University of Southern California Dissertations and Theses

On the index of elliptic operators

(USC Thesis Other)

On the index of elliptic operators

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content ON THE INDEX OF ELLIPTIC OPERATORS by Fan Yang A Thesis Presented to the FACULTY OF THE USC DORNSIFE UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF ARTS (MATHEMATICS) May 2021 Copyright 2021 Fan Yang Acknowledgements I would like to thank my advisor Professor Aravind Asok. I would like to thank my committee members Professor Francis Bonahon and Professor Sheel Ganatra. I would like to thank my family. Finally, I would like to thank all of my friends. ii Table of Contents Acknowledgements ii Abstract iv Chapter 1: Introduction 1 Chapter 2: Vector bundles and K-theory 2 2.1 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Background of K-theory: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Equivalence of two definitions of K-theory with compact support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Properties of K c () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Pushforward with compact support . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 3: Differential Operators 10 3.1 Differential operators : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Pseudodifferential Operators and Elliptic Operators . . . . . . . . . . . . . . . . . 11 3.3 Dirac Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 4: Analytic and Topological Index 16 4.1 Analytic Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Topological Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Chapter 5: Atiyah-Singer Index Theorem 18 Chapter 6: Proof of The Theorem 20 6.1 Overview of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.2 Proof of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 References 30 iii Abstract The Atiyah-Singer Index Theorem plays an important role in connecting elliptic differenital oper- ators over a compact manifold with some topological data. The topological data is defined to be the analytic index. The theorem states that the analytic index of the elliptic operator is equal to the topological index. Here we want to use the K-theory approach to state the theorem and overview the proof given in [1]. Also we prove some basic theorems in K-theory with compact support. iv Chapter 1 Introduction The Atiyah-Singer index theorem is a relatively old theorem, but it has a lot of applications and they are all useful in today’s math research. There are two versions of the index theorem. One is concluded in this paper. This is the K-theory version. The main theorem and its proof is established mainly based on the K-theory. However, there is another version and is concluded in [1]. This version is a relatively well known version given by Todd classes and Chern classes. The basic goal to write this paper is to learn some applications of topological K-theory. Also, we are interested in the idea of the proof of Atiyah-Singer Index Theorem. The whole paper is helpful to have a good review of K-theory and K-theory of compact support. Additionally, the proof process is also reviewed in this paper It will be a good review of as- sociate vector bundles and understand principal symbols of elliptic operators over compact mani- folds. 1 Chapter 2 Vector bundles and K-theory 2.1 Vector Bundles To talk about the K-theory, we need to establish the background of vector bundles first. Definition 2.1.1. Assume X is a topological space, then a vector bundle (complex) over X is a topological space E with: 1. A continuous surjection mapp : E! X; 2. A finite dimensional vector space on each fiber, i.e. E x =p 1 (x) for x2 X; 3. E is locally trivial, in particular, for any x2 X there is a neighborhood U of x such that there is a homeomorphismf : UC n !p 1 (U) for some n, satisfying for all p2 U;v2C n , the map v!f(p;v) is a linear isomorphism betweenC n andp 1 (p). Similarly for real vector bundle. Remark: If we do not specialize the term ”real vector bundles” in this paper, we will focus on complex vector bundles. Now we need to define the bundle map or bundle homomorphism: Definition 2.1.2. A bundle map from p : E! X to f : F! X is a continuous map f : E! F, satisfying: 1. ff = p; 2. For any x2 X,f : E x ! F x is a linear map between vector space. 2 Note that vector bundles over X together with bundle maps actually define a category of vector bundles, denoted as Vect(X). Now we want to recall what equivariant spaces and equivariant vector bundles are under the action of a compact Lie group G. First we should recall the group action: Definition 2.1.3. A (left) group G with identity e acts on a set S is a function: f : G S! S which maps(g;s) to g s satisfying 1. For g;h2 G, g(h s)=(gh) s. 2. e s= s. Similarly for right group action. Definition 2.1.4. Assume X is topological space, we say X is equipped with compact topological group action of G if G X! X,(g;x)! g x the map is continuous. Remark: Equivalently, we say there is a mapf : G! Homeo(X), which maps g2 G to a home- omorphism of X,f(g). And the map G X! X,(g;x)!f(g) x is a continuous map. Definition 2.1.5. Assume E! X is a vector bundle, then an equivariant vector bundle under the compact topological group action of G is a pair given by a vector bundle E! X and a lifting of group action G X! X to G E! E so that the projection map E! X is equivariant and for each g2 G, x2 X the map E x ! E gx is a linear map. Note: A bundle map between G-vector bundles is a bundle map compatible with G action. Sim- ilarly, the G-vector bundle over G-space X together with bundle maps also defines a category of G-vector bundles, denoted as Vect G (X). Remark: X is a compact topological space equipped with a group action of a compact topological group G, then we set Vect G (X) to be the set of all isomorphism class of equivariant vector bundle over X, and(Vect G (X);) gives us a structure of abelian semi group. Now we focus our attention on complex vector bundles in the remainder if without specific notation. 3 2.2 Background of K-theory: Here we use the usual definition of topological K-theory. Just a quick recall: Let F(Vect G (X)) be the free abelian group generated by the element of Vect G (X). Let E(Vect G (X)) be the subgroup of F(Vect G (X)) generated by the element with the form a+ a 0 (a a 0 ) where+ denote the addition in F(Vect G (X)) and denote the addition in Vect G (X). Definition 2.2.1. The K-theory is defined to be the quotient group K G (X)= F(Vect G (X))=E(Vect G (X)). Remark: If G is just trivial group, we can get K-theory as usual, that is K(X)= F(Vect(X))=E(Vect(X)). Then we can introduce what the reduced K-theory is: Definition 2.2.2. Fix Y be a pointed compact topological space; then the reduced the K-theory of this Y is just the kernel of the map f : K(Y)! K(pt). 