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 ∈ 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 ′ that Lγ is independent of the homotopy class of γ rel endpoints, and that Lγγ′ ≃ Lγ′ 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 ∗ ∗ → ∗ ∈ 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 E∞ page yields filtration quotients E∞ = 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 E∞ page is a free, (graded-commutative) bigraded algebra, then H (E; R) is isomorphic to the total complex of E∞ 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 E∞ page. From the above discussion and Theorem 2, the statement above really is ‘self-evident’, as is claimed in ([MT68], p. 61). For a spelled-out proof, see ([Wei94] p.159), though note that there appears to be a problem with his statement of the theorem (the elements he describes need not necessarily be summands). Remark. The cohomology Bockstein spectral sequence also has an algebra structure, with the Bockstein homomorphisms being derivations ([Hat02],§3.G).

2 CHERN CLASSES

We first consider generalities concerning characteristic classes: DEFINITION. A characteristic class is a function c : Vect(B) → Hn(B; R) on isomorphism classes of (real, oriented, or complex) vector bundles E → B that is natural with respect to pullbacks of vector bundles and cohomology classes. As every vector bundle is isomorphic to a pullback of some universal vector bundle over a Grassmannian, we see that all characteristic classes arise as a cohomology classes of a Grassmannian. These cohomology classes may be computed easily

3 using the spectral sequence machinery presented in the previous section. For now we restrict attention to the complex case. ∞ ∗ ∞ ∼ THEOREM. The cohomology of the Grassmannian Gn(C ) is given by H (Gn(C ); Z) = Z[c1, . . . , cn] where |ci| = 2i. We will first need to determine the cohomology of some auxiliary spaces. ∗ ∼ LEMMA. The cohomology of the unitary group U(n) is given by H (U(n); Z) = ΛZ(x1, x3, x5, . . . , x2n−1), the exterior algebra with generators x2i−1, i = 1, . . . , n of degree 2i − 1.

∼ 1 ∗ ∼ Proof. We argue by induction. In the case U(1) = S we obviously have H (U(1); Z) = ΛZ(x1). Now consider the fibre 2n−1 bundle U(n − 1) → U(n) → S (see [Hat02], 4.55); the E2 page of the Serre spectral sequence is concentrated in − 0,q ∼ q − Z ⊗ Z{ } 2n−1,q ∼ q − Z ⊗ Z{ } columns 0 and 2n 1, where we will have E2 = H (U(n 1); ) 1 and E2 = H (U(n 1); ) h , for h the generator of H2n−1(S2n−1; Z), as illustrated below in the case n = 3:

x1x3 x1x3x5 4

x3 x3x5 3

x1 x1x5 1

x5 0 0 1 2 3 4 5

∗ Because the generator with the largest degree in H (U(n − 1); Z) is x2n−3, any non-zero differential in the spectral sequence would have to have horizontal degree 2n−1 and hence vertical degree −(2n−2) < −(2n−3). Hence none of p,q the classes that generate E2 as a differential bigraded algebra have any non-zero differential. By the derivation property of the differentials, we see that all of the differentials must therefore be zero and hence the spectral sequence collapses on the E2 page. The result in Theorem 1 (c) completes the proof. ■

With this in hand we can now prove Theorem 2:

