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Characteristic Classes, with Spectral Sequences

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Characteristic Classes, with Spectral Sequences CHARACTERISTIC CLASSES, WITH SPECTRAL SEQUENCES Maxim Jeffs December 12, 2016 INTRODUCTION This report presents the theory of characteristic classes using the machinery of spectral sequences. Generalities concerning the Serre and Bockstein spectral sequences are treated in the first section. The second section uses the Serre spectral sequence to study Chern classes; the Gysin sequence and the Thom isomorphism are derived using spectral sequences in §3 and used to study the Euler class. The fourth section presents the Stiefel-Whitney and Pontrjagin classes, making use of the Bockstein spectral sequence. I would like to thank Vigleik Angeltveit for supervising this reading course and initiating me into the dark arts of spectral sequences. I would also like to thank Jack Davies for kindly lending me some of his TikZ code for spectral sequences. 1 SPECTRAL SEQUENCES i π Throughout, F −! E −! B denotes a fibre bundle over a base space B with fibre F . We assume for simplicity that B; F are connected and have the homotopy type of CW complexes. Let γ : I ! B be a path in B; we can define a map gt : Fγ(0) ! B by mapping the entire fibre Fγ(0) to γ(t) for each t 2 I. Since F has the homotopy type of a CW complex and fibre bundles have the lifting property with respect to all CW paris ([Hat02], 4.48), we have a diagram of the form: E B g~ i g F i0 F × I giving a lift g~t : Fγ(0) ! E with g~t(Fγ(0)) ⊆ Fγ(t). Hence we have a natural map Lγ : Fγ(0) ! Fγ(1). It is easy to show 0 that Lγ is independent of the homotopy class of γ rel endpoints, and that Lγγ0 ' Lγ0 Lγ for a pair of paths γ; γ ([Hat02], p. 405). Hence Lγ gives a homotopy equivalence of Fγ(0) with Fγ(1). We now say that E is simple with respect to the ∗ ∗ ! ∗ 2 coefficient ring R if the isomorphism Lγ : H (Fγ(0); R) H (Fγ(0); R) is the identity for all loops γ π1(B). Note that all sphere bundles are simple with respect to the coefficients R = Z2, and all fibre bundles over a simply connected space are simple with respect to any system of coefficients. We will henceforth assume that all our fibre bundles are simple with respect to the given system of coefficients, some Noetherian ring R. We now recall the basic facts concerning the Serre spectral sequence that will be needed in subsequent sections. p;q p;q p+r;q−r+1 THEOREM. There exists a natural first-quadrant spectral sequence of algebras (Er ; dr : Er ! Er ) associated to any fibre bundle as above, converging as an algebra to H∗(E; R), such that: p;q ∼ p q (a) The second page is given by E2 = H (B; H (F ; R)), pictured below, which is isomorphic as a differential bigraded algebra to p q H (B; R) ⊗R H (F ; R) when both cohomology rings are free R-modules of finite type ([McC01], Theorem 5.2, Proposition 5.6). In fact, if any two of F; E; B have cohomology that is finitely generated in each dimension, so does the third ([McC01], Example 5.A) p;q ∼ p p+q p+1 p+q p n p ∗ (b) The E1 page yields filtration quotients E1 = F H (E; R)/F H (E; R) where F H (E; R) = F H (E; R) \ i∗ Hn(E; R) is the filtration of H∗(E; R) given by F pH∗(E; R) = ker(H∗(E; R) −! H∗(E(p−1); R)), for E(p−1) = 1 π−1(B(p−1)) the lift of the CW skeleta of B to a filtration of E. ([McC01], Proposition 5.3) ∗ p;q (c) Whenever the E1 page is a free, (graded-commutative) bigraded algebra, then H (E; R) is isomorphic to the total complex of E1 as a bigraded algebra ([McC01], Example 1.K). 0;5 1;5 2;5 3;5 4;5 5;5 E2 E2 E2 E2 E2 E2 0;4 1;4 2;4 3;4 4;4 5;4 E2 E2 E2 E2 E2 E2 0;3 1;3 2;3 3;3 4;3 5;3 E2 E2 E2 E2 E2 E2 0;2 1;2 2;2 3;2 4;2 5;2 E2 E2 E2 E2 E2 E2 0;1 1;1 2;1 3;1 4;1 5;1 E2 E2 E2 E2 E2 E2 0;0 1;0 2;0 3;0 4;0 5;0 E2 E2 E2 E2 E2 E2 We will also use some elementary facts about the cohomology Bockstein spectral sequence which we shall consider below in the case of the prime p = 2. Due to the lack of an adequate reference, we spend some time summarising the construction, which is somewhat atypical. n n+1 n Let X be a space and define Bockstein homomorphisms β~ : H (X; Z2) ! H (X; Z) and β = β1 : H (X; Z2) ! n+1 H (X; Z2) to be the connecting homomorphisms in the long exact sequences in cohomology coming from the short 2 exact sequences of coefficient groups 0 ! Z −! Z ! Z2 ! 