2.3 Equivalence of two definitions of K-theory with compact support Now we want to introduce the definition of K-theory with compact support. Definition 2.3.1. (First Definition) If X is a paracompact Hausdorff space, then the K-theory with compact support of X is defined to be the reduced K-theory of its one-point compactification, i.e., K c (X)= ˜ K(X + ). Here X + is the one-point compactification of X. To get our second definition, we may define the “L-type” first. Construction of “L-Type”: Assume X is a paracompact space. For each integer n 1, consider the element(V 0 ;V 1 ;:::; V n ;s 1 ;s 2 ;:::;s n ) where V 0 ;V 1 ;:::;V n are vector bundles over X and where the sequence 0! V 0 s 1 ! V 1 s 2 !::: s n ! V n ! 0 is a sequence of bundle maps. And we say this element is a V-element if the bundle map sequence is exact on the complement of some compact set. We say two V- elements V=(V 0 ;V 1 ;:::;V n ;s 1 ;s 2 ;:::;s n ) and V 0 =(V 0 0 ;V 0 1 ;:::;V 0 n ;s 0 1 ;s 0 2 ;:::;s 0 n ) are isomorphic 4 if there are bundle isomorphisms f i : V i ! V 0 i over X such that the diagram: V i1 V i V 0 i1 V 0 i s i f i1 f i s 0 i commutes. Also we say a V-element V=(V 0 ;V 1 ;:::;V n ;s 1 ;s 2 ;:::;s n ) is elementary if there is an i such that: 1. V i = V i1 and s i = id 2. V j = 0 f or j6= i or i 1. Also we can define the operation for two V-element, V-elements V=(V 0 ;V 1 ;:::;V n ;s 1 ;s 2 ;:::;s n ) and V 0 =(V 0 0 ;V 0 1 ;:::;V 0 m ;s 0 1 ;s 0 2 ;:::;s 0 m ). In particular, without loss of generality we may assume m n then VV 0 =(V 0 V 0 0 ;V 1 V 0 1 ;:::;V m V 0 m ;V m+1 ;:::;V n ;s 1 s 0 1 ;:::;s m s 0 m ;s m+1 ;:::;s n ). Then V=(V 0 ;V 1 ;:::;V n ;s 1 ;s 2 ;:::;s n ) and V 0 =(V 0 0 ;V 0 1 ;:::;V 0 n ;s 0 1 ;s 0 2 ;:::;s 0 n ) are said to be equivalent if there are elementary element E 1 ;:::E m ;F 1 ;:::;F k such that V E 1 ::: E m = V 0 F 1 ::: F k . This is indeed an equivalence relation. Reflexivity: V = V given by definition. Symmetry and transitivity is given by the same properties of isomorphism. Definition 2.3.2. We denote the set of all V-elements as L n (X) c , also we denote the set of all equivalence classes[V 0 ;V 1 ;:::;V n ;s 1 ;s 2 ;:::;s n ] as L n (X) c . Then we introduce an important lemma to understand the group structure of L n (X) c . Lemma 2.3.1. If X is a paracompact, Hausdorff, and has finite covering dimension, then for any vector bundle E over X, there exists a vector bundle E 0 such that E E 0 is trivial. Proof. Since X is paracompact we can choose a locally finite open covering, sayfU i g i2I such that Ej U i for any i is trivial. Since X also has a finite covering dimension, without loss of gen- erality, we may assume k. Then there is a refinement offU i g sayfV j g j2J , which satisfying V j 1 \V j 2 ;:::;V j k+1 = / 0. Also, since for any j2 J there is a i2 I such that V j U i , Ej V j is trivial. Now we can choose an open coveringfW j g j2J such that ¯ W j V j . Letl J : X!R be a continuous func- tion that takes value 1 on ¯ W j and the value 0 outside V j . Then for each non-empty finite subset S of 5 J we can form a set M(S) X, such that for any x2 U(S), we have Min j2S l J (x)> Max j= 2S l J (x). Let M n be the union of sets M(S) for which S has n elements. Then since V j 1 \V j 2 :::V j k+1 = / 0 and W j V j , we have the maximal value of n is just k. So we have X = M 1 [ M 2 [:::M n . Now for any given x2 X, if we have n positivel J (x), then x2 M n . If j2 S, we have M(S) V j . This implies thatfM 1 :::M k g is locally finite. And since Ej V j is trivial, Ej M(S) is also trivial. This implies Ej M n are also trivial. In particular, there is a local trivialization f i : M i C n ! Ej M i for i2 I is finite. Since X is manifold hence paracompact, there is a partition of unity f i with supp( f i ) M i . Then we can define y i : Ej M i ! M i C n , such that for each v2 Ej M i , x2 M i , y i (v)= f i (x)f 1 i (v). Now by extending by zero y i is a map E! M i C n . Then the direct sum of y i gives us the desire homomorphism, i.e L i y i : E! X( L i C njIj ). The injection is given by the non-vanishing of f i . Then by [2, Theorem 1.3],“if F! C is a vector bundle over a paracompact C, and F 0 F is a subbundle then their exsits a vector bundle F 1 such that F 0 F 1 = F ”, we have a E 0 such that E E 0 is trivial bundle. Now for a fixed n, the group structure of (L n (X) c ;;I;V 1 ) is obvious, here I is the identity element which stands for the equivalence class of elementary elements in L n (X) c and the binary operation is just the direct sum. Indeed, associativity is given by the associativity of. Since I is defined to be the equivalence class of elementary elements, by our definition of L n (X) c , I V= V I = V. By our lemma, each element V in L n (X) c has an inverse as a direct summand of the trivial bundle which is V 1 . Definition 2.3.3. (Second Definition) If X is a paracompact Hausdorff space, then elements[V 0 ;V 1 ;s 1 ]2 L 1 (X) c forms a group K 0 c (X). Note that since we have natrual isomorphism L 1 (X) c = ! L 2 (X) c = !:::L n (X) c , we can just use L 1 (X) c to define the K-theory here. Construction: Assume X is a manifold, then we construct a map as follows: Let us denote the one point compactification of X as X + . Let ˆ E and ˆ F are vector bundles over X + and ˆ Ej X = E, ˆ Fj X = F 6 and FF 0 = n, similarly ˆ F ˆ F 0 = n then the map is given as follows: f([ ˆ E][ ˆ F])=[EF 0 ;n;s] and the inverse is given byf 1 [(E;F;s)]=[ ˆ E ˆ F 0 ]. Theorem 2.3.2. The mapf : K c (X) = ! K 0 c (X) is an isomorphism. Proof. The group homomorphism is easy to see since both groups have sum as binary operation. First of all, if we suppose ˆ E and ˆ F are vector bundles over X + , every element in e K(X + ) can be expressed as [ ˆ E][ ˆ F]. Since X + is compact, there are ˆ E 0 and ˆ F 0 such that ˆ E 0 ˆ E = ˆ F 0 ˆ F = n by [2, Propostition 1.4], ”For each vector bundle M! B with B a compact Housdorff, there is a vector bundle M’ such that MM 0 is trvial.” . Here ”n” stands for trivial bundle. Then[ ˆ E][ ˆ F]= [ ˆ E]+[ ˆ F 0 ][ ˆ F 0 ][ ˆ F]=[ ˆ E]+[ ˆ F 0 ][n]. Since ˆ E and ˆ F 0 are all vector bundles, ˆ E ˆ F 0 j fptg = nj fptg which implies that there is a compact set A such that ˆ E ˆ F 0 j (XA) = nj (XA) . This is the map that we want. The map is injection, since for the same triple they must have the same vector bundle class over X + , hence the preimage is the same. For the inverse direction, we want to show this map is surjection. Assume A is a compact set in X, X + is one-point compactification of X and E, F are vector bundle over X, a : E! F is isomorphism outside A. By our Lemma 2.3.1 there is a E’ on X such that E E 0 = n on X. Now we can replace thr triple (E;F;a) with (E E 0 ;F E 0 ;a Id), and we can get F E 0 j (XA) = E E 0 j (XA) = n. Which impLies that F E 0 and E E 0 can be extended to X + , say ˆ F and ˆ E. Then [ ˆ E][ ˆ F] is an element in K(X + ). Playing the same trick we can get that [ ˆ E]+[ ˆ F 0 ][n] is in K(X + ). By the definition of reduced K-theory e K(X + )= ker(K(X + )! K(pt)) over X + . Then by the isomorphism K(X + ) = ! e K(X + )Z which maps[G][n] to([G];rank x (G) n) we get our desire element in K c (X). So the map we construct is actually an isomorphism. 