Characteristic classes of vector bunles Yoshinori Hashimoto 1 Introuction Let be a smooth, connecte, compact manifol of imension n without bounary, an p : E be a real or complex vector bunle of rank k over x. Characteristic classes associate to each vector bunle E over a cohomology class ce H k ; R R is an appropriate coefficient ring as we will see later such that the association is natural, i.e. if we consier a map f : Y between smooth manifols an a pull-back bunle f E over Y, the characteristic class of the pull-back bunle cf E H k Y ; R is equal to the pull-back of the cohomology class f ce H k Y ; R. The power of characteristic classes is that it can efine an invariant of an isomorphism class of vector bunles over. Suppose p 1 : E 1 an p 2 : E 2 are two isomorphic vector bunles over, with the bunle isomorphism given by f : E 1 E 2, so that E 1 = f E 2. Suppose also for the sake of simplicity that c is in the top egree with Z as the coefficient ring, i.e. ce 1, ce 2 H n ; Z. Then, by coupling with the funamental class of, we get a number ce 1 an ce 2 corresponing to each vector bunle E 1 an E 2. Since we assume E 1 = f E 2, we see that noting that f acts as an ientity on the base ce 1 = cf E 2 = f ce 2 = ce 2 by virtue of naturality. This means that, if there are two vector bunles p : E an p : E which have ifferent numbers ce an ce they cannot possibly be isomorphic. Thus, a number ce can be powerful enough to tell an isomorphism class of a vector bunle. Often, a number obtaine by integrating a characteristic class is calle a characteristic number. Later, we will see that the tautological bunle over CP 1 cannot be isomorphic to the hyperplane bunle over CP 1 by using this principle. Of course, we have not shown if such characteristic classes exist at all, let alone explaining how to construct such an association E ce H k ; R. But an example of characteristic classes was alreay known in 1935, calle the Stiefel-Whitney classes w i E H i ; Z/2Z. In 194 s, Pontrjagin classes p i E H 4i ; Z were iscovere for a real vector bunle E, an Chern classes c i E H 2i ; Z were iscovere for a complex vector bunle E. Another famous characteristic class is the Euler class ee H k ; Z, integration of which is equal to the Euler characteristic χ if E = T. All of these characteristic classes are not inepenent of each other, i.e. there are certain relationships between various characteristic classes, an the way of efining these characteristic classes is not unique. The approach we take in this section is ifferential-geometric, which is calle Chern-Weil theory in a wier context. Mainly because of the limitation on length, we will only focus on the Chern classes mentione above. One reason for oing so is that Chern classes are inispensable in complex ifferential an algebraic This ocument is an excerpt from an assignment for SMSTC Geometry/Topology stream 211-12, complete by the author. 1
geometry, whereas other characteristic classes have more to o with much more subtle topological structure on a vector bunle, e.g. existence of spin structure, signature of the intersection form, to mention just a few. Another reason is that once we efine the Chern classes by means of the Chern-Weil theory, the efinition of the Pontrjagin classes goes parallel with only minor moification. The efinition of Euler class also follows a similar strategy. We also have to remark that the Chern-Weil theory cannot be use to efine the Stiefel- Whitney classes, since the Chern-Weil theory goes through e Rham theory an the Stiefel-Whitney classes are efine over Z/2Z. 2 Chern classes Let p : E be a complex vector bunle of rank k i.e. each fibre is a C-vector space with imension k over C efine on the base manifol. We efine a connection on E, which exists by the Exercise 3.B.2. Let R ΓEn C E 2 T be the curvature, i.e. R V, W = V W W V [V,W ]. Since R is a 2-form on with coefficients in En C E, we see that the following makes sense, where I is the ientity matrix in the enomorphism ring En C C k, by noting that is the usual erminant taken over the En C E sector of R, an we may treat 1 as a matrix polynomial since R is a 2-form an hence commutes with itself. The presence of the numerical constant 1/ will be explaine later. We give a provisional efinition of the Chern classes as follows. Definition 2.1. We efine ifferential 2i-forms c i E, Γ 2i T as 2π = which later will turn out to be what is known as Chern classes. Examples The main theorem that we wish to prove is the following. c i E, λ k i, c E, = 1 R. R c 1 E, = tr 2π Γ 2 T. 2 Theorem 2.2. c i E,, for i k, is a close form, an the cohomology class [c i E, ] H 2i ; C is inepenent of the connection initially chosen to efine c i E,. Moreover, [c i E, ] is in fact a real class, i.e. [c i E, ] H 2i ; R H 2i ; C Remark This theorem has a far-reaching generalisation, which is calle the Chern-Weil theory. The key feature is that the erminant is ajoint-action invariant ajoint action in the sense of Lie group theory. However, it is not possible to iscuss this in its full ails here, ue to the limitation on length. 1 2