0 and 0 ! Z2 ! Z4 ! Z2 ! 0 respectively. If ∗ ∗ ρ : H (X; Z) ! H (X; Z2) denotes the reduction homomorphism, then it is easy to see that β1 = ρ ◦ β~ by considering the naturality of the long exact sequence in cohomology with respect to the obvious morphism of short exact sequences of coefficient groups given by 0 Z Z Z2 0 0 Z2 Z4 Z2 0 By instead considering these homomorphisms as forming an exact triangle H∗(X; Z) 2 H∗(X; Z) ρ β~ ∗ H (X; Z2) and taking the associated (singly-graded) spectral sequence, we will have d1 = ρ ◦ β~ = β1 the differential on the ∗ ∗ Z E1 page E1 = H (X; 2). It is not too hard to see that the resulting spectral sequence will converge (strongly) to ∗ (H (X; Z)/torsion) ⊗ Z2 ([McC01], Theorem 10.3). The crucial fact is that we may actually identify the higher-order n differentials in this spectral sequence by more generally considering Bockstein homomorphisms βr : H (X; Z2r ) ! n+1 H (X; Z2r ) coming from the short exact sequence 0 ! Z2r ! Z22r ! Z2r ! 0. We then have ∗ ∗ Z THEOREM. The page Er of the Bockstein spectral sequence may be identified with the subgroup of H (X; 2r ) given by multiples of r−1 ∗ ∗ 2 . Then the differential dr can be identified with the Bockstein homomorphism βr : H (X; Z2r ) ! H (X; Z2r ). For a proof, see [McC01], Proposition 10.4; also note that in a singly-graded spectral sequence, all of the differentials will indeed have degree 1. 2 To simplify some of the following homological algebra, we introduce a construction from ([Hat02], p. 305). Denote by C∗ the singular chain complex of X, which we assume to have finitely-generated cohomology in each dimension; we construct a new chain complex M as a direct sum over the following chain complexes. For each Z summand of Hn(C), take a chain complex · · · − 0 − Z − 0 −· · · concentrated in degree n, and for each Zk summand in Hn(C), take a k chain complex · · · − 0 − Z − Z − 0 ··· concentrated in degree n also. We then have an obvious chain map M ! C given by sending the various generators of the Z summands to appropriate representatives of the homology classes, and it not hard to see that this will be a quasi-isomorphism. By the universal coefficient theorem, this chain map will then induce isomorphisms on homology and cohomology with any coefficients. Hence we may as well work exclusively with M rather than C, where changes of coefficients become rather more transparent. From this construction, it is already clear that a Z summand in Hn(X; Z) will always correspond to a single Z2r sum- n Z Z Z ≥ mand in H (X; 2r ). To see what happens to a 2k summand of Hn(X; ) for k r under a change of coefficients, we dualise the relevant summand of M to get 2k ··· 0 Z2r Z2r 0 ··· n n+1 and we see that for k ≥ r, we will obtain two Z2r summands in adjacent dimensions H (X; Z2r ) and H (X; Z2r ). All this we already knew; the important point is that we may now calculate the Bockstein homomorphisms by explicitly writing out the short exact sequences of complexes 0 0 0 0 Z2r Z22r Z2r 0 2k 2k 2k 0 Z2r Z22r Z2r 0 0 0 0 k−r ≥ Z and finding the connecting homomorphism βr to be multiplication by 2 for k r. This shows that a 2k summand in Hn(X; Z) will yield a pair of Z2r summands in each Hn(X; Z2r ) group for r < k, with a trivial Bockstein between them, until we reach r = k where the Bockstein becomes an isomorphism between these two summands and hence kills this summand completely on the next page of the spectral sequence. In fact, we have THEOREM. An element of Hn+1(X; Z) generates a cylcic direct summand of order 2r if and only if the generators of the corresponding n n+1 r r+1 cyclic direct summands in H (X; Z2) and H (X; Z2) survive to the E page of the Bockstein spectral sequence, but not the E page. An element generates a Z summand of Hn+1(X; Z) if and only if it survives to the E1 page.
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