2.4 Properties of K c () Proper Homotopy Invariance Note that by our second definition, the K-theory with compact support can be viewed as a set of bundle maps that are isomorphisms outside a compact set. Then the proper homotopic maps between two manifold will give us the desired invariance. Recall, a 7 map is said to be proper, if the preimage of a compact set is compact, and a proper homotopy is a homotopy that is proper. This can be seen directly from the proper homotopy invariance of compactly supported cohomology. For detail proof can see [3, Proposition 6.6]. Functoriality Note that we can also view K c () as a functor, which maps the category of paracom- pact space to the category of groups, i.e. for a paracompact space X K c () : Vect(X)! K c (X)2 Grp where Grp is the category of groups. 2.5 Pushforward with compact support Here we want to introduce the pushforward maps of K-theory with compact support. First of all, let us consider two differentiable manifold X, Y with codimension k and a proper embedding f : X ,! Y . Then here we want to use Thom isomorphism as a tool. We introduce a relevant Thom isomorphism in K-theory. Theorem 2.5.1. Suppose p : E! X is a complex vector bundle with dimention k over a dif- ferentiable manifold X and the map i : X! E is the inclusion as the zero section. Then the map i ! : K c (X)! K c (E) is a Thom isomorphism of the form: i ! =L 1 p u, and where L 1 = [p L even C E;p L odd C E;s]. And heres is defined at each non-zero vector e in E bys e = e^(e )x. Recall: L (E) stands for exterior algebra and for a vector v2C n , vx:L p C C n !L p1 C C n is given by vx(v 1 ^ v 2 ^:::^V p )=å p i=1 (1) i+1 < v i ;v> v 1 ^:::^ ˆ v i ^:::^ v p and ˆ v i means deletion. The proof of this theorem can be found in [1, Appendix C Theorem C.8]. Now we have tools to construct the pushforward map. Assume X, Y are differentiable manifold, and f : X ,! Y is an differentiable proper embedding, and N is a normal bundle over X. By tubular neighborhood theorem, we have a commutative diagram: X Y N f 8 X is the 0-section in N and the map from N to Y is a proper embedding. Then consider the compostion of maps K c (X)! K c (N)! K c (Y), we get our desire map f ! : K c (X) ,! K c (Y). Note that the first map is given by the theorem and the second map is given by the diffeomorphism between a open neighborhood of X in Y and a convex open neighborhood of X in N. 9 Chapter 3 Differential Operators 3.1 Differential operators : Now we are in the stage to introduce differential operators. First of all, let us introduce the differential operator locally. Definition 3.1.1. Assume X is a differentiable manifold, then in local coordinates (x 1 ;:::;x n ) on X we can define the differential operators ¶ jaj ¶x a = ¶ jaj ¶x a 1 1 ¶x a 2 2 :::¶x a n n . Here for non-negative n numbers a =(a 1 ;:::;a n ), we definejaj=å k a k . Now we can define the differential operator over a differential manifold X. First of all, here is a notation, assume E is a vector bundle over X. We denote the space of smooth sections of E by G ¥ (E). For a differentiable manifold X, we define differential operators of order at most k inductively. First of all, we define 0-order differential operators. For arbitrary differentiable functions f and g, we denote d f (g)= f g. Definition 3.1.2. A differential operator D is said to have order 0 if[D;d f ]= 0, where[;] is the Lie bracket or commutator. Now we can define higher order differential operators. Definition 3.1.3. The differential operator at most of order k is an operator D such that[D;d f ] is a differential operator of order at most k 1 for any differentiable functions f. 10 Remark: This is well-defined since the Lie bracket of two vector fields is again a vector field. Based on this definition we can introduce our first very important definition principal symbol. Definition 3.1.4. For a manifold X, P is a differential operator, the sections(P)2G(( m T X) Hom(E;F)) is the principal symbol of the differential operator P , where stands for the symmetric tensor product. Note that since for any vector space, say V , the space n V is isomorphic to the space of homogeneous polynomial functions of degree n on V . Then if we consider the mapp : T X! X, the principal symbol gives us a vector bundle map p E!p F. This implies for each e2 T x X, the principal symbol gives a map s e (P) : E x ! F x ,since (p E) e s e (D) ! (p F) e and (p E) e = E x ;(p F) e = F x . Here we introduce an important property for principal symbol. Proposition 3.1.1. [1, Chapter 3, Propostion 1.7] Assume E, F , G are complex vector bundles over X. Let P :G(E)!G(F), ˜ P :G(E)!G(F) and ¯ P :G(F)!G(G) be differential operator over X and P; ˜ P share the same order. Then for any element e2 T X and for all a;b2R, we have: s e (aP+ b ˜ P)= as e (P)+ bs e ( ˜ P) ands e ( ¯ P ˜ P)=s e ( ¯ P)s e ( ˜ P). The proof can be found in [1, Chapter 3, Proposition 1.7]. 3.2 Pseudodifferential Operators and Elliptic Operators Now we are going to define a kind of special differential operator called seudodifferential Operator. First of all, we need to define symbols of order m. Definition 3.2.1. [1, Chapter 3, Definition 3.1] For m2R, a smooth function p(x;e) onR n R n is said to be a symbol of order m if for each a and b there is a constant number C ab such that jD a x D b e p(x;e)j C ab (1+jej) mjbj for all x and e. And we denote this space as Sym m . Based on this definition we are able to define Pseudodifferential operator. Construction: Let u2 C ¥ (R n ) then the Inverse Fourier Formula gives us 11 u(x)=(2p) ( n 2 ) R e i<x;e> ˆ u(e)de . For a differential operator P onR n with the form P=å A a(x) D a , Pu(x)=(2p) ( n 2 ) R e i<x;e> ˆ u(e)p(x;e)de where p(x;e) is the total symbol of P. Now if we replace p with a more general definition we can define the pseudodifferential operators. Lemma 3.2.1. For each p2 Sym m this Formula defines a linear operator P :S!S . If we also have p is compact x-supported, then P has a continuous extension P : L 2 s+m ! L 2 s for all s. Recall: HereS is Schwartz space given byS =fu2 C ¥ (R n ) : for anya, k, there is a number C a;k such thatjD a u(x)jC a;k (1+jxj) k onR n g, and L 2 i is Sobolev space. Also we have D a = i jaj¶ jaj ¶x a . The lemma’s proof can be found in [1, Chapter 3, Propostion 3.2]. Definition 3.2.2. We say an operator P is an pseudodifferential operator of order m onR n if it satisfies this lemma. The space of all such operators is denoted asYDO m . Now what we want to do is to apply pseudodifferential operator to vector bundles and mani- folds, say differentiable complex vector bundle E, F over a compact manifold X. Consider a new definition which is similar to how we define a differential operator. Recall: A linear map D, D :G(E)!G(F) is infinitely smoothing if it extends to a bounded linear map D : L 2 s (E)! L 2 s+m (F) for all s;m2R. Definition 3.2.3. [1, Chapter 3, Definition 3.15] For a linear map say P , P :G(E)!G(F) is called a pseudodifferential operator of order m if after modulo infintely smoothing operators, P= å P a is a finite sum where any P a can be expressed as in some local coordinates system x a : U a !R n and differentiable bundle trivialization as a pseudodifferential operator of order m with compact support. The linear space of all such things is denoted asYDO m (E;F). And for P;Q2 YDO m (E;F) and P Q if P Q is infinitely smoothing. Based on this definition we can define elliptic operators that are useful to define analytic index. 