Group cohomology, Steenrod operations and Massey higher products

Béatrice I. Chetard

December 9, 2016

Abstract We present detailed calculations of the cohomology ring of several small 2-groups, together with the action of the Steenrod algebra on said cohomology.

Introduction

The cohomology of a group G was first defined sometime during World War II, independently by Eilenberg and Mac Lane in Columbia and Chicago, Freudenthal in Amsterdam, and Hopf in Switzerland. Commu- nication was difficult then, and the fact that four mathematicians in three different countries came up with the same idea at the same time shows how pressing the need for the concept had become. A few years after the end of the war, in 1950, Hochschild used the term Galois cohomology for the first time in a paper where he studied local class field theory. Since then, Galois cohomology has become a prevalent part of abstract algebra and an active area of research. One of the great challenges of Galois theory is that of determining absolute Galois groups, and group co- homology provides us with tools to understand these. For example, let F be an arbitrary field and Fsep its separable closure. The absolute Galois group G := Gal(F F) of F can be written as the limit of the Galois F sep| groups of the finite subextensions K F of F , and its cohomology groups can be calculated as the colimits | sep of the cohomology groups of the (finite) Gal(K F). From an infinite problem, we reduce to a finite one. | The first step towards understanding absolute Galois groups is therefore to understand how to compute finite group cohomology, and our aim was to achieve this by exploring as wide an array of tools as possible, while keeping the complexity of the computations manageable. This paper presents a variety of hand computations of the cohomology of small 2-groups, with coefficients in F2, together with the ring structure and the action of the Steenrod squares. The cohomology of the smallest nontrivial group, C2, is computed directly, using a projective resolution and the definition of the cup product. This is the corner stone of our computations, as the result will be heavily n used in the sequel. We then move on to the finite product C2 , whose cohomology can be deduced directly from that of C2 by the Künneth formula. The cohomology of C4 is computed via the Lyndon-Hochschild- Serre spectral sequence associated to its normal subgroup C2, and the result is then generalised to C2n via the use of the universal coefficient theorem, and, once again, the subgroup C2. The last approach we chose is somewhat different, and is adapted from a method presented by Jon Carlson in Modules and group algebras ([Car96]): the cohomology is computed using the correspondence between elements of the n-th Ext group Extn (k, k) and homotopy classes of chain maps of degree n from a projective resolution of k to itself. kG − The action of the Steenrod squares on the cohomology rings is straightforward combinatorics for C2 and n C2 , because these are generated by classes of degree 1 and have no nilpotent elements. The computation 1 for C4 is more interesting, as it involves the definition of Sq as the Bockstein homomorphism. Finally, the

1 2 case D8 will require heavy use of the characterisation of the second cohomology groups H (G, A) as classes of abelian extensions of G by A. The main source I learned group cohomology from is [Wei94], with some occasional browsing of [NSW00] and [Bro82] for an additional point of view. The diagrammatic approach in section 2.5 is adapted from [Car96]. Historical facts come from [Wei99].

Contents

1 Definitions 2 1.1 Basic group cohomology ...... 2 1.2 Steenrod operations ...... 4 1.3 Massey higher products in low-dimensional group cohomology ...... 5

2 Calculating cohomology rings 6 2.1 Working directly from the definitions: the cyclic group C2 ...... 6 2.2 The Künneth formula and the product of cyclic groups ...... 7 2.3 The Lyndon-Hochschild-Serre spectral sequence: the cyclic group C4 ...... 7 2.4 The universal coefficient theorem applied to the group C2n ...... 9 2.5 Cohomology with pictures: Jon Carlson’s diagrams ...... 10

3 Steenrod operations 13 3.1 Operations on H∗(C2, F2) ...... 13 n 3.2 Operations on H∗(C2 , F2) ...... 13 3.3 Operations on H∗(C4, F2) ...... 13 3.4 Operations on H∗(D8, F2) ...... 14

4 Towards Massey higher products 15

1 Definitions

1.1 Basic group cohomology Let G be a group, k a commutative ring and A a kG-module.

Definition. We define i i H (G, A) := ExtkG(k, A) the i-th cohomology group of G with coefficients in A, and

kG Hi(G, A) := Tori (k, A)

the i-th homology group of G with coefficients in A

The definition of group cohomology is obviously valid for any action of G on A. However, in the sequel, unless mentioned otherwise, G will be understood to act trivially on A. If P k is a projective resolution, → an element x Exti (k, A) is an equivalence class of morphisms P k. In the sequel, we will often denote ∈ kG i → any representative of this equivalence class by x as well.