∞ ∞ Proof. (of Theorem 2) We consider now the fibre bundle given by U(n) → Vn(C ) → Gn(C ) (see [Hat02], 4.53). ∞ It is well-known that Vn(C ) is contractible, and hence has trivial cohomology, so we shall try to use the Serre spectral sequence in reverse. To do so, we shall invoke the following theorem, which we quote from [McC01], Theorem 3.27: ∗ ∗ p,q THEOREM. Let K be a field and V ,W be two graded K-algebras. Suppose Er is a first-quadrant spectral sequence of algebras p,q ∼ ∗ ⊗ ∗ with E2 = V K W as bigraded algebras, subject to the hypotheses 0,0 p,q 1. The E∞ page has E∞ = K and E∞ = 0 otherwise; ∗ 2. The algebra W is isomorphic to Λ(x1, . . . , xm) where xi is homogeneous of odd degree 2ri − 1; ≤ ≤ ̸ 3. For each i, we have dj(xi) = 0 for 2 j 2ri, and d2ri (xi) = 0, that is, the generators xi transgress. ∗ ∼ Then V = K[y1, . . . , ym], for yi = d2ri (xi). Note that the characteristic ≠ 2 condition cited in [McC01] is not strictly necessary: see Theorem 6.21 in [McC01]. ∞ Condition (1) is immediate from the contractibility of Vn(C ) and condition (2) follows from Lemma 1. Condition (3) C∞ also follows from the contractibility of Vn( ); if any of the generators xi did not transgress, then the kernel of d2ri would

be non-trivial and would hence yield an element on the E2ri+1 page surviving to the E∞ page. Finally, we may argue from the universal coefficient theorem to lift the generators yi from field coefficients to integer coefficients. ■

4 The proof of Theorem 2 involves ‘comparing’ the Serre spectral sequence to an algebraically constructed spectral se- ∞ quence. To illustrate the idea, we shall consider the special case n = 1 where we have the fibre bundle S1 → S∞ → CP and the Serre spectral sequence has E2 page given by

2 x xy xy2 xy3 1 y y2 y3 0 0 1 2 3 4 5 6

0,1 ∼ 1 1 Z ∼ Z{ } To derive this, we begin with the module E2 = H (S ; ) = x . Since the only non-zero differential mapping in or 0,1 2,0 ∼ Z{ } out of E2 is d2, we must have E2 = y for some class y, with d2(x) = y, in order for the E∞ page to contain no ∞ nontrivial classes in positive degree. This yields us a class y ∈ H2(CP ; Z) and hence there must also exist a non-trivial ⊗ ∈ 1,2 ′ ∈ 4,0 class x y E2 . Repeating the same argument as before, we find that there must be a non-trivial class y E2 ′ ′ 2 such that d2(xy) = y . But by the derivation property of d2, we have y = d2(xy) = d2(x)y − xd2(y) = y , yielding ∞ ∞ another class, in H4(CP ; Z). Continuing in a similar manner tells us that H∗(CP ; Z) is indeed isomorphic to Z[y] and Theorem 1 (c) tells us that this is actually an isomorphism of rings. ■

Now we may define the Chern classes: 2i ∗ DEFINITION. Given a rank n complex vector bundle E → B, the ith Chern class ci(E) ∈ H (B; Z) is defined to be f (ci) for ∞ 0 ≤ i ≤ n and 0 otherwise, where f : B → Gn(C ) is the classifying map for E. Using this description, the characteristic properties of the Chern classes are apparent, with the exception of the sum formula, which we prove in §4 for the Stiefel-Whitney classes.

3 THE EULER CLASS

We begin by deriving a number of basic results about the cohomology of fibre bundles using the Serre spectral sequence.

THEOREM. (Leray-Hirsch) Suppose F → E → B is a fibre bundle as in §1 and suppose there are cohomology classes c1, . . . , cn ∈ ∗ ∗ ∼ ∗ ∗ H (E; R) that form a basis for the cohomology of each fibre upon restriction. Then H (E; R) = H (B; R) ⊗R H (F ; R).