12 Definition 3.2.4. A pseudodifferential operator D with symbol p(x;e) is an elliptic operator if it satisfying: 1. There is a constant c such that for alljej c, p(x;e) is invertible. 2.jp(x;e)j c(1+jej) m . Similarly, we say D is an elliptic operator over a compact manifold, if D is elliptic for all local chart. 3.3 Dirac Operators We also want to review some basic information about Dirac operators and the relationship between Dirac operators and Dirac bundles. Firstly, we introduce associated bundles. Construction: We assume X is a riemannian manifold, V is a principal G-bundle over X i.e., p : V! X. Also, we assume F is another arbitrary space, and now we denote Homeo(F) as the homeomorphism group of F. (We equip Homeo(F) with suitable topology.) Now we can consider the group homomorphismf : G! Homeo(F) and construct a fiber bundle over X as follows: Consider the mapy : GVF! VF, which mapsy(g;v; f)=(vg 1 ;f(g) f) where g2 G, (v; f)2 V F. Now denote V f F to be the quotient space under G action. We get our desire fiber bundle p f : V f F! X. Note that here V f F is a fiber bundle over X with fibers F. Definition 3.3.1. The fiber bundle p f : V f F! X is the bundle associated to V byf. Based on this definition, we may define the Clifford bundles, which will help us define the Dirac operators over a compact manifold. Let us start with two special bundles. Assume E is an oriented real vector bundle over X and equipped with a riemannian structure. Now consider the principal O n -bundle and the fiber at each point is the set of orthonormal bases of E x . We denote this bundle as P O (E). Similarly we can define P SO (E). 13 Then note that each action of O n onR n will induce an action on Cl(R n ), and this implies that we can consider the map cl n : SO n ! Aut(Cl(R n )), since SO n is a subgroup of O n . And the map cl n is the map that we described when we define the associated bundles. Remark: Here the Cl(R n ) is the Clifford algebra ofR n . The detail can be checked in [1, Chapter 1]. Recall: Here we want to quickly recall a property of Cl(R n ). There is a canonical map w : Cl(R n )! Cl(R n ). This is the automorphism extending theZ 2 map with w 2 = 1. Now we can de- compose the Cl(R n ) into two parts, i.e. Cl(R n )= Cl 0 (R n )Cl 1 (R n ), where Cl 0 (R n ) =L even (R n ) and Cl 1 (R n ) =L odd (R n ). More detail can find in [1, Chapter 2]. Definition 3.3.2. The Clifford bundle of the oriented riemannian vector bundle E is defined as: Cl(E)= P SO (E) cl n Cl(R n ) where cl n is the map defined above. We can introduce the Dirac operator now. Assume X is a riemannian manifold together with Clifford bundle Cl(X). Also we construct a kind of bundles, say L, which is a vector bundle over X and for each x2 X, L x is a left module over Cl(X) x . Note that since L x is a vector space hence a abelian group, together with Cl(X) x we are able to form the left module. Additionally, we assume L is riemannian and equip with a remannian connection. Recall: Every Riemannian manifold is equipped with a connection called Levi-Civita connection by Foundamental theorem of Riemannian geometry [4]. A connectionO is a Levi-Civita connec- tion if for a Riemannian manifold(Y;g), and any vector fields A, B over it,O preserves the metric g andO A BO B A=[A;B] where[ ; ] is the Lie bracket. Definition 3.3.3. The first-order differential operator D :G(L)!G(L) is called Dirac operator of L if it satisfies: for a section l of L, D(l)=å e i O e i l at each x2 X, where e i is an orthonormal basis of T x (X), and whereO means the covariant derivative given by the connection, and Clifford multiplication gives. Actually, each Dirac operator is an elliptic operator in euclidean space, which is a direct result from [1, Chapter 2, Lemma 5.1]. 14 Similarly, we can also define Dirac bundles. Definition 3.3.4. A Dirac structure d over a Clifford bundle L is a pair (g;O) where g is the Riemannian metric andO is the Levi-Civita connection satisfying: 1. For all l 1 ; l 2 2 L x and all unit vectors e2 T x (X). < el 1 ;el 2 >=< l 1 ;l 2 >. 2. For alla2G(Cl(X)) and all l2G(L), we haveO(a l)=(Oa) l+a(Ol). Then a Clifford bundle is just the pair(L;d) where L is a Clifford bundle andd is a dirac structure. Note that since Dirac operators are always elliptic in Euclidean space, if we consider its an- alytic index which will be discussed in chapter 4 and chapter 5 in this paper, we can establish a relationship between the index of Dirac operators over a compact space Z and the K-theory of Z. Actually, the index of the Dirac operator gives us an element in K-theory, and this is easier to be understood by alternative definition of K-theory [5, Chapter 2, Section 6]. For detail see [1, Chapter 3, Lemma 8.4]. 15 Chapter 4 Analytic and Topological Index For a compact differentiable manifold X and an elliptic operator D, consider two maps: f : X ,!R n and q : TR n ! pt. 4.1 Analytic Index Firstly, we need to note that an elliptic operator over a compact manifold is Fredholm, hence it has finite dimensional kernel and cokernel. Definition 4.1.1. For D :G(E)!G(F) an elliptic operator. Then we can define the analytic index of D: ind ana (D)= dim(kerD) dim(coker(D)). 4.2 Topological Index Let X be a compact differentiable manifold of dimension n. Now consider the embedding f : X ,!R N into some euclidean space, and consider the induced pushforward map f ! : K c (T X)! K c (TR N ). Also consider the map q : pt ,!R N which also induced a pushforward map q ! : K c (TR N )! K(pt). Now for an elliptic operator D : G(E)! G(F), where E and F are dif- ferentiable complex vector bundles over X, the principal symbol gives us a class in K c (T X) as F(D)=[p E;p F;s(D)]2 K c (T X), wherep : T X! X. Definition 4.2.1. The topological index of D is defined as an integer: ind top (D) q ! f ! F(D). 16 Claim: This definition is independent of f . For a g= i f where i :R N ,!R M is an linear inclusion. So i also induced a shriek map i ! : K c (TR N )! K c (TR M ) which implies that i ! is also a Thom isomorphism by bott periodicity for the bundleC M !C N . Then for p : pt ,! TR M , we have p ! g ! = q ! f ! since p ! can also be viewed as a Thom isomorphism. Now for two embeddings f 0 : X ,!R N 0 and f 1 : X ,!R N 1 , we can consider j 0 f 0 : X ,!R N 0 +N 1 and j 1 f 1 : X ,!R N 0 +N 1 . Here j 0 and j 1 are two embeddings. Let M = N 0 +N 1 we can construct a homotopy by F t = t j 1 f 1 +(1t) j 0 f 0 . Then by the proper homotopy invariant of K c we are done since embeddings are always proper. 