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CHERN CLASSES Juan Moreno

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CHERN CLASSES Juan Moreno CHERN CLASSES Juan Moreno §1. INTRODUCTION A characteristic class associates to a vector bundle ! (E,¼,B) a cohomology class ® H¤(B), which Æ ! 2 respects bundle maps, that is, if f : ! ´ is a bundle map, then f ¤® ® . Chern classes, denoted ! ´ Æ ! ck(!) for k N, are characteristic classes associated to complex vector bundles which satisfy the 2 following properties: 0 2k • c0(!) 1 H (B), ck(!) H (B), k N. Æ 2 2 8 2 n • Let C denote the trival rank n complex vector bundle. Then ck(E) 0, k 0. Æ 8 È • If rank(!) n then ck(!) 0 for all k n. Æ Æ È P • Let c(!) denote the formal sum i ci(!) called the total Chern class. Then we have the Whitney Product formula c(! ´) c(!)c(´). © Æ One can give an axiomatic definition of Chern classes, however, we will instead focus on definitions which are more tractible for computations, and give intuition as to what the Chern classes measure. We will give two different perspectives on Chern classes: the first is uses an auxilliary characteristic class, while the second uses the geometry of a connection. These notes do not present a full intro- duction to Chern classes. In particular, we do not provide any proofs of these important properties described above. The purpose of these notes is simply to give explicit constructions of Chern classes so that one can compute them at least for simple examples. For a detailed account of Chern classes and the theory of characteristic classes in general see [Sta74]. In order to construct characteristic classes we must fix a particular cohomology theory to work with. In this note, we work exclusively with de Rham cohomology. For an introduction to the theory of differential forms and de Rham cohomology see [Pol10] or [Tu95] §2. CHERN CLASSES VIA THE EULER CLASS §2.1. ORIENTATIONS AND POINCARÉ DUALITY Let V be a real vector space of finite dimension n. We begin with a quick review of orientations. Definition 1. An orientation on V is an equivalence class of basis b : Rn V, where two bases n ! b, b0 : R V are equivalent if and only if there is an element A GLn(R) such that b0 b A and ! 2 Æ ± detA 0. È Remark 1. Note that there are precisely two possible orientations on a real vector space V corre- sponding to the two connected components of GLn(R). In fact the equivalence above could be re- stated as follows. Two bases b, b0 are equivalent if and only if there is a path γ : I GLn(R) such ! that γ(0) In and b γ(1) b0. Equivalently, if B(V) is the space of all bases of V,then B(V) is Æ ± Æ GLn(R)-torsor, and we can topologize it as such. Then an orientation is simply a choice of connected component of B(V). 1 Let M be a closed1 smooth n-manifold. Definition 2. An orientation of M is a choice of orientation [bx] of Tx M for each x M subject to 2 the following compatibility condition: For each x M there is a neighborhood U of x and a diffeomor- 2 phism n 1 b : U R ¼¡ (U) TM £ ! ½ such that [b(x,_)] [bx]. Æ Recall that choosing an orientation on a manifold allows one to integrate differential forms over the manifold. Now if M is a closed oriented manifold, Stokes’ Theorem implies that integration over M descends to cohomology. The proof is a simple computation: Z Z Z Z Z Z Z Z n n 1 ! d´ ! d´ ! ´ ! ´ !, ! ­ (M),´ ­ ¡ (M). M Å Æ M Å M Æ M Å @M Æ M Å Æ M 2 2 ; There results a bilinear pairing Z k n k : H (M) H ¡ (M) R. M ­ ! Z ! ´ ! ´ ­ 7! M ^ Theorem 1. (Poincaré duality)[Tu95] This pairing is nondegenerate, implying n k k H ¡ (M) (H (M))¤. Æ» Let i : S , M be an embedded oriented submanifold of M of dimension k such that i(S) is closed ! k R (as a subspace) in M. This determines a linear functional H (M) R given by [!] S i¤!. By n k ! 