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Characteristic classes and K-theory Oscar Randal-Williams https://www.dpmms.cam.ac.uk/∼or257/teaching/notes/Kthy.pdf 1 Vector bundles 1 1.1 Vector bundles . 1 1.2 Inner products . 5 1.3 Embedding into trivial bundles . 6 1.4 Classification and concordance . 7 1.5 Clutching . 8 2 Characteristic classes 10 2.1 Recollections on Thom and Euler classes . 10 2.2 The projective bundle formula . 12 2.3 Chern classes . 14 2.4 Stiefel–Whitney classes . 16 2.5 Pontrjagin classes . 17 2.6 The splitting principle . 17 2.7 The Euler class revisited . 18 2.8 Examples . 18 2.9 Some tangent bundles . 20 2.10 Nonimmersions . 21 3 K-theory 23 3.1 The functor K ................................. 23 3.2 The fundamental product theorem . 26 3.3 Bott periodicity and the cohomological structure of K-theory . 28 3.4 The Mayer–Vietoris sequence . 36 3.5 The Fundamental Product Theorem for K−1 . 36 3.6 K-theory and degree . 38 4 Further structure of K-theory 39 4.1 The yoga of symmetric polynomials . 39 4.2 The Chern character . 41 n 4.3 K-theory of CP and the projective bundle formula . 44 4.4 K-theory Chern classes and exterior powers . 46 4.5 The K-theory Thom isomorphism, Euler class, and Gysin sequence . 47 n 4.6 K-theory of RP ................................ 49 4.7 Adams operations . 51 4.8 The Hopf invariant . 53 4.9 Correction classes . 55 4.10 Gysin maps and topological Grothendieck–Riemann–Roch . 58 Last updated May 22, 2018. Corrections to [email protected]. Chapter 1 Vector bundles Throughout these notes, map means continuous function. 1.1 Vector bundles Let F be R or C. Definition 1.1.1. An n-dimensional F-vector bundle over a space X is a collection ` fExgx2X of F-vector spaces and a topology on the set E := x2X Ex so that the map π : E −! X e 2 Ex 7−! x satisfies the following local triviality condition: each x 2 X has a neighbourhood U 3 x and a homeomorphism n −1 'U : U × F −! EjU := π (U) n ∼ such that π('U (x; v)) = x and such that 'U gives a linear isomorphism fyg × F ! Ey for all y 2 U. We call such a U a trivialising open set, and 'U a trivialisation. We will usually refer to π : E ! X as a vector bundle, leaving the vector space −1 structure on the fibres Ex := π (x) implicit. A section of a vector bundle π : E ! X is a map s : X ! E such that π ◦ s = IdX . In particular, letting s0(x) be the zero element in the vector space Ex defines the zero # section s0 : X ! E. We write E := E n s0(X) for the complement of the zero section, i.e. the set of non-zero vectors. n Example 1.1.2. Projection to the first coordinate π1 : X × F ! X defines a vector n bundle, the trivial n-dimensional F-vector bundle, where Ex = fxg × F is given the n n evident F-vector space structure. To save space we will write this as F , or FX if we need to emphasise the base space. Definition 1.1.3. If π : E ! X is a vector bundle, then a subspace E0 ⊂ E is a subbundle if each E0 \ Ex is a vector subspace of Ex, and πjE0 : E0 ! X is locally trivial as in Definition 1.1.1. N Example 1.1.4. The Grassmannian Grn(F ) is the set of n-dimensional vector sub- N N N n spaces of F . If F rn(F ) ⊂ (F ) is the subspace of those sequences (v1; : : : ; vn) of vectors which are linearly independent, then there is a surjective map N N q : F rn(F ) −! Grn(F ) (v1; : : : ; vn) 7−! hv; : : : ; vniF 1 2 Chapter 1 Vector bundles given by sending a sequence of linearly independent vectors to its span, and we use this N to give Grn(F ) the quotient topology. Let γn;N := f(P; v) 2 Gr ( N ) × N j v 2 P g; F n F F and π : γn;N ! Gr ( N ) be given by projection to the first factor. The fibre π−1(P ) is F n F identified with P , so has the structure of a vector space. We claim that this is locally N N trivial. To see this, for P 2 Grn(F ) consider the orthogonal projection πP : F ! P : the set N U := fQ 2 Grn(F ) j πP jQ : Q ! P is an isog is an open neighbourhood of P (this can be checked by showing that q−1(U) is open), and the map γn;N j −! U × P F U (Q; v) 7−! (Q; πP (v)) is a homeomorphism: its inverse 'U gives a local trivialisation. Definition 1.1.5. It is conventional to call a 1-dimensional vector bundle a line bundle. 