17 Chapter 5 Atiyah-Singer Index Theorem Now we are in the stage to introduce the Atiyah-Singer theorem. First of all we should know that the Atiyah-Singer index theorem is a bridge to connect our analytic index and topological index. It gives us a way to consider the problem in both direction. The idea goes as follows: Let us go from analytic index firstly. For a differential operator D, when we apply it to a manifold, we can consider its principal symbol. However, if the principal symbol is special enough, in particular if it is invertible, it is an elliptic operator. Then if now the manifold is compact, we can find the value dim(ker(D)) dim(coker(D)) is invariant. Now we have a new problem, can we interpret this value in a topological way? The answer is yes by Atiyah-Singer Index Theorem. From the topological way, we will focus on K-theory with compact support and interpret the principal symbol as an element in K-theory. Let us suppose now D is an elliptic operator, s(D) is its principal symbol. By Chapter 2 of this paper we can see that the [s(D)] can be viewed as element in K c (T X) since it is invertible outside 0-section. So now if we want to construct a map, say f : K c (T X)!Z which maps [s(D)] to its analytic index of D. To construct the map f, we need to use Thom isomorphism as a tool. Basically, it tells us that K c (X) = K +k c (V) where V is a k dimensional vector bundle over X. Now consider the k dimensional normal bundle N! X and f : X ,! Y embedding X in Y . Then consider the chain f ! : K c (X)! K c (N)! K c (Y). Here the first arrow comes from Thom isomorphism and the second arrow comes from Tubular 18 neighborhood theorem. Note that the map f induced a map T f : T X ,! TY which implies that we can consider the map T f ! : K c (T X)! K c (TY). Now let Y be theR n so what we have now is a chain X f , !R n j !fptg. Then consider the correspondence chain of K-theory with compact support i.e. K c (T X) T f ! ! K c (TR n ) T j ! ! K c (T pt) =Z, we get our desire mapf. Theorem. (The Atiyah-Singer Index Theorem K-Theory version) For any elliptic operator P over a compact manifold, we have ind ana (P)= ind top (P). That is, the topological index and the analytic index are equal. 19 Chapter 6 Proof of The Theorem 6.1 Overview of the proof Firstly, we want to find an homomorphism ind : K c (T X)! K(pt) =Z which for a fixed n maps a class, say[a]2 K c (T X) to an integer ind ana (D) where D is an elliptic operator, D2YCO n (E;F), whose asymptotic principal symbol is a and the maps satisfying two properties “identity property” and “functoriality”, where YCO n (E;F) is the set of classical operators. Also, the first property, “identity property“, intuitively is that if the whole space X is just a point, then the homomorphism definded at the beginning should be the identity map. The second property, functoriality, is in particular we want the following diagram commute. K c (T X) K c (TY) K(pt) T f ! ind X ind Y Then we consider the embedding X q , ! S N and the map S N f , ! pt, and they induce two pushforward maps as we mentioned at the beginning of this section, say q ! and f ! . Now consider ind ana (D)= ind X (a)= ind S N(q ! a)= ind S N(( f ! ) 1 ( f ! )(q ! )a) property2 = ind pt (( f ! )(q ! )a) property1 = ( f ! )(q ! )(a)= ind top (D). We are done if we can establish these properties by the above idea. 20 Now let’s go through the proof under this idea. 6.2 Proof of the theorem Here we want to review the proof given in [1, Chapter 3, Section 13] and [6, Chapter 8] First of all, assume M and N are differentiable complex vector bundles over a compact riem- mannian manifold X. Definition 6.2.1. An operator P2YDO m (M;N) is a classical operator if its principal symbol is homogeneous of degree m ine outside of some compact subset of T X. In particular,P is classical ifs te (P)= t m s e (P) where t 1; e2 T X;kek a for some constant a. If X is not compact, then P is a classical operetor if P satisfies this property over every compact subset of X. Denoting the set of all such operators asYCO m (M;N). Definition 6.2.2. For an operator P2YCO m (M;N) we can define the asymptotic principal sym- bol: ˆ s e (P)= lim t!¥ s te (P) t m wheree2¶DX =fe2 T X :jjejj= 1g. Now forp :¶DX! X a bundle projection, we can consider the sequence: 0!YDO m1 (M;N)!YCO m (M;N) ˆ s !G(Hom(p M;p N))! 0. Theorem 6.2.1. The sequence above is exact. Proof. The second arrow is injection. Since the first arrow maps an element by adding an coordi- nate, if we pick two elements equal inYCO m (M;N) and their preimage inYDO m1 (M;N), they must be the same in theYDO m1 (M;N) otherwise they can not be equal inYCO m (M;N). Then we want to show that the map ˆ s is a surjection. First of all, we can pick a section, say a2G(Hom(p M;p N)). Then using bump function we can extend a to all of T X so that it is homogeneous of degree m in e forjjejj 1. Now we can pick an finite coverfU i g of X such that M and N can trivialize over each U i . Locally, we denote aj U i = a i . Then consider these two trivlizations, we can see that each a i is a matrix of complex valued functions and the rank 21 is determinded by the dimension of M and N. By multiplying a suitable continuous real valued function for each entries in a i , we can get the a new matrix b i . Note that this b i satisfying our Lemma 3.2.1, hence this gives us a pseudodifferential operator say h i . Letfc i g be a partition of unity subordinate to this cover. P=åc i h i 2YCO m (M;N) has the principal symbola. Let us now prove a lemma which is useful in the whole proof. Lemma 6.2.2. [1, Chapter 3 Lemma 13.3] Assume X is a compact manifold. Letp : B! X be a differentiable, real vector bundle over X. We also assume that E and F are vector bundles on X, and they are trivial bundle outside a compact set. Then every element in K c (B) can be represented as a triple (p E;p F;s)2 L(B) c where s :p E!p F is homogeneous of degree zero on the fibers of B where it is defined. Proof. First of all, by our definition, any element in K c (B) can be expressed as a triple(E 0 ;F 0 ;s 0 ) wheres 0 : E 0 ! F 0 is a bundle isomorphism outside some compact subset of B, say A B. Since B is again a manifold, then by our lemma 2.3.1, we can find ˆ E 0 such that E 0 ˆ E 0 = n. Now we can replace the triple(E 0 ;F 0 ;s 0 ) with( ˜ E; ˜ F; ˜ s)=(E 0 ˆ E 0 ;F 0 ˆ E 0 ;s 0 Id). We get the trivilization: t ˜ E : ˜ E!(B A)C m andt ˜ F : ˜ F!(B A)C m , so ˜ s =t 1 ˜ F t ˜ E . Let i : X ,! B be the inclusion of 0-section and we can pick a compact subset K of X such that A Bj K . Now let E = i ˜ E and F = i ˜ F. Now define a homotopy h : B[0;1]! B for h(b;t)= tb. Let E= h ˜ E and F= h ˜ F, and we can get Ej Bf0g =p E; Ej Bf1g = ˜ E and Fj Bf0g =p F; Fj Bf1g = ˜ F. After extending the flat connection over Bj (XK) , the parallel transport over b[0;1] gives two isomorphismsy E : ˜ E!p E andy F : ˜ F!p F. Also we can seey E =t 1 E t ˜ E andy E =t 1 E t ˜ E wheret E =t ˜ E j E andt F =t ˜ F j F . Based on this we can write down the bundle maps=y F ˜ sy 1 E which is defined on B A and is a constant map over Bp 1 (K). Now to complete the proof we need to redefines such that it is homogeneous of degree 0. Picking r> 0 such that Afb2 Bj K : jjbjj rg, we can defines in the setjjbjj r such thats te =s e . 