7! Poincaré duality, this corresponds to a unique element [´S] H ¡ (M) which is characterized by 2 Z Z k i¤! ! ´S, ! H (M). S Æ M ^ 8 2 This element [´S] is called the Poincaré dual to the submanifold S. §2.2 THE EULER CLASS OF A SMOOTH ORIENTED VECTOR BUNDLE Recall that y N is a regular value of a smooth map f : M N of manifolds if d fx has full rank for 1 2 ! all x f ¡ (y). The regular value theorem tells us that this condition guarantees that the solution set 2 1 {x M f (x) y} f ¡ (y) is a submanifold of M. The definition that follows generalizes this notion by 2 j Æ Æ giving us a sufficient condition for which the solution set {x M f (x) Z} for some submanfold Z N 2 j 2 ½ is a submanfiold of M. Definition 3. Let Z N be a submanifold and f : M N a smooth map. We say that f is transverse 1 ½ ! to Z if for all x f ¡ (Z), 2 d fxTx M T f (x)Z T f (x)N. Å Æ 1 Proposition 1. [Pol10] If f : M N is a smooth map transverse to a submanifold Z N, then f ¡ (Z) ! 1 ½ is a submanifold of M. Furthermore, the codimension of f ¡ (Z) in M is the same as the codimension of Z in N. Definition 4. Let ¼ : E M be a smooth vector bundle of rank r, and Z(E) E the image of the zero ! ½ section. A section s : M E is called generic if it is transverse to Z(E). The Euler class of E, denoted r ! 1 e(E) H (M) is the Poincare dual to s¡ (Z(E)), the zero-set of a generic section, s. 2 Remark 2. There are more general definitions of the Euler class for oriented non-smooth vector bundles, but this one is convenient in our setting. 1compact & boundaryless 2 §2.3 CHERN CLASSES We now give an inductive definition of Chern classes starting with the Euler class. Let ! (E,¼, M) Æ be a complex vector bundle. Forgetting the complex structure gives an underlying real vector bundle, which we denote by !R. Proposition 2. The underlying real vector bundle of a complex vector bundle has a canonical orien- tation. Proof. First, let V be a complex vector space and VR its underlying real vector space. Any complex basis v1,...,vn gives rise to an ordered real basis v1, iv1,...,vn, ivn which corresponds to the orien- tation on VR. This orientation is independent of the original basis of V since GLn(C) is connected. For a general complex vector bundle we can apply this construction fiberwise and get compatible orientations, again because GLn(C) is connected. Let rank(!) k. We define the top Chern class of !, ck(!), to be the Euler class of its underlying Æ 2k real vector bundle, e(!R) H (M). Now let B0 E \ Z(E), where Z(E) is the zero section of !. There 2 Æ is a natural vector bundle over B0 defined as follows. First, choose a Hermitian metric on !. This choice of metric is immaterial and simply used for conveniece. Now any point of B0 is a pair (x,v) 1 with x M and v Ex ¼¡ (x), so we may take the fiber above (x,v) to be v? Ex, the perpendicular 2 2 Æ ½ subspace of E. This defines a vector bundle !0 (E0,¼0,B0) of rank k 1. As such, it’s top Chern class 2(k 1) Æ ¡ 2(k 1) ck 1(!0) H ¡ (B0) is well defined, however, we wish to get a class in H ¡ (B). For this we need ¡ 2 the following fact [Sta74]: Let ! (E,¼,B) be an oriented vector bundle of real dimension n, and B0 E \ Z(E). Then there Æ Æ is an exact sequence i e i n ¼¤ i n i 1 H (B) [ H Å (B) H Å (B0) H Å (B) ¢¢¢ ! ¡¡! ¡! ! ! ¢¢¢ called the Gysin sequence. The map ¼¤ in the above sequence is simply the pullback via the restriction of ¼ to B0 E. Going ½ back to our situation above, since Hi(B) 0 for i 0, we have that Æ Ç i 2k i 2k ¼¤ : H Å (B) H Å (B0) ! 2(k 1) 2(k 2) is an isomorphism for i 1 so that ¼¤ : H ¡ (B) H ¡ (B0) is an isomorphism. At last, we can Ç¡ ! define 1 1 ck 1(!) (¼¤)¡ (ck 1(!0)) (¼¤)¡ (e(!R0 )). ¡ Æ ¡ Æ We can then repeat this procedure. At the ith step we get a vector bundle !(i) of rank 2(k i) over (i) (i 1) (i 1) ¡ B E ¡ \ Z(E ¡ ), as well as a sequence of isomorphisms Æ (i 1) 2(k i) ¼¤ 2(k i) (¼0)¤ 2(k i) (¼ ¡ )¤ 2(k i) (i) H ¡ (B) H ¡ (B0) H ¡ (B00) H ¡ (B ). ¡! ¡¡¡! ! ¢¢¢ ¡¡¡¡¡! (i) (i) We can then define c2(k i)(!) to be the inverse image of c2(k i)(! ) e(!R ) under these isomor- ¡ ¡ Æ phisms. Exercise 1. Suppose ! is a rank n complex vector bundle which admits k linearly-independent sections. Show that cn(!) cn 1(!) cn k 1(!) 0. Æ ¡ Æ ¢¢¢ Æ ¡ Å Æ 3 §3. CHERN CLASSES VIA CONNECTIONS The perspective on Chern classes of the previous section had the advantage of being manifestly a measure of the non-triviality of a complex vector bundle: If we could find a nowhere vanishing section, then the top Chern class vanishes by definition.
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