1.1.1 Morphisms of vector bundles If π : E ! X and π0 : E0 ! X are two vector bundles over the same space X, then a 0 0 map f : E ! E is a vector bundle map if it is linear on each fibre, i.e. fx : Ex ! Ex is a linear map. Hence two vector bundles are isomorphic if there are mutually inverse vector bundle maps f : E ! E0 and g : E0 ! E. 1.1.2 Pullback If π : E ! X is a vector bundle and f : Y ! X is a continuous map, we let f ∗E := f(y; e) 2 Y × E j f(y) = π(e)g; ∗ ∗ ∗ ∗ ∼ and define f π : f E ! Y by f π(y; e) = y. Then (f E)y = Ef(y) has the structure of a F-vector space. n ∼ −1 Suppose 'U : U × F ! EjU is a local trivialisation of π. Writing V := f (U), an open set in Y , the map n ∗ V × F −! (f E)jV (y; e) 7−! (y; 'U (f(y); e)) ∗ ∗ is a homeomorphism. Thus f π : f E ! Y is a F-vector bundle. 1.1 Vector bundles 3 1.1.3 Vector bundles via transition data Let π : E ! X be a vector bundle. As each point of X has a trivialising neighbourhood, we may find an open cover fUαgα2I by trivialising open sets. If x 2 Uα \ Uβ then we may form the composition −1 ' j ' jU \U n Uα Uα\Uβ Uβ α β n (Uα \ Uβ) × F −! EjUα\Uβ −! (Uα \ Uβ) × F ; which has the form '−1j ◦ ' j (x; v) = (x; (τ (x))(v)) for some continuous Uβ Uα\Uβ Uα Uα\Uβ α,β map n τα,β : Uα \ Uβ −! GL(F ) to the topological group of invertible n × n matrices over F, called the transition map. Given the data (fUαgα2I ; fτα,βgα,β2I ) we can recover the vector bundle π : E ! X up to isomorphism, as follows. Let 0 a n F := Uα × F ; α2I and try to define an equivalence relation ∼ on F 0 by 0 0 0 0 (x 2 Uα; v) ∼ (x 2 Uβ; v ) , x = x 2 X and τα,β(x)(v) = v : To see that ∼ is reflexive, we require that n τα,α(x) = I 2 GL(F ): (1.1.1) To see that ∼ is symmetric we require that −1 n τβ,α(x) = (τα,β(x)) 2 GL(F ): (1.1.2) In order for ∼ to be transitive, we require that n τβ,γ(x) ◦ τα,β(x) = τα,γ(x) 2 GL(F ) (1.1.3) when x 2 Uα \ Uβ \ Uγ. All of the above conditions are clearly true when the data (fUαgα2I ; fτα,βgα,β2I ) is constructed from a vector bundle. Let F := F 0= ∼, and define π([x; v]) = x : F ! X. The map φ0 : F 0 −! E (x 2 Uα; v) 7−! 'Uα (x; v) is constant on ∼-equivalence classes, and descends to a map φ : F ! E which is easily seen to be an isomorphism of vector bundles over X. However, this construction opens up the following possibility: given an open cover n fUαgα2I of X, and maps τα,β : Uα \ Uβ ! GL(F ) which satisfy (1.1.1), (1.1.2), and (1.1.3), then we can use the recipe above to define a vector bundle π : F ! X. 4 Chapter 1 Vector bundles 1.1.4 Operations on vector bundles Roughly speaking, any natural operation that one can perform on vector spaces, one can also perform on vector bundles. We shall not try to make this precise, but settle for the following examples. Sum of bundles 0 0 0 If π : E ! X is a n-dimensional F-vector bundle and π : E ! X is a n -dimensional 0 0 0 F-vector bundle, there is a (n + n )-dimensional vector bundle π ⊕ π : E ⊕ E ! X with E ⊕ E0 := f(e; e0) 2 E × E0 j π(e) = π0(e0)g 0 0 0 0 and (π ⊕ π )(e; e ) := π(e). Then (E ⊕ E )x = Ex × Ex has a natural F-vector space structure, and one can easily find local trivialisations over open sets given by intersecting a trivialising open set for E with one for E0. This is often called Whitney sum. Tensor product of bundles 0 0 0 If π : E ! X is a n-dimensional F-vector bundle and π : E ! X is a n -dimensional 0 0 0 F-vector bundle, there is a (n · n )-dimensional vector bundle π ⊗ π : E ⊗ E ! X. As a set 0 a 0 E ⊗ E = Ex ⊗F Ex; x2X but we must give this a topology making it into a vector bundle.
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