22 After we pick an element u2 K c (T X), we can represent this element by a triple(p M;p N;s) by lemma 6.2.1. For an integer m, we can pick P2YCO m (M;N) such that the asymptotic principal symbol iss by our exact sequence given at the beginning. Definition 6.2.3. We can define a homomorphism ind : K c (T X)!Z, ind(u)= ind ana (P). Theorem 6.2.3. The definition above is well-defined. Proof. The homomorphism is given by the direct sum operation defined in “L-type” and the defi- nition of analytic index. Then we want to show that this is well-defined. And this follows from the next claims. Claim: The definition is independent of the choice of P with a fixed asymptotic principal symbol. By [1, Chapter 3, Corollary 7.8], the index of an elliptic operator on a compact manifold only depends on its homotopy class. So for another P 0 2YCO m (M;N) satisfying ˆ s(P)= ˆ s(P 0 ),s(P) ands(P 0 ) are indeed homotopic. Claim: The definition is independent of the choice of the triple(p M;p N;s) of u. Suppose we have another triple(p M 0 ;p N 0 ;s 0 ), we may consider an element h=( ˜ M; ˜ N; ˜ s)2 L 1 (T X[0;1]) c such that hj T Xf0g =(p M;p N;s) and hj T Xf1g =(p M 0 ;p N 0 ;s 0 ). Then by lemma 6.2.2, we can replace h with a triple with the form(p ˜ M;p ˜ N; ˜ s) where ˜ M and ˜ N are vector bundle over X[0;1] and ˜ s is homogeneous of degree zero outside a compact a set. Now we can use(p ˜ M;p ˜ N; ˜ s) to construct a homotopy between(p M;p N;s) and(p M 0 ;p N 0 ;s 0 ), hence we get the same index here. Claim: The definition is independent of the choice of the positive number m. First of all, we should consider the simple case that m=0. Assume k = 0;1, and let u k = (p M k ;p N k ;s k ) be two different triple for associated differential operator P k by Lemma 6.2.2. Also we can pick P k for associated 0-order operators. By our definition of 2.3.2, we have L 1 (T X) c = K c (T X). So we can pick elementary elements e k =(p G k ;p G k ;Id) such that u 0 e 0 and u 1 e 1 are homotopic. Note that since the identity map will not change the index, ind(P k Id)= ind(P k ). So we finally get ind(P 0 )= ind(P 1 ). Then consider the case for non-zero m. 23 Then we can consider the case for non-zero m. For P2YCO m (M;N), we can fix Riemannian metric, and this will fix a Levi-Civita connection on E, and we can consider the associated laplacian on E. Let P i = P(1+Ñ Ñ) i 2 2YCO m+i (M;N). Since the leading term will not be changed through P i so we have ˆ s(P i )= ˆ s(P). Since(1+Ñ Ñ) i 2 is invertible we have indP i = indP. Combining claims above we can say that the analytic index gives us a well-defined homomor- phism: ind : K c (T X)!Z, ind : u! ind ana (P) for P the associated elliptic operator fixed by u. Now we can introduce our two main theorems, based on them we can prove the Atiyah-Singer Index Theorem. Theorem 6.2.4. [1, P .247, Property 1] When X=pt, then ind: K(pt)!Z is the identity Proof. Note that each element in K(pt) can be expressed as the form[C a ][C b ]. Then an ellitic operator P is just the map P :C a !C b , and we have ind ana P= a b. Theorem 6.2.5. [1, P .247, Property 2] If X, Y are compact manifold then the smooth embedding f : X ,! Y induced a homomorphism such that: ind(u)= ind( f ! u) for any u2 K c (T X) To prove this theorem, we use several lemmas to complete the second theorem. Lemma 6.2.6. [1, P .248, The Excision Property 13.4] Assume Y is an open manifold, and the maps f : Y ,! X and f 0 : Y ,! X 0 are two open embeddings from Y to compact manifold X and X’. We have ind f ! = ind f 0 ! on K c (T Y). Proof. Instead of calculating by different embeddings, here we just need to show that the calcula- tion is independent to the embedding that we select. We first pick an element y2 K c (T Y), then by lemma 6.2.2 we know that u can be represented by(p M;p N;s) where M and N are vector bundles over Y and trivial outside a compact set of Y . Also,s is homogeneous of degree 0 outside a compact set of T Y which implies that for a compact set A Y we have the trivilization: g M : Mj YA !(Y A)C n andg N : Nj YA !(Y A)C n . For(x;e)2 T (YA), we haves (x;e) =(g N ) 1 x (g M ) x =s x . This means that over T (YA), the 24 maps comes from a vector bundle mapf : E! F, and by our constructionf is the identity map for all point in Y A. Similarly to Theorem 6.2.1, we can construct an elliptic operator D2YCO 0 (M;N) such that the principal symbol s(D)=s outside a compact set in T Y and in Y-A the operator is just the identity map. Now if we have the open embedding f : Y ,! X, we can extend M, N trivially over X f(Y) since M, N are trivial outside a compact set in Y . Also after extend M, N we can extend D such that D is the identity there. This gives us an elliptic operator f ! D on X such that [s( f ! D)]= f ! [s(D)]= f ! u. Now since the support of element in ker( f ! D) is in A, ker( f ! D) ker(D) and by extending by 0, we have ker(D) ker( f ! D). So we can conclude that dim(ker(D))= dim(ker( f ! D)). Similarly we can get dim(ker(D ))= dim(ker( f ! D) ). Hence we can conclude that ind( f ! u)= ind( f ! D)= dim(ker(D))dim(ker(D )), and since dim(ker(D))dim(ker(D )) is independent of the choice of f, we are done. Lemma 6.2.7. [1, P .249, The Multiplication Property 13.5] (The Multiplicative Property) Assume X, Y are compact manifolds, then for any u2 K c (T X) and v2 K c (T Y) we have ind(u v)= (indu)(indv). Proof. First of all, we can represent u, v as the first order elliptic operator. A :G(M)!G(N) and B :G(M 0 )!G(N 0 ). Define D :G((M M 0 )(N N 0 ))!G((N M 0 )(M N 0 )) by D= 0 B @ A 1 1 B 1 B A 1 1 C A Note that since for A2YCO 1 (M;N), A 1 may not belong toYCO 1 (M M 0 ;N N 0 ) and similarly for other entries in the matrix. So here we need do some modifications. We use(A 1) e 2YCO 1 (M M 0 ;N N 0 ) instead of A 1 fore > 0 such that lim e!0 (A 1) e = A 1. Define a function y e (jxj;jhj) for (x;h)2 T X T Y as follows: Pick a C ¥ function f :R + ![0;1] such that f(t)= 0 for t 1 and f(t)= 1 for t 2. Now set y e (a;b)= 1 f(e p a 2 + b 2 )f( ea b ) for e > 0; a 0; b 0. Then we can get the (P 1) e by multiplying the principal symbol of P 1 byy e (jxj;jhj). Now we can do the same to all entries of the matrix D and we can get a family of D e and lim e!0 D e = D. Then by [1, Chapter 3, Corollary 7.4] , the 25 index is constant on the connected component of Fredholm space. We have ind(D e )= ind(D) for e > 0. Now for a compact set K T (XY), there is a constant c > 0, such that for all e c, s(D e )= s(D) on K. Then by our lemma 6.2.4 we have [s(D e )]=[s(D)]= u v for e small enough. Now we can focus on D to get the result. First calculate that: D D= 0 B @ A A 1+ 1 B B 0 0 AA 1+ 1 BB 1 C A DD = 0 B @ AA 1+ 1 B B 0 0 A A 1+ 1 BB 1 C A Then fora2G((M M 0 )(N N 0 )) if we have D Da = 0 