Proof. We claim that the Serre spectral sequence for F → E → B collapses on the E2 page, so that, schematically, the E2 page looks like

0 0 0 0

5 p,q ∼ p ⊗ q with all differentials zero. Then because E2 = H (B; R) R H (B; R) by Theorem 1 (a), we must then have ∗ ∼ ∗ ∗ H (E; R) = H (B; R) ⊗R H (F ; R) by Theorem 1 (b), giving the result. To see that the spectral sequence collapses, we begin by making some simple observations: 0,q • With the filtration from Theorem 1 (c), we have E∞ = F 0Hq(E; R)/F 1Hq(E; R), so that there are quotient 0,q maps Hq(E; R) → E∞ . 0,q 0,q 0,q • Since the Er terms are on the y-axis of the first-quadrant spectral sequence, every Er+1 is a submodule of Er because all of the incoming differentials are zero. 0,q 0,q 0,q ≥ • We must certainly have that E∞ = Eq+1, because all differentials in and out of Er for r q + 1 must go to the zero module. 0,q Collecting these observations and using a naturality argument to show that the resulting map Hq(E; R) → E∞ = 0,q ⊆ 0,q ⊆ · · · ⊆ 0,q ⊆ 0,q ∼ q ∗ q → q Eq+1 Eq E3 E2 = H (F ; R) obtained by composition is exactly i : H (E; R) H (F ; R) ([McC01], Theorem 5.9), we see that if i∗ is a surjection, as is the case under the hypotheses of the Theorem, then all of the inclusions in the previous sequence must actually be equalities. Thus the kernel of each dr for r ≥ 2 on the y-axis must be the entire module, and hence they must all be zero. Since the differentials dr are derivations, it follows that they are actually zero on the entire E2 page, since they are certainly zero on the x-axis as well. ■ THEOREM. (Gysin Sequence) Suppose Sn−1 → E → B is a fibre bundle as above. Then we have a long exact sequence

· ∗ · · · −→ Hp(B; R) −→e Hn+p(B; R) −→π Hn+p(E; R) −→ Hp+1(B; R) −→· · · for some cohomology class e ∈ Hn(B; R) the Euler class.

p,q p ⊗ q Proof. The Serre spectral sequence for this fibre bundle has E2 page E2 = H (B; R) R H (F ; R), which is given by p p H (B; R)⊗R R{1} when q = 0, by H (B; R)⊗R R{h} when q = n−1, and is zero otherwise, where h is the generator of Hn−1(Sn−1; R). Schematically, the spectral sequence looks like:

n − 1

Since the differential dr lowers the vertical degree by r − 1, the first non-zero differentials will be the dn differentials p,n−1 p+n,0 from En to En , called the transgression. Because these are the first non-zero differentials, we will in fact have p,q ∼ p,q ∼ ··· ∼ p,q E2 = E3 = = En . Since all subsequent differentials must be zero, the E∞ page will be the same as the En+1 p,0 ∼ p,0 p,n−1 ∼ page. Hence we will have E∞ = E2 /im dn and E∞ = ker dn, which we may organise into an exact sequence:

−→ p,n−1 −→ p,n−1 −→dn p+n,0 −→ p+n,0 −→ 0 E∞ E2 E2 E∞ 0

Now we consider the filtration on H∗(E; R). The only non-trivial filtration quotients will be

n−1 p+n−1 n p+n−1 p,n−1 F H (E; R)/F H (E; R) = E∞

6 − − − p+n,0 and F p+n 1Hp+n 1(E; R)/F p+nHp+n 1(E; R) = E∞ ; therefore we get a short exact sequence

p+n−1,0 p+n−1 p,n−1 0 −→ E∞ −→ H (E; R) −→ E∞ −→ 0

Using this short exact sequence to connect copies of the previous exact sequence therefore yields a long exact sequence

p+n−1 p dn p+n p+n p+1 · · · −→ H (E; R) −→ H (B; R) ⊗R R{h} −→ H (B; R) ⊗R R{1} −→ H (E; R) −→ H (B; R) ⊗ R{h} −→· · · which is of the same form as the Gysin sequence. To relate the differential map dn to the Euler class, define dn(1 ⊗ h) = e ⊗ 1. Since dn(x ⊗ 1) = 0 we can use the derivation property to compute that actually

dn(x ⊗ h) = dn((1 ⊗ h) · (x ⊗ 1)) = dn(1 ⊗ h) · (x ⊗ 1) + (1 ⊗ h) · dn(x ⊗ 1) = (e ⊗ 1) · (x ⊗ 1) = (ex) ⊗ 1 up to some signs, and that therefore, under the identifications of Hp(B; R) ⊗ R{h} with Hp(B; R) and Hp+n(B; R) ⊗ R{1} with Hp+n(B; R), the differential is indeed given by the cup product with the class e. ■

We take a brief detour to show how the Thom isomorphism may now be easily deduced from this result: COROLLARY. (Thom Isomorphism) Suppose Rn → E → B is a vector bundle with unit sphere bundle Sn−1 → S(E) → B satisfying the requirements of §1. Then there exists a class c ∈ Hn(D(E),S(E)) called the Thom class such that the map Φ: Hi(B) → Hi+n(D(E),S(E)) given by Φ(b) = p∗(b) ⌣ c is an isomorphism. Proof. We compare the long exact sequence in cohomology for the pair (D(E),S(E)) to the Gysin sequence for the sphere bundle Sn−1 → S(E) → B to yield the diagram

j∗ ··· Hi−1(D(E)) Hi−1(S(E)) Hi(D(E),S(E)) Hi(D(E)) Hi(S(E)) ···

p∗ = p∗ = p∗ ··· Hi−1(B) Hi−1(S(E)) Hi−n(B) ⌣e Hi(B) Hi(S(E)) ··· where the small squares may easily be seen to commute and where the vertical maps are all isomorphisms ([Hat02], p. 444). Taking i = n above, we have the diagram j∗ Hn(D(E),S(E)) Hn(D(E)) Hn(S(E))

p∗ = H0(B) ⌣e Hn(B) Hn(S(E))

where 1 ∈ H0(B) maps to e ∈ Hn(B). Then p∗(e) must map to 0 in Hn(S(E)) by exactness of the lower row, and hence p∗(e) = j∗(c) for some c ∈ Hn(D(E),S(E)) by exactness of the upper row. With this definition of c it is not difficult to see that by defining Φ: Hi(B) → Hi+n(D(E),S(E)) as in the statement of the Theorem, the diagram

j∗ ··· Hi−1(D(E)) Hi−1(S(E)) Hi(D(E),S(E)) Hi(D(E)) Hi(S(E)) ···

∗ ∗ p = Φ p = p∗ ··· Hi−1(B) Hi−1(S(E)) Hi−n(B) ⌣e Hi(B) Hi(S(E)) ··· must commute in all squares. The five lemma then completes the proof. ■ Remark. It is not difficult to show using Leray-Hirsch that the classes e and c are unique up to sign, depending on the choice of orientation of the vector bundle E. Also, we can observe from the construction in Theorem 7 that if E arises as the unit sphere bundle of a vector bundle with odd rank, then the Euler class will have order 2 ([Hat09],3.13(d)).

4 STIEFEL-WHITNEY AND PONTRJAGIN CLASSES

We first consider the simpler case of Z2 coefficients:

7 ∞ ∗ ∞ ∼ THEOREM. The cohomology of the Grassmannian Gn(R ) is given by H (Gn(R ); Z2) = Z2[w1, . . . , wn] where |wi| = i. In the proof of this fact one could largely follow the proof of the complex case, starting by deducing the cohomology of ∞ O(n) and then deducing the cohomology of Gn(R ) using a similar comparison argument to Theorem 2. The details in this case are more complicated; we refer the reader to [McC01] for this approach. Proceeding as before, we can now define a collection of characteristic classes: i DEFINITION. Given a rank n real vector bundle E → B, the ith Stiefel-Whitney class wi(E) ∈ H (B; Z2) is defined to be ∗ ∞ f (wi) for 0 ≤ i ≤ n and 0 otherwise, where f : B → Gn(R ) is the classifying map for E. Again, the proof of the characteristic properties is then simple, except for the sum formula:

THEOREM. (Whitney Sum Formula) If E1,E2 are real vector bundles, then w(E1 ⊕ E2) = w(E1)w(E2), where w = 1 + w1 + w2 + ... is the total Stiefel-Whitney class. Proof. We follow [McC01], Theorem 6.40 in using a Lemma ∞ ∼ ∞ ∞ LEMMA. Consider the map ϕ : Gn × Gm → Gn+m given by regarding R = R ⊕ R and mapping a pair of subspaces ⊕ ⊆ R∞ ⊕ R∞ ∗ ∗ Z → ∗ Z ⊗ ∗ Z (∑V,W ) to V W . Then the homomorphism ϕ : H (Gn+m; 2) H (Gn; 2) H (Gm; 2) takes wi to ⊗ r+s=i wr ws

Now suppose E1,E2 have classifying maps f : X → Gn and g : X → Gm respectively, and consider the pullback diagram: ∗ ∗ ∗ ∗ f En ⊕ g Em f En × g Em En × Em En+m

∆ f×g ϕ X X × X Gn × Gm Gn+m from which the result follows immediately by applying the Lemma and the Künneth Theorem. ■

The corresponding structure for Z coefficients is considerably more complicated. We devote the remainder of this section to proving ∗ ∞ ∼ THEOREM. We have H (Gn(R ); Z)/torsion = Z[p1, . . . , pk] for n = 2k or n = 2k +1 and classes pi of degree 4i. Moreover, all torsion is of order 2.

Proof. We argue by induction on n and begin by instead considering the oriented Grassmannian Gñ so that we may use the Serre spectral sequence with any system of coefficients; we will take R = Z[1/2] in order to remove all 2-torsion, and ∗ ∼ ∗ ∼ then claim that H (G˜2k+1; R) = R[p1, . . . , pk] and H (G˜2k; R) = R[p1, . . . , pk−1, e] for classes pi in degree 4i, and e in degree 2k. The sphere bundle S(Eñ) of the oriented canonical bundle π : Eñ → Gñ has a natural map p : S(Eñ) → ⊥ Gñ−1 given by sending a pair (V, v) of a subspace and a unit vector v ∈ V , to the orthogonal complement {v} ⊆ V in ∞ Gñ−1. This yields a fibre bundle p : S(Eñ) → Gñ−1 with fibre S and hence by the long exact sequence in homotopy, ∞ ∗ ∗ ∗ p is a weak homotopy equivalence because S is contractible. Therefore p : H (Gñ−1; R) → H (S(Eñ); R) is an ∗ ∗ isomorphism. Furthermore, we have an obvious splitting π Eñ = L ⊕ p Eñ−1 where L is the trivial line bundle on ˜ ∗ ˜C C ⊕ ∗ ˜C ∗ ˜C ∗ ˜C S(En). Complexifying, we see that π En = L p En−1 and hence π c2i(En ) = p c2i(En−1) by the analog of Theorem 9 for Chern classes. Now we consider the odd and even cases separately.

∗ When n = 2k is even, we shall assume by induction that H (Gñ−1; R) is generated as an algebra by the classes ˜C ∗ ∗ ˜ → ∗ ˜ c2i(En−1), and so, by the above discussion, we see that π : H (Gn; R) H (S(En); R) must be surjective. Hence the n−1 Serre spectral sequence for the fibre bundle S → S(Eñ−1) → Gñ schematically takes the form