In fact, there are 4 axioms which uniquely characterise Chern classes as efine in algebraic topology [H], [K]. In particular, these axioms characterise Chern classes as an integral class. We can show that [c i E, ] efine above satisfy all these axioms, although we o not provie ails in this section. We refer to [K] for ails. An important consequence of this is the following theorem. Theorem 2.3. [c i E, ] is in fact an integral class, i.e. [c i E, ] H 2i ; Z H 2i ; R Remark This is the reason for the numerical constant 1/ appearing in the efinition of the Chern classes - it ensures that [c i E, ] takes value in the integral class. These theorems finally enable us to efine the Chern classes. Definition 2.4. For a complex vector bunle of rank k over, i-th Chern class of E, written c i E, is the integral cohomology class efine by [c i E, ] H 2i ; Z. This is well-efine because [c i E, ] = [c i E, ] for any two connections an on E, by Theorem 2.2. Examples Let = CP 1 an E be the tautological line bunle L. Then, we can show CP 1 c 1 L =. This is in fact one of the 4 axioms in algebraic topology which characterise Chern classes. Let H be the hyperplane bunle over CP 1, i.e. the ual of L. By noting that the curvature of the ual bunle is given by R t an recalling 2, we see that CP 1 c 1 H = CP 1 c 1 L = +1. This shows that H is not isomorphic to L, as mentione in 1. We finally comment on how this efinition of the Chern classes satisfy the naturality 1 mentione in 1. Let f : Y be a map between smooth manifols an consier the pull-back bunle f E over Y. We efine an inuce connection f on f E as follows. For any local section s of E, we have a local section f s of f E an any section of f E arises this way, by the efinition of the pull-back bunle. Then, for any vector fiel v on Y, we efine f v f s := f f vs. We can prove that the curvature of f is given by f R pull-back of a ifferential form, by consiering a connection 1-form of connection 1-form is efine, for example, in [K]. Thus c i f E, f λ k i = λi Rf 2π = λi f R 2π = f c i E, λ k i an we see that the naturality of c i E, is satisfie, even at the level of ifferential forms i.e. even before taking the cohomology class. 3 Proof of Theorem 2.2. We follow the exposition in [K]. 1. c i E, is close. Proof. We remark that En C k E is a trivial line bunle, since an enomorphism bunle of any line bunle is trivial. Thus, the covariant exterior erivative on En C k E agrees with the usual 1 In fact, the naturality is one of the 4 axioms in algebraic topology which efine the Chern classes. 3
exterior erivative. Thus, by noting that λi R / is an En C k E-value ifferential form on, we see that 2π = 2π = 3 where we use the Bianchi ientity R = an the Leibniz rule of to erive the secon equality. This shows c i E, = for i. 2. [c i E, ] H 2i ; C is inepenent of the connection chosen. Proof. Let an 1 be two connections efine on E. Then, the ifference 1 is an En C E- value 1-form which we call α. Let t := 1 t + t 1 = + tα be a family of connections connecting an 1. Let t be the covariant exterior erivative efine by t. Then, the curvature R t := R t is given by R t = t t = + tα + tα. Then, R t = t + t = α t + t α = t α 4 where the last equality is the one as an element of ΓEn C E 2 T, i.e. α t ξ+ t αξ = t αξ for ξ ΓE. Note that λi R / acts on a section ξ 1 ξ k of k E by 2π ξ 1 ξ k = 2π ξ 1 2π ξ k. Similarly, we efine an En C k E-value ifferential form ϕ by ϕ := k 1 2π α λi Rt λi Rt 2π. ϕ = k 1 Therefore, by recalling 3, the Bianchi ientity, Leibniz rule, 4, an the funamental theorem in calculus, we compute 1 α λi Rt λi Rt = k = k which proves the claim. 1 α t λi Rt t α λi Rt λi Rt λi Rt = k 1 R t 2π λi Rt λi Rt 2π 1 = k λi Rt 2π λi Rt λi Rt 1 = λi Rt 2π λi Rt λi Rt = 1 2π 4
3. c i E, is a real form. Proof. Define a hermitian metric on E an take a connection which is compatible with the metric. They o exist on any complex vector bunle, as explaine in Chapter 5 of [M]. Then, by Gram-Schmi process, we can reuce the bunle structure group from GL k C to Uk, an the curvature R takes value in the Lie subalgebra uk = LieUk of the Lie algebra En C C k = gl k C. Thus, R is a ifferential 2-form with values in skew-hermitian matrices. Thus, assuming λ R without loss of generality, = = λi + t = λi t = λi which proves the claim, an hence completes the proof of Theorem 2.2. = λi 4 References H : Hirzebruch, F. Topological Methos in Algebraic Geometry, Springer, 1978. K : Kobayashi, S. Differential Geometry of Complex Vector Bunles, Worl Publishing Corp., 1987. M : Morita, S. Geometry of Differential Forms, AMS, 21. MS : Milnor, J.W., Stasheff, J.D. Characteristic classes, Princeton Univercity Press, 1974. 5