then(D Da;a)=(Da;Da)= 0 which implies that Da = 0. So kerD D= kerD. Then we may ssume a2G(M M 0 ). Let ˜ A= A 1;A 1;AA 1;A A 1 and ˜ B= 1 B;1 B ;1 BB ;1 B B. D Da= 0)( g A Aa;a)+ ( g B Ba;a)= 0)jj ˜ Aajj 2 +jj ˜ Bajj 2 = 0) ˜ Aa= ˜ Ba= 0. Similarly fora2 N N 0 . Also note that ker ˜ A\ ker ˜ B= kerA kerB, so we have kerD= kerD D =(kerA kerB)(kerA kerB ) and cokerD = kerD = kerDD =(kerA kerB)(kerA kerB ). Hence we can conclude that[kerD][cokerD]=([kerA][cokerB])([kerA][cokerB]) which implies that ind ana D = (ind ana A)(ind ana B) = (indu)(indv). So we can conclude ind(u v) = ind(D e )= indD=(indu)(indv). This lemma can be applied to a lot of situations, however here we want to focus on sphere bundles. For detail of the construction can viewed in [1, P.251] and [6, Chapter 8, Section 5] To conclude, let p : V! X be a principal O n -bundle over compact manifold X, and here O n stands for orthogonal group. Denote Z= V O n S n . Then we want to construct a map K c (T X) K O n (T S n ) c ! K c (T Z). This can be done in the following way: First of all, if we pick an metric in Z we can split T Z =p T X T(Z=X) where T(Z=X)= T Z=p T X. This splitting gives us an multiplication K c (T X) K c T(Z=X)! K c (T Z). Now consider the composition of maps K O n (T S n )! K O n (VT S n ) c ! K c (V O n T S n )= K c (T(Z=X)). Here the first arrow is just given 26 by the pullback of V T S n ! T S n . And the second arrow is just the isomorphism given by the definition. Hence we get our desire map K c (T X) K O n (T S n ) c ! K c (T Z). Lemma 6.2.8. [1, P .252, The Multiplicative Property for Sphere Bundle 13.6] or [6, Theorem 8.5.1] Assume Z is an S n -bundle defined as Z = V O n S n where V is a principal O n -bundle over compact manifold X, i.e., p : V! X. Then for any u2 K c (T X) and v2 K O n (T S n ) c , we have ind(u v)= ind(u ind O n v) Proof. First of all we represent u, v by first-order elliptic operators A and B. Here B is an O n equivariant operator on O n bundles. Now here the problem that we want to solve is to construct a ˜ A such that it is also O n invariant. Since X is compact, then by [7, P.77 Lemma 7.1] we can cover V by finitely manyfU i O n g i= 1;:::;N. Now we can lift A on each of thefU i O n g, and we can get ˜ A i . In particular, if M and N are vector bundles over X, then A can be viewed as A :G(M)!G(N). The projectionp : V! X induced the pullbacks of M and N say ˜ M and ˜ N. Then ˜ M V M. So we just need to consider the O n -action on first coordinate in V M. Similarly for ˜ N. Hence locally for each A i , we can get the lift ˜ A i :G( ˜ M)!G( ˜ N). Now for a partition of unity fy i g subordinates tofU i g on X, we can get ˜ A=åy i ˜ A i on B and it is O n equivariant. Now using the same idea in Lemma 6.2.7 we can consider the matrix: ˜ D= 0 B @ ˜ A 1 1 B 1 B ˜ A 1 1 C A on V S n . This is obvious an O n operator. Using the same step in Lemma 6.2.7, we can find that the principal symbol s(D) represent u v where defined in K c (T X) K O n (T S n ) c ! K c (T Z). So now if we can calculate the analytic index of D we are done. We calculate ˜ D firstly. After calculating like lemma 6.2.7 we have ker ˜ D= ker ˜ D ˜ D=(ker( ˜ A 1)\ ker(1 B))(ker( ˜ A 1)\ ker(1 B )). Now consider the first summand ker( ˜ A 1)\ ker(1 B). The idea here is not to calculate directly, but we can first consider ker( ˜ A 1) or ker(1 B). Then consider what is the other one inside it. And similarly for ker ˜ D . OnG(M M 0 ) the space ker(1 B) contains sections ˜ f such that ˜ fj fpgS n M p(p) kerB for p2 V . Then the section ˜ f satisfies: ˜ f(pg 1 ;gx)=r g ˜ f(p;x) for g2 O n where r is the natural representation of O n on kerB. This implies that ˜ f corresponds to a sectionf of vector bundle M B over X where 27 B= V r kerB. Similarly we can consider ker(1 B ). Or alternatively, we can say ˜ f satisfies: ˜ f(pg 1 ;gx)= ˜ f(pg 1 ;x) for g2 O n . Then the remainder is similar to what we have done here. For detail check [6, Theorem 8.5.1]. Now we have a clear understanding of ker(1 B), then consider the next two maps. A :G(M B)!G(N B) and A :G(M B )!G(N B ). Now if we restrict the operator ˜ D to sections of Z, we get the operator D. Then kerD= kerA kerA =(kerA B)(kerA B ). And similarly, kerD =(kerA B)(kerA B ) Hence we can conclude that ind ana (D)= ind ana [A (B B )]= ind(u ind O n v). Now we have our last lemma. Lemma 6.2.9. [1, Chapter 3, Lemma13.7] Since S n R n R, we can view S n as an O n manifold. In particular, it can be rotated around an axis and has two fixed point. Let i : pt ,! S n denote the inclusion of one of the two fixed point of this action then we have ind O n (i ! 1)= 1. Proof. Consider D : Cl 0 ! Cl 1 with respect to the standard metric on S n . Note that D is just de Rham-Hodge operator d+ d :L even !L odd . By Hodge Theory this is an O n operator, and we have that ind ana O n (D)=[H 0 ]+(1) n [H n ]. Since O n acts on[H 0 ]=fconstant f unctionsg is always trivial and on[H n ]=Rfthe n volume f ormg is trivial if and only if n is even. So we can conclude that ind ana O n (D)= 8 > > < > > : 2 if n is even 1r if n is odd wherer is the non-trivial 1-dimensional representation on O n . However in K O n (S n ), [s(D)]= 8 > > < > > : 2i ! (1) if n is even (1r)i ! (1) if n is odd . This can be directly calculated from [6, Lemma 8.4.8]. Compared this two results, we are done. Then we can complete the proof of the theorem. We prove the Theorem 6.2.5 at first. Recall X, Y are compact manifold and the map f : X ,! Y is an embedding. Proof. By Lemma 6.2.6 we can replace Y with a tubular neighborhood of X in Y , which is diffeo- morphism to the normal bundle of X in Y by tubular neighborhood theorem. So now we want to 28 show that for V a vector bundle over X and all u2 K c (T X), ind(u)= ind( f ! u) where f : X ,! V is the inclusion to the zero section. By Lemma 6.2.6 we can compactify V as V O n S n . Now by , ind(u i 1 1)= ind(u ind O n (i 1 1)). Then by our last lemma: ind(u i 1 1)= ind(u), compared with our definition f ! u= u i ! 1, we prove the Theorem. Now for embedding f : X ,! S N and embedding j : pt ,! S N by Theorem 6.2.5 we have ind(u)= ind( f ! u)= ind( j 1 ! f ! u), and by Theorem 6.2.4 we have ind j 1 ! = j 1 ! = q ! here q is the map that we used to define the topological index. We can conclude that ind(u)= ind ana (P)= ind top (P). Here P is the operator in Definition 6.2.3. Hence we complete the proof of the Atiyah-Singer Index theorem. 29 References 1. Lawson, H. Spin geometry (Princeton University Press, Princeton, N.J, 1989). 2. Hatcher, A.http://pi.math.cornell.edu/ ~ hatcher/VBKT/VBpage.html. 3. Iversen, B. Cohomology of sheaves (Springer, Berlin, 1986). 4. Carmo, M. P. d. Riemannian geometry (Birkh¨ auser, Boston, 1992). 5. Atiyah, M. K-theory (Addison-Wesley Pub. Co., Advanced Book Program, Redwood City, Calif, 1989). 6. Mukherjee, A. Atiyah-singer index theorem: an introduction (Hindustan Book Agency, New Dehli, 2013). 7. Walschap, G. Metric structures in differential geometry (Springer, New York, 2004). 30