2 H (G; k) has the structure of a graded ring, with multiplication given as follows: let P k be a pro- ∗ i → jective resolution of k. A projective resolution of k by G G-modules is given by the chain complex P P × ⊗ defined by (P P) := (P P ) ⊗k n i ⊗k j i+Mj=n together with the differentials a ∂(a b) = ∂a b + ( 1)| |a ∂b ⊗ ⊗ − ⊗ There is a natural homomorphism

µ : HomkG(P, k) HomkG(P, k) HomkG G(P P, k) ⊗ −→ × ⊗ (µ( f f )(a b) = f (a) f (b) ⊗ 0 ⊗ 0 where a P , b P for some i, j. This homomorphism induces a homomorphism : H (G, k) H (G ∈ i ∈ j × ∗ → ∗ × G, k) on cohomology, the cross product. The cohomology product ^, called the cup product1 is then defined as the composite of the cross product with the cohomology of the diagonal map

∆∗ : H∗(G G; k) H∗(G; k) × →

This gives to H∗(G, K) the structure of a graded commutative ring, that is

a b a ^ b = ( 1)| |·| |b ^ a − where a = n whenever a Hn(G, k). | | ∈ Low-dimensional homology and cohomology There are additional characterizations of low-dimensional cohomology groups, that we will make use of later.

Proposition 1.1 ([Wei94, Prop. 6.1.11]). For any group G and trivial kG-module A,

G H1(G, A) = Hom (G, A) = Hom , A ∼ Grp ∼ Ab [G, G]   Theorem 1.2 ([Wei94, Th. 6.6.3]). Equivalence classes of extensions of G by A

0 A E G 1 → → → → 2 are in 1-1 correspondence with the cohomology group H (G, A), where two extensions E and E0 are equivalent if there is a homomorphism E E making the following diagram commute: → 0 E >

0 / A G / 1 ?

 E0

The correspondence is constructed as follows: for an extension

π 0 A E G 1 → → −→ → 1In the sequel, we will often omit the symbol ^ and write xy for x ^ y.

3 choose a set-theoretic section s : G E of π such that s(1) = 1, and define the factor set determined by E → and s as [] : G G A s × → 1 ([g, h]s = s(g)s(h)s(gh)− it can be shown that [] corresponds uniquely to a 2-cocycle in the bar resolution of A, a particular projective resolution2 of A. This means, in particular, that a split extension corresponds to the element 0 H2(G, A), ∈ since then s can be chosen to be a homomorphism and []s = 0.

1.2 Steenrod operations Consider R, S as additive groups. A cohomology operation of type (R, n; S, m) is a natural transformation

θ : Hn( ; R) Hm( ; S) − → − Bockstein homomorphism Consider the short exact sequence of abelian groups

i j 0 G H K 0 (1) → −→ −→ → This short exact sequence yields a long exact sequence

i i ∂ Hn(X; G) ∗ Hn(X; H) ∗ Hn(X; K) Hn+1(X; G) · · · → −→ −→ −→ → · · · The map ∂ : Hn(X; K) Hn+1(X; G) is called the Bockstein homomorphism associated to the short exact → sequence (1). In particular, the sequence

i j 0 Z/p Z/p2 Z/p 0 (2) → −→ −→ → yields a cohomology operation of type (Z/p, n ; Z/p, n + 1), often denoted β. This particular instance of the Bockstein homomorphism has two interesting properties:

If p > 2 then ββ = 0 • If ^ denotes the cup product on H (X; Z/p), β(a ^ b) = β(a) ^ b + ( 1)dim a ^ β(b) • ∗ − Steenrod squares

The Steenrod squares are a family of cohomology operations Sqn of type (Z/2, m ; Z/2, m + n), defined { } as the unique operations satisfying the following axioms:

(i) Sqn is a cohomology operation of type (Z/2, m ; Z/2, m + n)

(ii) Sq0 is the identity homomorphism

(iii) Sqn is the cup square on classes of dimension n (that is, Sqn(u) = u ^ u)

(iv) If u < n then Sqn(u) = 0 | | (v) Cartan formula: Sqn(u ^ v) = ∑ (Sqiu) ^ (Sqjv) i+j=n

2See [Wei94, §6.5].

