Some Background Material

Some background material

Steve Mitchell September 24, 2011

This is background material for my topics course “Equivariant cohomology of ﬁnite group actions”. Much of it will be discussed also in the lectures.

1 Topological interpretation of group cohomology

Let G be a discrete group, M a G-module. Then group cohomology with coeﬃcients in M is by deﬁnition

H∗(G; M) = Ext∗ ( ,M). ZG Z Here the Ext groups can be computed using either an injective resolution of M, or a projective resolution of the trivial module Z. Using the former version, we see that the group cohomology is given by the right derived functors of the invariants functor M 7→ M G. On the other hand, there is the following topological interpretation in terms of the classifying space BG:

Theorem 1.1 Let G be a discrete group. Then H∗(G; M) ∼= H∗(BG; M), where M is the local coeﬃcient system associated to M.

We remark that there are various definitions of cohomology with local coefficients, but the simplest goes like this: Let X be a path-connected space with a universal cover X˜. Then a local coefficient system is just a module M over the fundamental group π1X, and we define

∗ ∗ ˜ H (X; M) = H (HomZπ1X (C∗X,M)).

Here C∗ denotes the singular chain complex. With this definition the theorem is almost a tautology, since for discrete groups the universal cover of BG is the contractible space EG, whose singular chain complex therefore gives a free resolution of Z. If M is a trivial G-module (meaning the action of G on M is trivial), we just get ordinary cohomology with coefficients in M. This is the case we are primarily interested in, with M = Fp. This topological interpretation is very powerful. Among other things, it allows us to bring compact Lie groups and their classifying spaces into the picture. The advantage of this is that for classical groups such as G = U(n) or G = O(n), the cohomology of BG has a much simpler structure than that of a typical finite group, and there are functo- rial generators: Chern classes, Stiefel-Whitney classes, Pontrjagin classes. In particular

1 we can define Chern/Stiefel-Whitney/Pontrjagin classes of a representation as follows: If θ : G−→U(n) is a complex representation of G, ck(θ) = ck(ξ), where ξ is the vector bundle defined by Bθ : BG−→U(n). Explicitly, ξ can be realized as the Borel construction n EG ×G C . Stiefel-Whitney/Pontrjagin classes are defined similarly, using real representations. This construction is especially useful for computing induced maps. Suppose we have a homomorphism φ : G−→H and want to compute φ∗ : H∗BH−→H∗BG. If α ∈ H∗BG ∗ has the form α = ck(θ)) for some representation θ, then φ α = ck(θ ◦ φ). We also recall one crucial distinction that must be kept in mind: Let EG denote the universal principal G-bundle over BG as usual, so EG is contractible. If G is discrete, then EG is the universal cover and BG is also known as a K(G, 1)-space, i.e. π1BG = G and πnBG = 0 for n > 1. But if G is not discrete, say G is a compact Lie group of positive dimension, then EG is certainly not the universal cover and BG is not a K(G, 1). Furthermore, we no longer have the purely algebraic description of H∗BG as an Ext group. Some basic facts about BG (G a compact Lie group, although all of these results hold much more generally):

Proposition 1.2 Suppose θ : G−→G is an inner automorphism. Then Bθ : BG−→BG is homotopic to the identity. In particular, Bθ∗ is the identity on H∗.

Corollary 1.3 Let H be a closed subgroup of G. Then the conjugation action of NGH on ∗ ∗ ∗ H BH factors through W = NGH/H, and the image of H BG−→H BH lies in the ring of invariants (H ∗ BH)W .

Proposition 1.4 If H is a closed subgroup of G, then there is a ﬁber sequence

G/H−→BH−→BG, where BH−→BG is induced by the inclusion H ⊂ G. If H is normal in G, inclusion H ⊂ G and projection G−→G/H induce a ﬁber sequence

BH−→BG−→B(G/H).

Here A−→B−→C is a ﬁber sequence if there is a homotopy-commutative diagram

ABC- -

? ? ? - - XYZq such that the vertical maps are weak equivalences and q is a Serre fibration with fiber X, or an analogous diagram with the vertical arrows reversed. Thus we can treat A−→B−→C as though B−→C was a fibration with fiber A; in particular we have a Serre spectral sequence H∗(C; H∗A) ⇒ H∗B. In the first case of the proposition, the spectral sequence for G finite is uninteresting; by Shapiro’s lemma it degenerates to the identity H∗H = H∗H. But it will be very useful in the case G is a connected Lie group such as U(n). When G is finite, the spectral sequence associated to the second fiber sequence of the proposition is the same as

2 the Hochschild-Serre spectral sequence of the group extension H−→G−→G/H. The latter can be constructed by pure homological algebra, most elegantly by using Grothendieck’s composite functor spectral sequence (the G-invariants functor M−→M G factors as M 7→ M H 7→ (M H )G/H ).

2 The transfer

Throughout this section G is a ﬁnite group, H is a subgroup of G, and M is a G-module. We have a restriction homomorphism on group cohomology

i(H,G)∗ : H∗(G; M)−→H∗(H; M). In topological terms, i∗ is induced by the map on classifying spaces BH−→BG. The transfer is a homomorphism going the other way:

τ(H,G): H∗(H; M)−→H∗(G; M). Thinking of group cohomology as derived functors of the invariants functor, the transfer is H G P induced by the obvious abelian group homomorphism M −→M given by m 7→ g∈G/H gm. Here the notation g ∈ G/H is short for “take a set of coset representatives”; the choice of H representatives is immaterial since m ∈ M . If we use a projective resolution P·Z to compute group cohomology, τ is induced by the chain map

H G HomZH (P·,M) = (HomZ(P·,M)) −→(HomZ(P·,M)) = HomZG(P·,M). P again using the operator g∈G/H g. The following key fact is then obvious:

Proposition 2.1 τ(H,G) ◦ i∗(H,G) is multiplication by [G : H].

Taking H to be the trivial group, we obtain:

n n Corollary 2.2 |G| · H (G; M) = 0 for all n > 0. In particular, H (G; Q) = 0 for all n > 0.

Since the rational cohomology is trivial, we will focus on mod p cohomology for p prime.

∗ ∗ Corollary 2.3 If [G : H] is prime to p, H (G; Fp)−→H (H; Fp) is injective. In particular, n (i) this is true for H the p-Sylow subgroup of G, and (ii) if |G| is prime to p then H (G; Fp) = 0 for all n > 0.

In the next corollary we again take Fp-coeﬃcients.

Corollary 2.4 Suppose i(H,G)∗ is onto and p divides [G : H]. Then τ(H,G) is identically zero. In particular, if G is a p-torus and H is a proper subgroup then τ(H,G) = 0.

The proposition itself is actually a corollary of the stronger assertion:

3 Proposition 2.5 Taking coeﬃcients in a commutative ring R, τ(H,G) is a map of H∗(G; R)- modules:

τ(ai∗b) = (τ(a))b.

Proof: We use the singular chain complex C∗EG as our free resolution of R. Using the standard cochain formula for the cup product, we see that the (non-commutative) diﬀerential graded algebra C∗EG is a G-algebra; i.e. g · (xy) = (g · x)(g · y). It is then immediate that the cochain level transfer (C∗EG)H −→(C∗EG)G deﬁned above is a map of both left and right C∗BG = (C∗EG)G-modules.

We turn next to the double coset formula, which concerns composition in the reverse order: transfer followed by restriction. It looks somewhat complicated at ﬁrst glance, but in fact it is easy to prove and easy to use. Although there is a more conceptual explanation of the formula that we’ll give later, the explicit form given here is very useful. Suppose K,H ⊂ G, and consider the double cosets K\G/H, thought of as the K-orbits of the natural K-action on G/H.

Proposition 2.6

X −1 −1 −1 ∗ i(K,G) ◦ τ(H,G) = τ(K ∩ xHx ,K) ◦ i(K ∩ xHx , xHx ) ◦ cx−1 , x∈K\G/H where the sum is over a set of double-coset representatives.

Proof: We work on the cochain level, using injective resolutions. The lefthand side is just H τ G K P the composite I −→ I −→I , where τα = g∈G/H gα. A typical term on the right is given by the composite

IH −→x IxHx−1 ⊂ IK∩xHx−1 −→τx IK , P −1 where τx(α) = k kα and the sum is over k ∈ K/(K ∩ xHx ). Hence the composite is given by α 7→ P(kx)α. But K ∩ xHx−1 is the isotropy group in K of xH. So in this last sum kx is ranging over the elements of the orbit KxH/H. Since G/H is the disjoint union of the K-orbits, the result follows. P The case K = H is of particular interest. For any ﬁnite group L, let L = x∈L x ∈ FpL.

Proposition 2.7 Suppose H is normal in G. Then i(H,G)∗τ(H,G)(a) = G/Ha for all a ∈ H∗H.

Proof: H acts trivially on G/H, so this is immediate from the double coset formula.

Corollary 2.8 If H is normal in G and [G : H] is prime to p, then i∗ : H∗G−→H∗H is an isomorphism onto the ring of invariants (H∗H)G/H .

4 In general, HxH = xH if and only if x ∈ NGH, so the ﬁxed points of the H-action on G/H correspond to the quotient group W = NGH/H. Hence we always have

i(H,G) ◦ τ(H,G)(a) = W a + E, where the term E corresponds to double cosets HxH such that xHx−1 6= H. Here is one ad hoc but useful case where E = 0 even for non-normal H:

Proposition 2.9 Fix m > 0. Suppose H is a p-subgroup of G, and for every proper sub- conjugate xHx−1∩H ⊂ H the restriction i(xHx−1∩H,H)∗ on Hm is either zero or surjective. Then

i(H,G) ◦ τ(H,G)(a) = W a.

Proof: One easily checks that the other terms in the double-coset formula vanish.

Corollary 2.10 Suppose H is as in the proposition and [G : H] is prime to p. Then HmG−→HmH is an isomorphism onto the invariants (HmH)W .

Remark: The proposition can be useful even when p divides |W |, but if p divides CGH/H it is of no use since then W = 0 on H∗H.

As another application, suppose [G : H] is prime to p. Then the restriction map i∗ is injective and we wish to describe the image. Call α ∈ H∗H stable if for all x ∈ G,

−1 −1 −1 i(H,H ∩ xHx )α = i(xHx ,H ∩ xHx ) ◦ cxα.

Theorem 2.11 Stable element theorem. Suppose [G : H] is prime to p. Then α ∈ H∗H is in the image of i∗ if and only if α is stable.

Proof: Suppose α is stable and let β = τ(H,G)α. Take H = K in the double coset formula. Then

i∗β = X τ(H ∩ xHx−1,H) ◦ i(H,H ∩ xHx−1)α = X[H : H ∩ xHx−1]α = [G : H]α. x x Hence α = i∗([G : H]−1β. The converse is a trivial consequence of the fact that conjugation by x induces the identity map on H∗G.

Concluding Remarks: 1. In fact the transfer applies to inﬁnite groups as well; we only need the index of H in G to be ﬁnite. 2. There is no map of spaces BG−→BH inducing the transfer. This is clear because the transfer is never a ring homomorphism, except in the trivial case H = G, since τ(H,G) is multiplication by [G : H] on H0. However, there is a morphism of suspension spectra t :Σ∞BG−→Σ∞BH such that t∗ = τ(H,G). In particular, the transfer commutes with Steenrod operations.

5 3. In fact the transfer ﬁts into a much broader context of wrong-way maps arising from the Pontrjagin-Thom construction. In algebraic geometry the induced cohomology homomorphisms are usually called “Gysin” maps, the analog of Proposition 2.5 is called the “projection formula”, and the double-coset formula corresponds to “naturality”. This is not the place to go into details, but a brief comment on “naturality” is in order. The general context is a suitable pullback diagram

f AX-

π0 π ? ? - BYg

∗ 0 ∗ 0 Then naturality for cohomology can be written in the form g π∗ = π∗f , where π∗, π∗ are covariant transfer/Gysin homomorphisms. The double-coset formula then comes from the pullback square

- ∼ EK ×K (G/H) EG ×G (G/H) = BH

? ? BKBG-

Here the right vertical map is a covering map, while the left vertical map is the union of covering maps indexed by the double cosets K\G/H; hence it makes sense that “naturality” as given above leads to the double coset formula. More surprising is that the whole business can be extended to the case of compact Lie groups G. Here the natural Gysin homomorphism amounts to “integrating over the ﬁber”, and therefore doesn’t seem to work since it lowers degrees by dim G/H. But there is a modiﬁcation, the Becker-Gottlieb transfer, that yields a degree-preserving homomorphism ∞ ∞ and indeed is induced by a map of suspension spectra Σ (BG+)−→Σ (BH+). There is a double-coset formula in this setting too, but it takes some work to describe and we will not give it here.

3 Computations

3.1 Cyclic groups

Let Cd denote a cyclic group of order d, identiﬁed with the group of d-th roots of unity. Then there is a ﬁber sequence

1 ∼ 1 ∞ S = S /Cd−→BCd−→CP . This follows from the general fact xxx discussed above, or can be seen directly by observing ∞ 1 d ∞ that BCd = S /Cd is homeomorphic to the S -bundle of ⊗ λ, where λ ↓ CP is the tautologous line bundle. The Gysin sequence then yields immediately:

6 ∗ ∼ Proposition 3.1 H (BCd; Z) = Z[y]/(dy), where |y| = 2.

1 In fact y can be taken to be the ﬁrst Chern class of the representation Cd−→S . Next we consider mod p cohomology. Hence we may as well assume d = pk for some k.

∗ ∼ Proposition 3.2 a) If p is odd or p = 2 and k > 1, then H (BCpk ; Fp) = Fp[y] ⊗ Fphxi, where |y| = 2 and |x| = 1. ∗ ∼ If p = 2 and k = 1 then H (BC2) = F2[x], where |x| is the ﬁrst Stiefel-Whitney class of = the sign representation C2 −→ O(1). b) Inclusion Cpk ⊂ Cpk+1 induces an isomorphism on even-dimensional cohomology and the zero map on odd dimensional cohomology. 1 m c) Surjection Cpk+1 −→Cpk induces an isomorphism on H and the zero map on H for m > 1.

The dichotomy between the two cases is encapsulated in the motto: “Four is an odd prime”. Part (a) of the proposition follows immediately from the previous one, except for deciding when x2 = 0. Postponing this last item, one can then easily verify b-c. Finally, consider x2. If p is odd then x2 = 0 by anti-commutativity. If p = 2 and k = 1 then x2 6= 0 (this is part ∗ ∞ of the well-known calculation of H RP , which can be done using the Gysin sequence in mod 2 cohomology of the S0-bundle associated to the tautologous line bundle). If p = 2 and 2 k > 1, consider the short exact sequence 0−→C2−→C4−→C2 and deduce x = 0 from b-c.

3.2 p-tori and groups with p-toral Sylow subgroup ∼ n Let A be a p-torus of rank n. Thus A = (Z/p) , and hence by the Kunneth formula there is an isomorphism of algebras

∗ ∼ H A = Fp[y1, ..., yn] ⊗ Fphx1, ..., xni for p odd, while for p = 2

∗ ∼ H A = F2[x1, ..., xn].

Here |yi| = 2, |xi| = 1. We express these isomorphisms in coordinate-free terms as follows: 1 ∼ ∗ ∗ There is a natural isomorphism H A = A , where A denotes the Z/p-dual. Moreover, the 2 Bockstein β associated to the short exact coeﬃcient sequence 0−→Z/p−→Z/p −→Z/p−→0 deﬁnes an injection H1A−→H2A, and we have a natural isomorphism

H∗A ∼= S(βA∗) ⊗ E(A∗) for p odd,

H∗A ∼= S(A∗) for p = 2.

7 Proposition 3.3 Suppose A is a p-torus and Sylow subgroup of G, and let W = NGA/A. Then the restriction map i∗ is an isomorphism

∼ H∗G −→= (H∗A)W .

Proof: As noted earlier, i∗ is injective since [G : A] is prime to p, and maps into the W - invariants (since inner automorphisms act trivially on cohomology). Now if a ∈ (H∗A)W , ∗ then i τ(a) = |W | · a, since all terms in the double coset formula not coming from NGA vanish (Lemma xxx). Hence a = i∗(τ(a/|W |)), completing the proof.

∗ ∼ Example: Let Sp denote the symmetric group on p elements. For p odd, H Sp = Fp[a]⊗Fphbi, where |a| = 2p − 2 and |b| = 2p − 3. The proof is left as a highly-recommended exercise.

∗ d Example: H GL2Fq at the characteristic prime p. Let q = p be a prime power. As one would expect, the mod p cohomology and the mod ` cohomology, ` 6= p, behave quite diﬀerently. Here we are concerned with the mod p cohomology. First of all, as p-Sylow subgroup A we take the upper triangular unipotent subgroup

1 ∗ ! 0 1

Note that A indeed a p-torus, of rank d. The normalizer of A is the Borel subgroup B of all triangular matrices, while W can be identiﬁed with the diagonal matrices or equivalently × × −1 Fq × Fq . Moreover the action of W on A is given by (α, β) · a = αβ a. This yields:

Proposition 3.4 Let A denote the underlying additive group of Fq (a p-torus of rank d). × Let Fq act on A by scalar multiplication in Fq. Then

∼ × ∗ = ∗ Fq H GL2Fq −→ (H A) .

The ring of invariants on the right is computable, at least as a vector space, by tensoring × up to Fq in the coeﬃcients and diagonalizing the action of Fq . It is also a ﬁnitely-generated module over the ring of GLdFp-invariants. For an explicit example, consider q = 4. We write ∗ H of the 2-torus as F2[x, y].

Proposition 3.5

∗ ∼ 2 3 H (GL2F4; F2) = F2[D2,D3, z]/(z + D3z + D2) ⊂ F2[x, y],

3 2 3 2 2 where z = x + x y + y , and D2, D3 are the Dickson invariants D2 = x + xy + y , 2 2 D3 = x y + xy .

The proof is not diﬃcult, although somewhat lengthy, and is left to the interested reader. Notice, once more, that the Krull dimension of the ring is 2, i.e., the rank of a maximal 2-torus.

8 3.3 Dihedral groups

All cohomology groups in this section have coeﬃcients F2. n We write Dn for the dihedral group of order 2 , n ≥ 3, and view it as deﬁned by generators s, t with relations s2n−1 = 1 = t2, tst = s−1. The center is generated by r = s2n−2 , while the 0 2 0 commutator subgroup Dn is generated by s ; thus Dn/Dn = C2 × C2 with basis s, t. Let a, b 1 denote the dual basis for H Dn, and let θ denote the standard representation of Dn in O(2). Then b = w1(θ). Finally, up to conjugacy there are two maximal 2-tori in Dn: K, with basis r, t, and T , with basis r, st.

∗ Proposition 3.6 H Dn = F2[a, w1, w2]/a(a + w1), where wi = wi(θ).

∗ 2 Lemma 3.7 iK maps a, w1, w2 to 0, y, x + xy respectively, where x, y is the dual basis to r, t; ∗ 2 iT maps a, w1, w2 to y, y, x + xy respectively, where x, y is the dual basis to r, st.

The proof is by easy direct checking; for w2 consider how the representation θ restricts to K,T .

Proof of proposition: Consider the spectral sequence of the ﬁber sequence

1 S = O(2)/Dn−→BDn−→BO(2). One could view this as a Gysin sequence, but I want to illustrate some spectral sequence 1 1 1 2 ideas. Since H S = Z/2, the local coefficient system is trivial. Since H Dn = (Z/2) , the 0,1 2,0 ∗ differential d2 : E2 −→E2 is zero and the spectral sequence collapses. It follows that H Dn is a free module over H∗BO(2) on 1, a (because this is true for the associated graded object E∞). 2 2 Now in E∞ we have a = 0, but this only tells us that a = 0 modulo elements of higher 2 2 filtration. Without further information, all we can say is that a = jaw1 + kw1 for some j, k ∈ Z/2. Restricting to K,T and using the lemma, however, shows that k = 0 and j = 1. This completes the proof.

Remark: The spectral sequence used above was chosen because it fits so well with the Stiefel-Whitney classes of the standard representation. But there are several other group extensions one could use, staying within the realm of finite groups. Taking the dihedral group D3 of order 8 for simplicity, there are three obvious extensions C4−→D3−→C2, C2−→D3−→C2 × C2, and C2 × C2−→D3−→C2. It’s an interesting exercise to determine the behaviour of the corresponding spectral sequences (note that the third one has a nontrivial local coefficient system).

∗ Remark: Using the lemma again, we see that K,T form a “detecting family” for H Dn; in ∗ ∗ ∗ ∗ other words, the restriction map j : H Dn−→H K × H T is injective. With an eye toward ∗ later developments, note that the Krull dimension of H Dn is 2, i.e. equals the rank of a maximal 2-torus, there are no nilpotent elements, and there are two minimal prime ideals ∗ ∗ in H Dn. In geometrical terms, the aﬃne variety V deﬁned by H Dn is the union of two planes in 3-space intersecting a line, and the map j∗ is induced by the quotient morphism from the disjoint union of two planes to V .

9 3.4 Generalized quaternion groups

n+1 Let Qn+1 denote the generalized quaternion group of order 2 , n ≥ 2. It has generators s, t with relations s2n−1 = t2, t4 = 1, tst−1 = s−1, and there is an obvious faithful representation 3 2 Ψ: Qn+1 ⊂ S in the unit quaternions. The center is generated by t and there is a central extension

π C2−→Qn+1 −→ Dn, where the s, t notation is compatible with that of the previous section. When n = 2, D2 is 0 ∼ to be interpreted as C2 × C2. The abelianization Qn+1/Qn+1 = C2 × C2 with basis s, t; hence 1 H Qn+1 has basis a, b where again a, b correspond to the notation of the previous section, abusing notation so that π∗a = a and so on.

∗ ∼ Proposition 3.8 H Qn+1 = F2[a, b, w4]/R, where w4 = w4(Ψ) and

( (a2 + ab + b2, a2b + ab2) if n = 2 R = (a2 + ab, b3) if n > 2

3 Here w4(Ψ) really means w4 of the real representation Qn+1 ⊂ S ⊂ O(4). In the proof we will see an instructive example of a spectral sequence in which the differentials are non-trivial, but easy to compute because the cohomology of the total space is known in ∗ 3 ∼ advance. First of all we have H BS = F2[z], where z is the mod 2 reduction of the Chern class c2 of the standard 2-dimensional complex representation arising from the identification S3 = SU(2). Since the even Chern classes reduce mod 2 to the corresponding Stiefel-Whitney 3 classes of the underlying real representation, z is also w4 of S ⊂ O(4). Second, recall ∗ ∼ that H BSO(3) = F2[w2, w3], where w2, w3 are the Stiefel-Whitney classes of the universal ∞ 3 bundle. Now consider the fiber sequence RP −→BS −→BSO(3) induced by the central 3 ρ group extension C2−→S −→ SO(3); here we recall that ρ is the adjoint representation, or equivalently the action on the span of i, j, k induced by conjugation in the quaternions. 1 ∞ Finally, let x ∈ H RP be the generator.

∞ i 3 ρ Lemma 3.9 In the spectral sequence of RP −→ BS −→ BSO(3), we have d2x = w2 and 2 ∗ 4 d3x = w3. The spectral sequence collapses at E4 and i is an isomorphism onto F2[x ].

i 3 Proof: Since H BS = 0 for 0 < i < 4, the diﬀerential d2x = w2 is forced. Since the 2 2 diﬀerentials are derivations, x survives to E3 and we have E3 = F2[x ] ⊗ F2[w3]. Again 2 4 d3x = w3 is forced, x survives to E4 and the result follows.

We now prove the proposition. There is a commutative diagram of representations

Ψ - 3 Qn+1 S

π ρ

? ? - Dn SO(3) ζ

10 where ζ = θ⊕det θ and the vertical maps are central extensions with kernel C2. This induces a pullback diagram of ﬁbrations

- 3 BQn+1 BS

? ? - BDn BSO(3)

∞ 2 where the vertical arrows have fiber RP . An easy check shows that w2(ζ) = w2(θ) + b 2 and w3(ζ) = w2(θ)b (where w2(θ) is replaced throughout by a + ab when n = 2). Since the differentials in the Serre spectral sequence are natural with respect to pullbacks, we conclude 4 2 that in the spectral sequence of the lefthand fibration, x is a permanent cycle, d2x = w2 +b , 2 2 d3x = w2b, and it collapses at E4. The proposition now follows easily, noting that w2 + b and w2b are not zero divisors in the E2 and E3-terms, respectively.

∗ Remark: Note that the nilradical of H Qn+1 is (a, b), which in turn is exactly the kernel of the restriction to the unique maximal 2-torus C2. Note also that the Krull dimension of H ∗ Qn+1 is one, i.e. the rank of the maximal 2-torus. Finally note the periodicity of period 4 given by multiplication by w4.

Exercise: a) Every subgroup K ⊂ Qn+1 is either cyclic or generalized quaternion. 4 3 b) If K is nontrivial then H (iK ) is an isomorphism; if K 6= Qn+1 then H (iK ) = 0.

The next proposition shows that groups with generalized quaternion 2-Sylow subgroup have periodic mod 2 cohomology.

Proposition 3.10 Suppose G has 2-Sylow subgroup H isomorphic to some Qn+1. Then ∗ 0 2 iH is an isomorphism onto a free F2[w4] submodule containing 1 ∈ H and x = ab , the generator of H3.

Proof: This follows from Proposition xxx and the previous exercise.

3.4.1 The semi-dihedral group The purpose of this section is to show how generalized quaternion groups embed in semi- dihedral groups, a technical point that will be needed in the next section. ∼ Consider a cyclic group C = C2n+1 , n ≥ 2. Then Aut C = C2 ×C2n−1 . In particular, there are three distinct automorphisms of order 2, given by x 7→ x−1, x 7→ x2n−1, and x 7→ x1+2n , and hence three ways to make a semi-direct product/split extension C−→G−→C2. The n+2 second option deﬁnes the semi-dihedral group SDn+2 of order 2 . It has a presentation with generators x, y and relations x2n+1 = 1 = y2, yx = x2n−1y. Thus there is a homomorphism : SDn+2−→C2 deﬁned by (x) = −1 = (y) (writing C2 = ±1).

11 Lemma 3.11 The kernel of is isomorphic to Qn+1.

Proof: This is easy on inspection: The kernel is generated by x2, xy, and xy has order 4, acting by conjugation on x2 by x2 7→ x−2.

3.5 Mod 2 cohomology of SL2Fq ∗ Let q be an odd prime power. To compute H SL2Fq (where mod 2 coeﬃcients are understood 2 throughout), we ﬁrst determine the 2-Sylow subgroups. Recall that |SL2Fq| = q(q − 1). Let νpn denote the exponent of p in the prime factorization of an integer n.

Lemma 3.12 The 2-Sylow subgroups of SL2Fq are generalized quaternion groups Qn+1, 2 where n + 1 = ν2(q − 1).

Proof: For q = 1 mod 4, let H be the subgroup generated by (i) diagonal matrices of deter- minant 1 and 2-power order, and (ii) the order 4 matrix

0 −1 ! 1 0

Then H is generalized quaternion and has the correct order, hence is a 2-Sylow subgroup. If q = 3 mod 4, we will first show that the 2-Sylow subgroups of GL2Fq are semi-dihedral.√ 2 Identify Fq with Fq2 , and note the latter field is obtained from Fq by adjoining −1. The × action of the group of units defines an inclusion Fq2 ⊂ GL2Fq, and in particular we get a n+1 cyclic subgroup A of order 2 . Let H denote the subgroup of GL2Fq generated by A and the Frobenius σ. Then H is a 2-Sylow subgroup (since it has the correct order), and fits into a split extension

A−→H−→C2, with σ acting on A by σ(a) = aq. Since q = 2n − 1 mod 2n+1 by assumption, we see that H n+2 is semi-dihedral of order 2 as claimed. √ Now consider the restriction of det to H. We have det σ = −1 (use the basis 1, −1), and if a ∈ A then det a = a · σ(a) = aq+1. Hence if a is a generator, det a = −1. In otherwords, det corresponds to the homomorphism in Lemma 3.11, with kernel Qn+1. This completes the proof.

Theorem 3.13 Let q be an odd prime power. Then

∗ ∼ H (SL2Fq; F2) = F2[y] ⊗ F2hxi, where |y| = 4, |x| = 3.

12 Proof: In view of Proposition 3.10 and Lemma 3.12, it only remains to show that H1 = 2 H = 0. If K is any ﬁeld, an exercise in linear algebra shows that SL2K is generated by the upper and lower triangular unipotent matrices. When K = Fq these elements have order p, 1 and since p is odd we conclude that H SL2Fq = Hom (SL2Fq, Z/2) = 0. n Now let An denote the 2-component of the ﬁnite abelian group H (SL2Fq; Z), and consider the short exact sequences

n 0−→An/2−→H SL2Fq−→An+1[2]−→0, where for an abelian group A, A[2] = {a ∈ A : 2a = 0}. Taking n = 1 shows that A2[2] = 0 5 and hence A2 = 0. Since H SL2Fq = 0 by periodicity, it follows that A5/2 = 0 and hence A5 = 0. We deduce

2 ∼ A4/2 = Z/2 ⇒ A3/2 = 0 ⇒ A3 = 0 ⇒ H SL2Fq = A2/2 = 0.