Abstract (if available)

Abstract The Atiyah-Singer Index Theorem plays an important role in connecting elliptic differential operators over a compact manifold with some topological data. The topological data is defined to be the analytic index. The theorem states that the analytic index of the elliptic operator is equal to the topological index. Here we want to use the K-theory approach to state the theorem and overview the proof given in Spin Geometry by H. Blaine Lawson and Marie-Louise Michelsohn. Also, we state definitions and prove some basic theorems in K-theory with compact support.

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Entry times statistics on metric spaces

PDF

On foliations of higher dimensional symplectic manifolds and symplectic mapping class group relations

PDF

Some stable splittings in motivic homotopy theory

PDF

2d N = (0, 2) SCFTs from M5-branes

PDF

On sutured construction of Liouville sectors-with-corners

PDF

On the Giroux correspondence

PDF

Computations for bivariant cycle cohomology

PDF

On the homotopy class of 2-plane fields and its applications in contact topology

PDF

Congestion in negative curvature: networks to manifolds

PDF

Growth of torsion in quadratic extensions of quadratic cyclotomic fields

PDF

Cornered Calabi-Yau categories in open-string mirror symmetry

PDF

Information geometry and optimal transport

PDF

Essays on the econometric analysis of cross-sectional dependence

PDF

On the torsion structure of elliptic curves over cubic number fields

PDF

The geometry of motivic spheres

PDF

Applications of contact geometry to first-order PDEs on manifolds

PDF

Coexistence in two type first passage percolation model: expansion of a paper of O. Garet and R. Marchand

PDF

On the Chow-Witt rings of toric surfaces

PDF

Categorical operators and crystal structures on the ring of symmetric functions

PDF

Resolving black hole horizons: with superstrata

Asset Metadata

Creator Yang, Fan (author)

Core Title On the index of elliptic operators

School College of Letters, Arts and Sciences

Degree Master of Arts

Degree Program Mathematics

Publication Date 04/15/2021

Defense Date 03/18/2021

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

Tag Atiyah-Singer index theorem,elliptic operators,K-theory,OAI-PMH Harvest,vector bundles

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Asok, Aravind (committee chair), Bonahon, Francis (committee member), Ganatra, Sheel (committee member)

Creator Email fyang399@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-445170

Unique identifier UC11668707

Identifier etd-YangFan-9449.pdf (filename),usctheses-c89-445170 (legacy record id)

Legacy Identifier etd-YangFan-9449.pdf

Dmrecord 445170

Document Type Thesis

Rights Yang, Fan

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA