolo gy of BG 2 withZ 2 -c o ecients is given by Z 2 [τ,y 0,2 ,w alg 4 ,w alg 6 ,w alg 7 ,τ − 2 (w alg 4 ) 2 ,τ − 2 (w alg 6 ) 2 ,τ − 1 (w alg 7 ) 2 ,τ − 1 w alg 4 w alg 6 ,τ − 1 w alg 4 w alg 7 ,τ − 1 w alg 6 w alg 7 ]/I wher e de gr e es ar e given by deg(τ )=(0,1) deg(y 0,2 )=(4,2) deg(w alg 4 )=(4,3) deg(w alg 6 )=(6,4) deg(w alg 7 )=(7,4) and I is the ide al gener ate d by the fol lowing r elations: τy 0,2 =0 y 0,2 w alg 4 =0 y 0,2 w alg 6 =0 y 0,2 w alg 7 =0 y 0,2 τ − 1 (w alg 7 ) 2 =0 y 0,2 τ − 1 w alg 4 w alg 6 =0 y 0,2 τ − 1 w alg 4 w alg 7 =0 y 0,2 τ − 1 w alg 6 w alg 7 =0 7.5 Code W ritten for Thesis The follo wing co de p osted has b een p osted on [Gith ub] and w as written b y me to test and v erify results in this pap er ([Gith ub]): JULIA CODE: SL_2-test.jl demonstrates that randomly constructed matrices of the form in Corollary 2.2.5 are in G 2 and b eha v e as exp ected, in particular that it xes the subalgebra of split-quaternions (see Subsection 2.2.2) SL_3-test.jl demonstrates that randomly constructed matrices of the form in Lemma 2.2.4 are in G 2 and b eha v e as exp ected, in particular that it preserv es the subalgebra of split-complex n um b ers (see Subsection 2.2.2) nd_steenro d_p olynomial_for_thesis_result.jl searc hes for p olynomials satisfying relations re- lated to reduced p o w er op erations. The parameters for computation are input on lines 329 and 330. The arra y non_zero_generator_ags can generally tak e en tries in {0,1,2}. F or prime_to_use = 2, the i th en try b eing zero means that Sq i of the p olynomial m ust b e 0; a v alue of one means the en try m ust b e non-zero; a v alue of t w o means it can b e either. In this case, [0 0 0 1 0 1] means that Sq i m ust b e non-zero for i = 4,6 but also automatically for i = 0,7 b y prop erties of the Steenro d homomorphism (see Theorem 5.5.2) zorn-relations.jl prin ts the m ultiplication tables of the rank 1,2, and3 forms of Zorn’s construction, i.e. split-complex n um b ers, Zorn’s quaternions and Zorn’s o ctonions. It also prin ts and sorts all the relev an t relations coming from analogs of Theorem 2.2.3 (see Subsection 2.2.2) 122 MA TLAB CODE: classical_BSO4_nd_generators.m nds generators of H ∗ (BSO 4 ;Z) (see Section 5.3) Field_BSO4_nd_generators.m nds generators of CH ∗ (BSO 4 ) (see Section 5.3) higher_c ho w_to_motivic_pairing.m is used to visualize the isomorphism from higher Cho w groups to motivic cohomology groups (see Figure 4.1) motivic_to_higher_c ho w_pairing.m is used to visualize the isomorphism from motivic cohomol- ogy groups to higher Cho w groups (see Figure 4.1) regular_to_split_SOnC_isomorphism_searc h.m uses gradien t descen t to nd solutions to the system J =g T g needed for the isomorphism from SO ′ n to SO n (see Sections 5.2 and 7.2) Y agita_BSO4_nd_generators.m nds generators of H ∗ ,∗ ′ (BSO 4 ;Z 2 ) (see Section 5.3) 123

Abstract (if available)

Abstract Motivic cohomology is a powerful tool in algebraic geometry that is notoriously difficult to use but so universal that it's worth the effort. Its associated realization maps directly give important information about the relations between different cohomological invariants of schemes and their classifying spaces. The problem of computing general cohomological invariants of classifying spaces is ongoing. In [Tot99], Totaro constructed the Chow ring of a classifying space in general and used this to study symmetric groups. In [Gui07], Guillot did the same examination for the Lie groups G₂ and Spin(7). More recently in [Tot17], Totaro described the computation of the Hodge and de Rham cohomology groups of classifying stacks. ❧ The classifying space BG₂ is derived from the simplest exceptional Lie group and thus makes for a good starting point for testing computational methods. This thesis covers the computation of the motivic cohomology of BG₂ with integral coefficients. The primary approach is a restriction/transfer map statement inspired by the previous works of [Gui07], [Vez00] and [Yag10]. Yagita conjectured the mod-2 motivic cohomology based on the existence of a motivic transfer map involving G₂ and its maximal torus SO₄. Vezzosi's method for construction of a transfer is adaptable to the motivic setting due to the fact that Bloch's higher Chow groups are a model for motivic cohomology which directly extend Guillot's work. This work also covers the derivation of a polynomial relating the generators of the singular mod-2 cohomology of BG₂ through Steenrod squares.

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Creator Port, Alexander Maxwell (author)

Core Title Motivic cohomology of BG₂

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Degree Conferral Date 2021-12

Publication Date 09/29/2021

Defense Date 08/27/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag BG₂,injective restriction,motivic cohomology,OAI-PMH Harvest,principal bundle,SO₄,Steenrod polynomial

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Advisor Asok, Aravind (committee chair), Bonahon, Francis (committee member), Friedlander, Eric (committee member), Guralnick, Robert (committee member), Jonckheere, Edmond (committee member)

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