8 n − 1 h

for some unknown classes along the x-axis (localisation of modules commutes with tensor products). Each map π∗ : q q H (G˜n; R) → H (S(E˜n); R) is surjective and may be identified with the obvious map (see [McC01], Theorem 5.9): q ˜ q,0 ↠ q,0 ↠ ··· ↠ q,0 ↠ q,0 q,0 → q ˜ H (Gn; R) = E2 E3 Eq Eq+1 = E∞ , H (S(En); R)

q,0 q Because this composition is surjective, we must hence have E∞ = H (S(Eñ); R). Therefore only the classes along the x-axis of the Serre spectral sequence can survive to the E∞ page and hence every class of the form h ⊗ x for non-zero ∗ ∗ ∼ ∗ x ∈ H (Gñ; R) must transgress. It follows immediately that H (Gñ; R) = H (Gñ−1; R) ⊗R R[dn(h ⊗ 1)] as algebras using Theorem 1 (a), along with the multiplicative structure to show that dn(h⊗x) = xdn(h). Moreover, by the argument in the proof of Theorem 7, we may identify the transgression dn(h ⊗ 1) with the Euler class e of S(Eñ).

Now we breify sketch the odd case n = 2k + 1, which is rather more subtle. By Remark 2, the Euler class e(Eñ) has order 2 with Z coefficients and hence must be 0 with R = Z[1/2] coefficients. Identifying the transgression with · ⌣ e, n−1 we see that the Serre spectral sequence for S → S(Eñ−1) → Gñ collapses on the E2 page, giving, schematically:

n − 1 h

0 0

∗ ∗ Using Theorem 1, we can deduce from this diagram that H (Gñ; R) can be identified with a subring of H (Gñ−1; R), ∗ consisting of those classes not containing terms with odd multiples of e ∈ H (Gñ−1; R), identified here with the class h ˜C using the previous case in the induction. Moreover, under this identification we observe that c2i(En ) are indeed algebra generators; the only tricky part is the identification of the squared Euler class e2 with the top degree Chern class of the complexified bundle, which can be proved using an identical Serre spectral sequence argument to the above, but in the complex case ([Hat09], 3.15(b)). This completes the last step in the induction argument.

9 Now we wish to recover information about the unoriented Grassmannian, which we can do by considering the two- sheeted covering space Gñ → Gn and applying some basic results on transfer homomorphisms; Proposition 3G.1 from [Hat02] ∗ implies that, when the coefficient ring is R = Z[1/2], the algebra H (Gn; R) will be isomorphic to the subalgebra of classes ∗ in H (Gñ; R) that are left invariant under a reversal of orientation of Gñ. By Remark 2, we see that the Euler class e ˜C changes sign under a reversal of orientation, but the classes pi above are equal to c2i(En ) (up to sign) and hence must be ∗ ∼ invariant. Hence H (Gñ; R) = R[p1, . . . , pk] for n = 2k, 2k + 1, as claimed.

Because any p-torsion for p ≠ 2 will actually survive in cohomology with coefficients R = Z[1/2], it follows that there ∗ is none in H (Gn; Z) and hence it remains to show that all 2-torsion has order 2; this is a fairly trivial application of ∞ the Bockstein spectral sequence. First we determine the form of the Bockstein spectral sequence for RP ; we know that ∗ ∗ RP∞ Z ∼ Z ∗ RP∞ Z ∼ Z E1 = H ( ; 2) = 2[α] for some class α in degree 1, and also that H ( ; ) = [x]/(2x) for some x in degree ∞ 2. Because every class in Hi(RP ; Z), i > 0 has order 2, the Bockstein spectral sequence must collapse (in positive degrees) on the second page and by Theorem 3 we see that the first page takes the form:

0 1 2 0 3 1 4 0 0 → Z2{1} −→ Z2{α} −→ Z2{α } −→ Z2{α } −→ Z2{α } −→· · · where the Z2 in degree 0 surviving to the ∞ page corresponds to the Z summand in degree 0 of the integral cohomology. ∞ ×n To deduce the Bockstein spectral sequence of Gn, we use the embedding f :(RP ) → Gn given by regarding ∞ ∼ ∞ ∞ ∞ R = R ⊕R ⊕· · ·⊕R and sending a collection of lines to the corresponding n-dimensional subspace they span. By ∗ ∞ ×n ∼ the Künneth Theorem, we have H ((RP ) ; Z2) = Z2[α1, . . . , αn] and it is not hard to see [Hat09] using the Whitney ∗ sum lemma above that the induced map f on Z2 cohomology sends wi to σi(α1, . . . , αn), the ith elementary symmetric polynomial in the classes α1, . . . , αn. By the naturality of the Bockstein spectral sequence, we may compute β1 instead on ∗ ∞ ×n the cohomology H ((RP ) ; Z2). Using the derivation property (see Remark 1) of the Bockstein homomorphism β1 as ∞ well as the first page of the Bockstein spectral sequence for RP found above, we can see that β1(σk) = σ1σk+(k+1)σk+1, and hence that β1(w2i+1) = w1w2i+1 and β1w2i = w2i+1 + w1w2i.

We may easily find the image of the integral classes in the Z2 cohomology, since as we have claimed in the course of ˜C the argument above, pi = c2i(En ), and since Chern classes map to the Stiefel-Whitney classes under the coefficient ≡ ˜C ≡ ˜C ˜ ⊕ ˜ ˜ 2 homomorphism ρ ([Hat09], Proposition 3.8), we must have pi (En ) w4i(En ) = w4i(En En) = w2i(En) Z ∗ Z Z 2 2 mod 2. Hence the image of the classes in H (Gn; 2) must be exactly 2[w2, . . . , w2k]. Now by Theorem 3 in §1 it will suffice to show that these classes are the only ones surviving on the second page of the Bockstein spectral sequence. It is convenient to instead consider the Z2 cohomology as generated by the classes w1, w2, βw2, . . . , w2k, βw2k in the ∗ case where n = 2k + 1 is odd, and by w1, w2, βw2, . . . , w2k for n = 2k even. Then it is not hard to see that the E1 complex decomposes as a tensor product chain complex; in the odd case, as the tensor product of Z2[w1] and factors of the form Z2[w2i, βw2i], and in the even case as the tensor product of Z2[w1, w2k] with Z2[w2i, βw2i]. By the general Künneth Theorem from homological algebra (see [Hat02], Theorem 3B.5), since we are working over the field Z2 where all the Tor terms will vanish, we see that the Bockstein cohomology will be given by the tensor product of the Bockstein cohomologies of the tensor components, which may be computed as follows.

∞ For the complex of the form Z2[w1], the computation is exactly the same as for RP shown above, and hence we have no Bockstein cohomology in positive degrees. For the various Z2[w2i, βw2i] factors, we will have differentials β1 given by ℓ m ℓ−1 m+1 β1(w2i(βw2i) ) = ℓw2i (βw2i) using the derivation property. This can be displayed as

10 5

βw2i 1

w2i 0 0 1 2 3 4 5

where the non-zero Bockstein differentials are pictured as thick arrows. From this, it is easy to see that the only surviving 2 4 6 Z classes will be 1, w2i, w2i, w2i,... in even degrees along the x-axis. Finally, for the 2[w1, w2k] case, we will have βw2k = ℓ m ℓ+1 m w1w2k and hence β(w1w2k) = (ℓ + m)w1 w2k. Drawing a similar diagram to above:

w2k 1

w1 0 0 1 2 3 4 5

2 4 and from this we can see that the only surviving classes are those on the y-axis, of the form 1, w2k, w2k,... . This completes the proof. ■ ∗ ∞ Remark. Note that with a bit more effort, we can use this to deduce some information about the 2-torsion of H (Gn(R ); Z). Note also that this proof effectively ‘cheats’ by using the Chern classes constructed earlier to circumvent the most difficult part of the proof, proving the surjectivity of the map π∗. One might hope that some proof along the lines of the one given above would apply to the Stiefel-Whitney classes.

These cohomology classes pi now give rise to a series of characteristic classes, the Pontrjagin classes, and we know that these, along with the Stiefel-Whitney and Chern classes, must exhaust the set of possible characteristic classes.

REFERENCES

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