4 (vi) Sq1 is the Bockstein homomorphism associated to the short exact sequence

0 Z/2 Z/4 Z/2 0 → → → → Define the formal operator ∞ Sq(x) := ∑ Sqi(x) i=0 (note that this makes sense, as in practice Sqi(x) = 0 for i > x ). Then (v) above translates as | | Sq(u ^ v) = Sq(u) ^ Sq(v)

1.3 Massey higher products in low-dimensional group cohomology

In [Kra66], David Kraines defines Massey higher products as a generalization of the triple Massey product3. General Massey higher products are multi-valued operations (a , , a ) a , a on H (G, k). 1 ··· n 7→ h 1 ··· ni ∗ We are interested in a particular case, namely Massey products of cohomology classes of degree 1. Let u , , u H1(G, k). 1 ··· n ∈ Definition. A defining system for the product u , , u is a system A = (a ), for 1 i < j k + 1 and h 1 ··· ni i,j ≤ ≤ (i, j) = (1, n + 1) such that: 6 (i) a is a representative of u for all 1 i n, a Hom(P , n) i,i+1 i ≤ ≤ ij ∈ 1 j 1 − (ii) dai,j = ∑ ail alj l=i+1 Note that a defining system for the product may fail to exist. If there is a defining system for the product, we say that it is defined, and the value of the higher Massey product of u1, , un relative to this defining n ··· system is the class u1, , un A of the element ∑ a1l al,n+1. The Massey product u1, , un , is, when it h ··· i l=2 h ··· i is defined, the set of all classes that can be realised as u , , u for some defining system A. h 1 ··· niA If we try to fit the defining system into a matrix as follows:

1 a a 1 1,3 · · · ∗ 0 1 a2 a2,n+1 . ··· . M = ......  . . . . .    0 0 1 ak   ···  0 0 1   ······    we obtain an element of U¯ n+1(k), the quotient of the multiplicative group Un+1(k) of matrices that agree with the identity on the diagonal, by the subgroup Zn+1(k) of matrices that are zero everywhere except on the diagonal (where they agree on the identity) and in the top right corner. Note that Zn+1(k) is a subgroup of the centre of Un+1(k), so that the quotient U¯ n+1(k) is well-defined. There is actually a correspondence between these matrices and the defining systems for the Massey product:

Theorem 1.3 ([Dwy75, Th. 2.4]). Let (u , , u ) be elements of H1(G, k). There is a one-to-one correspondence 1 ··· n between defining systems M for the higher Massey product u , , u and group homomorphisms h 1 ··· ni φ : G U (k) → n+1 that have u , , u as near-diagonal components. − 1 ··· − n 3See [Kra66] for details of definitions and proofs, and [McC01, §8.2] for a discussion of the triple Massey product

5 2 Calculating cohomology rings

2.1 Working directly from the definitions: the cyclic group C2 n Let C2n be the cyclic group of order 2 , generated by σ. To calculate H∗(C2n , F2), we need a projective resolution of F2 by F2C2n modules. Consider the element of F2C2n :

2 n 1 N := 1 + σ + σ + + σ − ···

and note that (σ + 1)N = 0 in F2C2n . There is a projective resolution of F2 by F2C2n -modules given by:

N (σ+1) N (σ+1) e F C n F C n F C n F C n F 0 ··· −−→ 2 2 −−−→ 2 2 −−→ 2 2 −−−→ 2 2 −→ 2 → We then apply the functor Hom ( , F ) to this sequence and chop off the last term to obtain C2n − 2

N (σ+1) N (σ+1) Hom (F C n , F ) Hom (F C n , F ) Hom (F C n , F ) Hom (F C n , F ) 0 ··· −−→ C2n 2 2 2 −−−→ C2n 2 2 2 −−→ C2n 2 2 2 −−−→ C2n 2 2 2 → that is, 0 0 0 0 F F F F 0 ··· −→ 2 −→ 2 −→ 2 −→ 2 → Since the differentials are all 0, we have

i H (C n ; F ) = F , i 0 2 2 ∼ 2 ≥

We now restrict to the case n = 1, that is, we determine the cohomology ring H∗(C2, F2). Note that in the projective resolution above, all boundary maps are multiplication by (σ + 1). Let x H1(C , F ) be ∈ 2 2 a generator (that is, any nonzero homomorphism F C F ). We want to show that the cup power xn 2 2 → 2 generates Hn(C , F ), which boils down to showing that the morphism ^ x : Hi(C , F ) Hi+1(C , F ) 2 2 − 2 2 → 2 2 is injective. Now let y : C F F be a cocycle of degree n. By definition of the cup product, we have 2 2 → 2

y ^ x = ∆∗(y x) × that is, yx is the composition

∆ Hn(C , F ) H1(C , F ) × Hn+1(C C , F ) ∗ Hn+1(C , F ) 2 2 ⊗ 2 2 −−→ 2 × 2 2 −−→ 2 2

The cocycle y x is the class of a homomorphism i n F2C2 F2C2 F2, whose values on the × ≤ ⊗ −→ generators of i n F2C2 F2C2 are given by ≤ ⊗ L L y x(a b) = y(a)x(b) × ⊗ So, in particular, y ^ x(1) = y x(1 1) = y(1) (since x is a nonzero homomorphism one must have × ⊗ x(1) = 1). To determine whether y ^ x is zero in the cohomology, we consider an arbitrary homomorphism z : F C F C F . Then 2 2 ⊗ 2 2 −→ 2 L dz(a b) = z(da b) + z(a db) ⊗ ⊗ ⊗ = z((σ + 1)a b) + z(a (σ + 1)b) ⊗ ⊗ = z(σa b) + z(a σb) ⊗ ⊗ 1 = z(a σ− b) + z(a σb) ⊗ ⊗ = 0

6 Thus there are no nonzero coboundaries, and since y ^ x(1) = y(1), the cup product yx is non zero n n whenever y is nonzero. So x generates H (C2, F2). We have proven:

H∗(C2, F2) ∼= F2[x] where x = 1. | | Remark. It is usually straightforward enough to find projective resolutions of F2 for the small groups that are discussed in this paper; however there are much more interesting methods that can be applied to determine the cohomology ring structure.

2.2 The Künneth formula and the product of cyclic groups

N We now propose to calculate the cohomology ring of the product C2 . To do this, we can use the Künneth formula, a general homological algebra statement about the homology of tensor products of chain com- plexes. Applied to projective resolutions, it yields the following group cohomology statement:

Proposition 2.1. Let G and H be any two groups. Then for every n 1 there is a split exact sequence: ≥ 0 Hp(G, k) Hq(H, k) × Hn(G H, k) Tork(Hp(G, k), Hq(H, k)) 0 → ⊗k −→ × → 1 → p+Mq=n p+qM=n+1 where the first morphism is the cross product . × Note that when k is a field, the torsion term vanishes and the cross product is actually an isomorphism of algebras. The calculation can be done via a straightforward induction on N. For N = 2, we have:

Hn(C C , F ) Hn p(C , F ) Hp(C , F ) 2 2 2 ∼= − 2 2 F2 2 2 × p n ⊗ M≤ ∼= F2 i+Mj=n The isomorphism is given by the cross product and so Hn(C C , F ) is generated by classes of the form 2 × 2 2 xp yn p, where x H1(C C , F ) is a projection on the first factor and y H1(C C , F ) on the × − ∈ 2 × 2 2 ∈ 2 × 2 2 second. This induces on H (C C , F ) a graded ring structure, and: ∗ 2 × 2 2 H∗(C C , F ) = F [x, y] 2 × 2 2 2 with x = y = 1. By induction we obtain: | | | | N H∗(C , F ) = F [x , , x ] 2 2 ∼ 2 1 ··· N

2.3 The Lyndon-Hochschild-Serre spectral sequence: the cyclic group C4

As seen in section 2.1, the group structure of the cohomology of C4 is the following:

Hn(C , F ) = F , n 0 4 2 ∼ 2 ≥

To compute the cohomology of C4, we will make use of the Lyndon-Hochschild-Serre spectral sequence:

Proposition 2.2 ([Wei94, Prop. 6.8.2]). Let G be any group and H a normal subgroup of G. Then there is a convergent first quadrant spectral sequence p,q E = Hp(G/H, Hq(H, A)) Hp+q(G, A) 2 ⇒

7 Where the abelian group Hq(H, A) is made into a G/H-module by the action of conjugation by an element of G/H (see [Wei94, Cor. 6.7.9]). C4 contains the normal subgroup C2, so we can apply the Lyndon-Hochschild-Serre spectral sequence to the extension 0 C C C 0 −→ 2 −→ 4 −→ 2 −→ and obtain p,q E = Hp(C /C , Hq(C , F )) = Hp+q(C , F ) 2 ∼ 4 2 2 2 ⇒ 4 2 Since C2 is in the center of C4, the conjugation action of C4/C2 on the cohomology of C2 is trivial. Thus we p,q p p,q have E2 = H (C2, F2) with C2 acting trivially on F2, that is, E2 = F2 for any p, q.

q H (C2, F2)

4 y F2 F2 F2 F2 F2 0 0 0

3 y F2 F2 F2 F2 F2 ∼= ∼= ∼=

2 y F2 F2 F2 F2 F2 0 0 0 ···

1 y F2 F2 F2 F2 F2 ∼= ∼= ∼=

0 p y F2 F2 F2 F2 F2 H (C2, F2) x0 x1 x2 x3 x4

p,q By definition, each group E is generated by xp yq. Observe the following: 2 ⊗ d2 is a derivation, so for y2n H2n(C /C , F ) • ∈ 4 2 2 d2(y2n) = 2d2(yn)yn = 0

0,2q So all the groups E2 survive on page 3.

1,0 1 Since there are no nonzero differentials coming in or out of it, E2 survives until E∞, and H (C4, F2) = • 0,1 2 2 F2, so that E∞ = 0. Thus the differential d0,1 must be injective, whence an isomorphism. So d (y) = x2.

Repeating this reasoning and keeping in mind that the differential respects the multiplicative struc- • ture, that is, d2(a b) = d2a b + a d2b, we obtain the following page 3: ⊗ ⊗ ⊗ which allows us to conclude F2 [x, y] H∗(C , F ) = 4 2 ∼ (x2) with x = 1, y = 2. | | | | Remark. The spectral sequence can also be used to calculate the multiplicative structure of H (CN, F ): • ∗ 2 2 p,q F for N = 2, E2 ∼= 2 and the sequence converges to p+q N F F H (C2 , 2) ∼= 2 i+jM=p+q

so that all groups of page 2 survive until E∞. Thus the sequence collapses on page 2 and we obtain the desired multiplicative structure.

8 y2 F F 2 2 · · ·

· · · · ·

y1 F F 2 2 · · · ···

· · · · ·

0 y F2 F2 x0 x1 · · ·

n The result can be extended to any group of the form C n , n 2. Then we have H (C n , F ) = F for • 2 ≥ 2 2 2 n 0 and the spectral sequence has exactly the same form as that of C . We obtain ≥ 4

F2 [x, y] H∗(C n , F ) = 2 2 ∼ (x2)

with x = 1, y = 2. | | | |

2.4 The universal coefficient theorem applied to the group C2n Let G be an arbitrary group and consider the short exact sequence

incl exp 0 Z , C C∗ 1 −→ −−−→ −−−→ −→ This induces a long exact sequence in cohomology:

n n n n+1 H (G, Z) H (G, C) H (G, C∗) H (G, Z) · · · → → → → → · · · By [Wei94, Th. 6.1.10], if G is a finite group, then Hi(G, C) = 0 i 1 since the order G of G is invertible ∀ ≥ | | in C. Thus we have isomorphisms i i+1 H (G, C∗) ∼= H (G, Z) for all i 1, and in particular, H2(G, Z) = H1(G, C ) = Hom(G, C ). ≥ ∼ ∗ ∼ ∗ To compute the cohomology of C2n , we will use the restriction homomorphism: any group map ρ : G H induces an exact functor ρ? : G mod H mod, and there is a natural injection AG (ρ? A)H → − → − → extending to a ring homomorphism

G res : H∗(G, A) H∗(H, A) H → In our case, ρ will be the inclusion of a subgroup H into G. Then by the “restriction of an element to H” we mean the image of this element by the restriction homomorphism induced by the inclusion of H in G.

n 2iπ/2n Now let G = C2n with generator σ. Then Hom(C2n , C∗) ∼= Z/2 (generated by σ e ), so that the n 7→ restriction Hom(C n , C ) Hom(C , C ), which is the natural projection Z/2 Z/2 is surjective. We 2 ∗ → 2 ∗ → now use the universal coefficient theorem:

Proposition 2.3 (Universal coefficient theorem). There is a split exact sequence

1 n 0 Ext (Hn 1(G, k), A) H (G, A) HomAb(Hn 1(G, k), A) 0 → k − → → − →

9 And conclude 2 H (C n , Z/2) = Hom(C n , C∗) Z/2 2 ∼ 2 ⊗ The restriction is in fact an isomorphism

2 2 H (C2n , F2) ∼= H (C2, F2) 2 But the restriction map is a ring homomorphism, and the even part H ∗(C2, F2) is the polynomial ring F2 [y] with y = x2 a generator of degree 2. Thus

2 H ∗(C2n , F2) ∼= F2 [y] with y = 2. | | It remains to check the structure of the odd part. Consider the operation Sq1:

On the one hand, Sq1 is the Bockstein homomorphism induced by the sequence 0 Z/2 Z/4 • 1 →1 → → Z/2 0, so that Sq (x) = 0 if and only if x arises as the image of an element y H (C n , F ) under → ∈ 2 2 the map induced by the quotient map Z/4 Z/2. → 1 1 In other words, using the identification H (C2n , A) ∼= Hom(C2n , A), Sq (x) = 0 if, and only if the homomorphism x can be lifted to Z/4. This is always the case when n 2, so Sq1(x) = 0 ≥ 1 2 2 On the other hand, since x = 1, Sq (x) = x , so x = 0 in the cohomology of C n . • | | 2 k Finally, we investigate the products zy, where z H (C n , F ) is an arbitrary class of odd degree and y ∈ 2 2 is the generator of degree 2. Let P F be the minimal resolution of F by F C n -modules, and define → 2 2 2 2 Q = P P k i ⊗ j i+Mj=k If zy = 0, then there is a t Hom(Q , F ) satisfying dt = zy. Thus t has even degree, and: ∈ k+1 2 2n 1 − dt(1 1) = t(N 1 + 1 N) where N = ∑ σi ⊗ ⊗ ⊗ i=0 = t(N 1) + t(1 N) ⊗ ⊗ 2n 1 2n 1 − − = ∑ t(σi 1) + ∑ t(1 σi) i=0 ⊗ i=0 ⊗ 2n 1 2n 1 − i − i = ∑ t(1 σ− ) + ∑ t(1 σ ) i=0 ⊗ i=0 ⊗ = 0 so there are no nonzero coboundaries in H (C n , F ). By observing that, for example, z y(σ σ) = 1, we ∗ 2 2 × ⊗ can conclude that zy = 0, and thus we obtain an isomorphism 6

F2 [x, y] H∗(C n , F ) = 2 2 ∼ x2

2.5 Cohomology with pictures: Jon Carlson’s diagrams In [Car96], Jon Carlson presents a very pictorial way to calculate group cohomology, which relies on the n description of ExtkG(k, A) as chain homotopy classes of chain maps of degree n. Indeed, any element of Extn (k, A) can be represented by a map γ : P k, where P k is a projective resolution of k, satisfying kG n → → dγ = 0. Using this and the projectivity of P , one can lift the map into a chain map γ : P P of degree n, i → −

10 n and it can be shown that two maps represent the same element in ExtkG(k, A) if and only if they are chain homotopic. The cup product of two such classes α, β Extn (k, k) is then represented by the composition of the ∈ kG chain maps α, β. So computing the cohomology ring of a group G boils down to determining the chain maps P P of degree n, for each n. → − As an example, we consider the group D8 with the following presentation:

D = x, y x2 = y2 = (xy)4 = 1 8 | D E Let A = x + 1 and B = y + 1 in F2D8. Then F A, B F D = 2 h i 2 8 (A2, B2, ABAB + BABA) where F A, B denotes the polynomial ring in noncommuting variables A, B over F . 2 h i 2 We can represent F2D8 by the following diagram:

A • B

• • B A • • A B • • B A • The point at the top is 1 and each arrows represents multiplication by either A or B. Now, to form the projective resolution of F2 by F2D8-modules, we splice the successive short exact sequences:

•

• • • •

• • • • •

• • • •

• •

• •

• • • • • • • •

• • • • • • • • L • • • • • • • • •

• • • • •

···

u1 u2 x1 xn+1 • • • • s1 s2 • • • • • • • • v1 vn • • • • • • ··· • • • • L L t1 tn 1 − • • ··· • • • • • • • • ··· • •

• • • • • • • • • •

• • • •

Were, in the n-th sequence

11 u Ax , u Bx and v BABx + Ax • 1 7→ 1 2 7→ n+1 i 7→ i i+1

x1 s1, xn+1 s2 and xi ti 1 for 2 i n • 7→ 7→ 7→ − ≤ ≤ We thus obtain a resolution P F with P = (F D )i+1 (pictured below). Let b , b be generators of → 2 i 2 8 1 2 P . The diagram below represents the chain map x associated to the generator x H1(D , F ) mapping b 1 ∈ 8 2 1 to 1 and b2 to 0.

c1 c2 c3 b1 b2 a • • • • • •

• • • • • • • • • • • •

• • • • • • • • • • • • • L L L • • • • • • • • • • • •

• • • • • •

• • • •

• • • •

• • • • •

• • • b1 b2 a • • •

• • • • • •

• • • • • • • L • • • • • •

• • •

• •

• •

• •

•

By iterating the process, we can see that xn = x x sends the leftmost generator of P to a P ◦ · · · ◦ n ∈ 0 and all other generators to 0. It can be shown, with some additional diagram chasing, that xn is not chain homotopic to zero. Thus xn represents a nonzero element of Extn (F , F ). Repeating the same process F2D8 2 2 with the map y sending b2 to 1 and b1 to 0, we obtain a map y representing a non-nilpotent element of degree 1 in H∗(D8, F2). It is then easy to see that xy = 0 in the cohomology. Finally, if P is generated by c , c , c , then x2 corresponds to the chain map sending c to a P , and y2 2 1 2 3 1 ∈ 0 sends c3 to a. The map z sending the generator c2 to a is not (chain homotopic to) zero, and not represented 2 2 by any linear combination of x and y since both of these are 0 on c2. Thus z must be a third independent generator. By playing a little more with the chain maps, we see that in degree n,the map xn sends the leftmost generator n+1 n 2 of Pn = (F2D8) to a, and all other generators to 0. The map x − z sends the next generator on the right to a, and all others to 0, and so on. Similarly, yn sends the rightmost generator to a and the rest to 0, and n 2 y − z the next generator on the left to a. This way, we can generate the entire cohomology of D8 with products of x, y and z, and we obtain:

F2 [x, y, z] H∗(D , F ) = 8 2 ∼ (xy) where x = y = 1 and z = 2. | | | | | |

12 3 Steenrod operations

3.1 Operations on H∗(C2, F2) Recall that H (C , F ) = F [x] with x = 1. Here is a very straightforward, combinatorial approach to ∗ 2 2 ∼ 2 | | determining the Steenrod squares on a cohomology ring: consider the operator Sq applied to the generator x:

Sq(x) = Sq0(x) + Sq1(x) + 0 = x + x2 = x(x + 1) thus for any k 1, ≥ k k Sq(xk) = Sq(x)k = (x + 1)kxk = xk+i ∑ i i=0   whence k Sqi(xk) = xi+k i  

n 3.2 Operations on H∗(C2 , F2) Write H (Cn, F ) = F [x , , x ]. Then, as computed above, Sq(x ) = x (x + 1) for each i. A similar ∗ 2 2 ∼ 2 1 ··· n i i i calculation yields n t k1 kn ki ki+li Sq (x1 xn ) = ∑ ∏ xi ··· l + +l =t i=1 li ! 1 ··· n  

3.3 Operations on H∗(C4, F2) F [ ] The ring H (C , F ) = 2 x,y presents the added interest of the generator y of degree 2. We have: ∗ 4 2 ∼ (x2)

Sq(x) = x since x2 = 0. Thus Sq(xk) = xk and Sqi(xk) = 0 for any i = 0. • 6 Sq(y) = y + Sq1(y) + y2 • so we need to determine Sq1(y). For this, we use the alternative definition of Sq1 as the Bockstein homo- morphism associated to the sequence

0 Z/2 Z/4 Z/2 0 −→ −→ −→ −→ In our case, Sq1 is the map denoted β in the (exact) sequence below:

α β H2(C , Z/2) H2(C , Z/4) H2(C , Z/2) H3(C , Z/2) (3) 4 −→ 4 −→ 4 −→ 4 −→ · · · 1 2 1 and so Sq (y) = 0 if and only if y = α(z) for some z H (C4, Z/2). We know that the map H (C4, Z/2) 2 1 ∈ 4 2 → H (C4, Z/2) is zero, since it is Sq on classes of degree 1. Using the fact that H (C4, Z/4) = Z/4, (3) above becomes: α β 0 Z/2 Z/4 Z/2 Z/2 −→ −→ −→ −→ 4See [Wei94, Th. 6.2.2] for the detailed computation.

13 thus α must be surjective and β = 0. Hence Sq1(y) = 0 and Sq(y) = y + y2 = y(y + 1), and we can apply the usual reasoning to get: k k Sq(yk) = yk+i ∑ i i=1   2i k k k+i 2i+1 k so Sq (y ) = (i)y , Sq (y ) = 0. We can conclude:

Sq2i+1(xyk) = 0 k Sq2i(xyk) = xSq2iyk = x yk+i i  

3.4 Operations on H∗(D8, F2) F We showed that ( , F ) 2[x,y,z] , where is the dual homomorphism to the generator σ of , H∗ D8 2 ∼= (xy) x D8 y is dual to τ, and z is a choice of generator of degree 2 that we will make precise later. So there are three generators on which the Steenrod squares act.

Since x is a non-nilpotent generator of degree 1, we have Sq1(x) = x + x2 so we apply the same reason- i k k k+i ing as in section 3.1 and obtain Sq (x ) = (i)x and similarly for y. To compute the action of the squares on z, we will use the restriction homomorphism. We first make a choice of generator: let z H2(D , F ) be a non-trivial element such that its restrictions to the subgroups ∈ 8 2 σ and τ are both zero. Consider the extension: h i h i 0 C D D 1 → 2 → 16 → 8 → any set-theoretic section of D D sends the generating involutions σ, τ of D to involutions, thus it 16 → 8 8 is a homomorphism on the subgroups σ and τ . In other words, the restriction of this extension to h i h i the subgroups σ and τ has a section, and is thus split, which means that it corresponds to 0 in the h i 2 h i cohomology group H (D8, F2). We just proved that z corresponds to the extension above. Now let us look at the restriction of z on the subgroup H = C C generated by σ and the commutator 2 × 2 [σ, τ]. Observe that the preimage of H in D16 is isomorphic to D8, so that the restriction of z to H corresponds to the extension 0 C D C C 1 → 2 → 8 → 2 × 2 → where one of the preimages of a generator of C C has order 4 in D and the other has order 2. 2 × 2 8 The cohomology group H2(H, F ) is generated by a2 (where a H1(H, F ) is dual to σ), b2 (b dual to 2 ∈ 2 [σ, τ]) and ab. Write G 2 2 resH(z) = u.a + v.b + w.ab with u, v, w coefficients in F . We already established that the restriction of z to σ is 0, so we must have 2 h i u = 0. Now b2 is the restriction of z to [σ, τ] , which must be nontrivial since the preimage of [σ, τ] in h i h i D16 is not an involution (and thus the extension is non-split). Thus v = 1. Finally, since σ and [σ, τ] do not commute in D , the restriction of z to the subgroup σ [σ, τ] is nontrivial as well. So w = 1 and 16 h × i G 2 resH(z) = a + ab Applying Sq1, we obtain:

1 1 2 2 2 G Sq (res(z)) = Sq (a + ab) = a b + ab = b(resH(z)) Similarly, if H = τ, [σ, τ] , we have 0 h i Sq1(resG (z)) = c(resG (z)) H0 H0

14 1 where c is dual to τ in H (H0, F2) Since the Steenrod squares are natural transformations, they commute with the restriction, so that 1 G G 1 1 3 Sq (resH(z)) = resH(Sq (z)). So Sq (z) is an element of H (D8, F2) that restricts to both of these elements on H and H0. Since restriction is a ring homomorphism, a straightforward inspection yields:

Sq1(z) = (x + y)z

Thus Sq(z) = z + (x + y)z + z2. Straightforward combinatorics then yield

t i k l k k+s l l+t r r r Sq (x z ) = x z − (x + y ) ∑ s t ∑ s+t=i     r=1 and similarly for ykzl.

4 Towards Massey higher products

A dual problem to that of computing GF for an arbitrary field F is that of determining, for an arbitraty profinite group G, whether G can be realised as the absolute Galois group of some field F. Recent advances in this field have been made using Massey products. For example, it is shown in [MT13] that whenever G can be realised as the absolute Galois group of a field F, the triple Massey product of any classes of degree 1 in the cohomology of G with coefficients in F2 must contain zero.

In [Kra66], after defining Massey higher products, Kraines shows how they can be linked to Steen- rod operations and Bockstein homomorphisms: restricting the definition of the higher Massey product by adding conditions on defining systems turns the n-fold product of a class with itself into a well-defined cohomology operation. Cohomology operations are classified by the cohomology groups of Eilenberg-Mac Lane spaces, which are generated by Steenrod operations and Bockstein homomorphisms. Thus the Massey n-fold product can be shown to be the composition of a Bockstein homomorphism and a Steenrod power.

Unfortunately, this theorem is only true for coefficients in a finite field of odd order; even in these cases, the exact structure of general Massey products is not yet well determined. Understanding Massey products and how they relate to cohomology operations in a general setting would bring us one step closer to solving the inverse Galois problem.

References

[Bro82] Kenneth S. Brown. Cohomology of groups, volume 87 of Graduate Texts in Mathematics. Springer- Verlag, New York-Berlin, 1982.

[Car96] Jon F. Carlson. Modules and group algebras. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996. Notes by Ruedi Suter.

[Dwy75] William G. Dwyer. Homology, Massey products and maps between groups. J. Pure Appl. Algebra, 6(2):177–190, 1975.

[Kra66] David Kraines. Massey higher products. Trans. Amer. Math. Soc., 124:431–449, 1966.

[McC01] John McCleary. A user’s guide to spectral sequences, volume 58 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2001.

15 [MT13] Jan Minac and Nguyen Duy Tan. Triple Massey products and Galois theory, 2013. To appear in the Journal of the European mathematical society.

[NSW00] Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg. Cohomology of number fields, volume 323 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2000.

[Wei94] Charles A. Weibel. An introduction to homological algebra, volume 38 of Cambridge Studies in Ad- vanced Mathematics. Cambridge University Press, Cambridge, 1994.

[Wei99] Charles A. Weibel. History of homological algebra. In History of topology, pages 797–836. North- Holland, Amsterdam, 1999.

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