Artin’s Braid Group and the Homology of Certain Subgroups of the Mapping Class Group

F. R. Cohen^{∗}

Department of Mathematics University of Kentucky Lexington, KY 40506 U.S.A.

1 Statement of Results

By work of B. Wajnryb [21], the mapping class group Γ_{g,k}^{0}, k = 0, 1, is generated by 2g +1 Dehn twists; a set of generators is given by Dehn twists α_{1}, …, α_{g}, β_{1}, …, β_{g}, and δ about embedded circles with the same names as in the picture below.

The group Δ_{g} is the subgroup of Γ_{g,0}^{0} generated by α_{1}, …, α_{g}, β_{1}, …, β_{g}, and α_{g+1}. In case g = 2, then Δ_{2} = Γ_{2,0}^{0} (Birman and Hilden [4]). It is easy to see that Δ_{g} is strictly smaller than Γ_{g,0}^{0} if g ≥ 3 (Birman and Hilden [4]). There is a non-split central extension

1 → Z/2 → Δ_{g} → Γ_{0,0}^{2g+2} → 1

by Birman and Hilden [4]. The group Γ_{0,0}^{n} has elements of infinite order if n ≥ 4. Many of the results given below about Δ_{g} and Γ_{0,0}^{n} were stated in Cohen [8] without proof.

Theorem 1.1 The reduced integral homology of Γ_{0,0}^{n} is all p-torsion for primes p with p ≤ n. The reduced integral homology of Δ_{g} is all p-torsion with p ≤ 2g + 2.

∗Partially supported by a National Science Foundation grant

Ronnie Lee and Steve Weintraub [16] prove that the reduced integral homology of Δ_{2} = Γ_{2,0}^{0} is all 2, 3, and 5 torsion.

Theorem 1.2 There is a homomorphism α : Z/5 → Γ_{2,0}^{0} which induces an isomorphism on the 5-primary component in homology, and thus

(i) H_{2q+1}(Γ_{2,0}^{0}; Z) = Z/5 ⊕ (2 or 3 torsion) (ii) H_{2q}(Γ_{2,0}^{0}; Z) is 2 or 3 torsion for q > 0.

If p is a prime, there is a homomorphism α : Z/p → Γ_{0,0}^{p+1} which induces an isomorphism on the p-primary component in homology. Specializing yet further, one has

Theorem 1.3 H_{2}(Γ_{2,0}^{0}; Z) = Z/2 ⊕ A where A is a (possibly zero) 2-group. If n ≡ 2(4), then H_{2}(Γ_{0,0}^{n}; Z) has Z/2 as a summand, and if g ≡ 0(2), then H_{2}(Δ_{g}; Z) has Z/2 as a summand.

These theorems are proven by considering quotient maps

γ : B_{n} → Γ_{0,0}^{n}

and

γ : B_{2g+2} → Δ_{g}

where B_{n} is Artin’s braid group on n strings. It will be checked that the center of the braid group has trivial image in Γ_{0,0}^{n} and Δ_{g}. Thus there are induced maps γ̄ : B̄_{n} → Γ_{0,0}^{n} and γ̄ : B̄_{2g+2} → Δ_{g} where B̄_{n} = B_{n}/center. Let F_{p} denote the field of p elements with p a prime.

Theorem 1.4 The mod-2 cohomology of B̄_{n} is isomorphic to H^{∗}(B_{n}; F_{2})⊗H^{∗}(CP^{∞}; F_{2}). The map γ^{∗} is always an epimorphism in mod-2 cohomology as is the map γ̄^{∗} if n ≡ 2(4).

There is a natural map

λ : Γ_{0,0}^{n} → Σ_{n}

where Σ_{n} is the symmetric group on n letters; the map is specified by the action of a homeomorphism on the n punctures and is discussed in Section 2. Let K_{n} denote the kernel of λ. Thus K_{n} is the pure mapping class group.

Theorem 1.5 The integral cohomology of K_{n} is torsion free with Poincaré series

[1 + 2t][1 + 3t] ⋯ [1 + (n − 2)t].

Furthermore, H^{∗}(K_{n}; Z) is generated as an algebra by one-dimensional cohomology classes

B_{ij}, n ≥ i > j ≥ 2, i ≥ 4.

A basis for H^{q}(K_{n}; Z) is given by

B_{i_{1}j_{1}} … B_{i_{q}j_{q}}, 4 ≤ i_{1} < i_{2} < ⋯ < i_{q} ≤ n.

The relations in H^{∗}(K_{n}; Z) together with the action of Σ_{n} are given in Section 3; they are all consequences of B_{42}B_{43} = 0 together with equivariance.

Let e_{2} denote ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ) + ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) + ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) in the group ring of GL_{2}(F_{2}); e_{2} is the second Steinberg idempotent (Mitchell and Priddy [20]). Observe that GL_{2}(F_{2}) acts on RP^{∞} × RP^{∞} and thus e_{2} may be regarded as a self-map of Σ(RP^{∞} ∧ RP^{∞}) with addition taken with respect to the suspension coordinate. Let ΣM(2) denote the homotopy direct limit of e_{2} : Σ(RP^{∞} ∧ RP^{∞}) → Σ(RP^{∞} ∧ RP^{∞}). Next recall that Σ(K(Z/3, 1)) is homotopy equivalent to A∨B where H_{2}(A; Z) ≅ Z/3 and H_{4}(B; Z) ≅ Z/3 (Holzsager [15]).

Theorem 1.6 The classifying space BΓ_{0,0}^{4} is stably equivalent to BΣ_{4} ∨ ΣM(2) ∨ Σ^{−1}B.

Theorem 1.7 If 1 ≤ q ≤ 2, then H_{q}(Γ_{2,0}^{0}; F_{3}) = 0 and

H_{3}(Γ_{2,0}^{0}; F_{3}) = F_{3} ⊕ H_{1}(Σ_{6}; H_{2}(K_{6}; F_{3})).

Furthermore, H_{3+4q}(Γ_{2,0}^{0}; Z) contains a Z/3-summand.

We remark that most of the calculations here are directed at Γ_{2,0}^{0}; related results on Γ_{0,0}^{n} will be given in the sequel.

We thank the Sonderforschnungsbereich ”Geometrie und Analysis“ for partial support while this paper was in preparation and C.-F. Bödigheimer for his suggestions.

2 Presentations

A presentation for Artin’s braid group on n strings is given as follows (see Artin [2], Birman [3]):

(i) There are generators σ_{1}, …, σ_{n−1}.

(ii) A complete set of relations is given by

σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1} for all i
σ_{i}σ_{j} = σ_{j}σ_{i} if |i − j| ≥ 2.

The symmetric group on n letters Σ_{n} has a presentation given by (i) and (ii) above together with

σ_{i}^{2} = 1 for all i.

By work of Magnus [17] a presentation for Γ_{0,0}^{n} is given by (i) and (ii) above together with

(iii) (σ_{1}σ_{2} … σ_{n−1})(σ_{n−1} … σ_{2}σ_{1}) = 1 = (σ_{1}σ_{2} … σ_{n−1})^{n}.

The group Δ_{g} is a central extension

1 → Z/2 → Δ_{g} → Γ_{0,0}^{2g+2} → 1

where Δ_{g} has a presentation given by (i) and (ii), together with (iv) the element Ω = (σ_{1} … σ_{2g+1})(σ_{2g+1} … σ_{1}) is in the center and satisfies Ω^{2} = 1.

Furthermore, (σ_{1} … σ_{2g+1})^{2g+2} = 1. See Birman and Hilden [4].

By inspection of the above presentations, there are natural quotient maps γ : B_{n} → Γ_{0,0}^{n}, π : B_{n} → Σ_{n} and λ : Γ_{0,0}^{n} → Σ_{n}. The kernel of π is the pure braid group P_{n} (Artin [2], Birman [3]). Magnus identifies the kernel of γ. It is isomorphic to Z × F_{n−1} where F_{n−1} is a free group on (n − 1) generators. Write K_{n} for the kernel of λ. Notice that there is a commutative diagram of short exact sequences (where nine copies of the trivial group have been deleted)

Z × F_{n−1} → P_{n} → K_{n}
↓  ↓  ↓
Z × F_{n−1} → B_{n} →^{γ} Γ_{0,0}^{n}
↓  ↓π  ↓λ
1  → Σ_{n} → Σ_{n}.

In what follows, our main computations will be to apply the Lyndon-Hochschild-Serre spectral sequence to the group extension 1 → K_{n} → Γ_{0,0}^{n} → Σ_{n} → 1. Thus we need to compute the cohomology of Σ_{n} with coefficients in the cohomology of K_{n}; this being the E_{2}-term of the Lyndon-Hochschild-Serre spectral sequence.

3 H^{∗}(K_{n}; Z)

A computation of the integral cohomology of K_{n} as an algebra over the group ring of Σ_{n} is given in this section. The method of proof is to show that α : P_{n} → K_{n} induces a monomorphism in cohomology. Since the cohomology of P_{n} is understood and α^{∗} commutes with the Σ_{n}-action, Theorem 1.5 will follow. Thus we first recall the cohomology of P_{n} (Arnol’d [1]; Cohen-Lada-May [11, pp.250–266]); in this last reference, the notation α_{i+1,j} is used for A_{ij} in the following theorem.

Theorem 3.1 The integral cohomology of P_{n} is torsion free with Poincaré series

(1 + t)(1 + 2t) ⋯ (1 + (n − 1)t)

and is generated as an algebra by one-dimensional classes

A_{ij}, n ≥ i > j ≥ 1.

A complete set of relations is given by

A_{ij}^{2} = 0
A_{ij}A_{ik} = A_{kj}(A_{ik} − A_{ij}) if j < k.

The action of Σ_{n} is determined by

σA_{ij} = A_{σ(i)σ(j)} for σ ∈ Σ_{n}

with the convention that A_{ji} = A_{ij}. A basis for H^{q}(P_{n}; Z) is given by

A_{i_{1}j_{1}} … A_{i_{q}j_{q}}, 2 ≤ i_{1} < i_{2} < ⋯ < i_{q} ≤ n.

It is important to identify the elements A_{ij} more carefully. Let ϕ : P_{n} → H_{1}(P_{n}; Z) denote the Hurewicz homomorphism (abelianization). Define an element in P_{n} by

Ā_{ij} = (σ_{i−1}σ_{i−2} … σ_{j+1})(σ_{j}^{2})(σ_{j+1}^{−1} … σ_{i−2}^{−1}σ_{i−1}^{−1}), n ≥ i > j ≥ 1.

It was shown in Cohen-Lada-May [11, pp.250–266] that {ϕĀ_{ij} | n ≥ i > j ≥ 1} is a basis for H_{1}(P_{n}; Z); furthermore, {A_{ij} | n ≥ i > j ≥ 1} is the dual basis to {ϕĀ_{ij} | n ≥ i > j ≥ 1}.

Next recall the group extension

1 → Z × F_{n−1} → B_{n} → Γ_{0,0}^{n} → 1

and the map λ : Γ_{0,0}^{n} → Σ_{n}. Magnus gave a presentation for Z × F_{n−1} in terms of the generators σ_{i} for B_{n} (Magnus [17, 18], Magnus-Karrass-Solitar [19]). We require the following specific information from his presentation:

z = (σ_{1} … σ_{n−1})^{n}
τ_{1} = (σ_{1} … σ_{n−2})^{n−1}
τ_{2} = (σ_{1} … σ_{n−3})^{n−2}σ_{n−1}^{−2}
⋮
τ_{q} = (σ_{1} … σ_{n−q−1})^{n−q}(σ_{n−q+1} … σ_{n−1})^{−q}
⋮
τ_{n−1} = (σ_{2} … σ_{n−1})^{−n+1}

where z generates Z and {τ_{1}, …, τ_{n−1}} is a generating set for F_{n−1}.

The first technical lemma is

Lemma 3.2 The structure of H^{∗}(Z×F_{n−1}) as a Γ_{0,0}^{n}-module factors through the quotient map λ : Γ_{0,0}^{n} → Σ_{n}. Thus the Lyndon-Hochschild-Serre spectral sequence for 1 → Z × F_{n−1} → P_{n} → K_{n} → 1 has trivial local coefficients.

The proof of this lemma is held in abeyance until Section 10 where the following lemma is also proven which contains information about the behavior of the short exact sequence

1 → Z × F_{n−1} →^{i} P_{n} →^{α} K_{n} → 1 in homology.

Lemma 3.3 (1) i_{∗}ϕ(z) = ∑_{n≥i>j≥1} ϕ(Ā_{ij}), and

(2) i_{∗}ϕ(τ_{L}) = ∑_{n−L≥i>j≥1} ϕ(Ā_{ij}) − ∑_{L≥i>j≥1} ϕ(Ā_{n−L+i,n−L+j}).

Write {z^{∗}, τ_{1}^{∗}, …, τ_{n−1}^{∗}} for the basis of H^{1}(Z × F_{n−1}; Z) dual to the basis {ϕ(z), ϕ(τ_{1}), …, ϕ(τ_{n−1})}.

Direct dualization of Lemma 3.3 gives Lemma 3.4 The following formula holds in H^{∗}(Z × F_{n−1}; Z):

i^{∗}(A_{ij}) = \{ \begin{matrix}z^{∗} + ∑_{n−i≥k≥1} τ_{k}^{∗} & if j = 1 and \\ z^{∗} + ∑_{n−i≥k≥1} τ_{k}^{∗} − ∑_{n−1≥k≥n−j+1} τ_{k}^{∗} & if j ≥ 2. \end{matrix}

Corollary 3.5 The map i induces an epimorphism in cohomology. Hence α^{∗} is a monomorphism of Σ_{n}-modules and there is an isomorphism of graded abelian groups

H^{∗}(P_{n}; Z) → H^{∗}(K_{n}; Z) ⊗ H^{∗}(Z × F_{n−1}; Z).

Proof By Lemma 3.2, the Lyndon-Hochschild-Serre spectral sequence for 1 → Z×F_{n−1} →^{i} P_{n} →^{α} K_{n} → 1 has trivial local coefficients. By Lemma 3.4

i^{∗}(A_{n1}) = z^{∗},
i^{∗}(A_{qj} − A_{q−1,j}) = τ_{n−j+1}^{∗} if j ≥ 2, and
i^{∗}(A_{q1} − A_{q−1,1}) = τ_{n−q}^{∗} if q ≥ 3.

Thus i^{∗} induces an epimorphism on H^{1}. Since H^{∗}(Z × F_{n−1}; Z) is generated by H^{1} as an algebra, i^{∗} is onto. Thus the spectral sequence collapses and so α^{∗} is a monomorphism of Σ_{n}-modules. Note that H^{∗}(K_{n}; Z) is torsion free as it embeds in H^{∗}(P_{n}; Z) which itself is torsion free by Theorem 3.1. Corollary 3.5 follows.

Next, notice that Corollary 3.5 gives a split short exact sequence

0 → H^{1}(K_{n}; Z) →^{α^{∗}} H^{1}(P_{n}; Z) → H^{1}(Z × F_{n−1}; Z) → 0.

Lemma 3.6 There exist unique elements B_{ij} in H^{1}(K_{n}; Z), n ≥ i > j ≥ 2 and i ≥ 4, such that

α^{∗}(B_{ij}) = \{ \begin{matrix}A_{ij} − A_{i,j−1} + A_{j−1,1} − A_{j1} & if j ≥ 3 \\ A_{i2} − A_{i1} + A_{31} − A_{32} & if j = 2, i ≥ 4. \end{matrix}

Next recall that A_{ij}A_{ik} = A_{kj}(A_{ik} − A_{ij}) by Theorem 3.1. Thus

α^{∗}(B_{i_{1}j_{1}} … B_{i_{q}j_{q}}) = A_{i_{1}j_{1}} … A_{i_{q}j_{q}} + ∑ coefficient.A_{i′_{1}j′_{1}} … A_{i′_{q}j′_{q}}

with 4 ≤ i_{1} < i_{2} < ⋯ < i_{q} ≤ n, 2 ≤ i′_{1} < ⋯ < i′_{q} ≤ n and at least one of the following two inequalities is satisfied:

(1) i′_{q} < i_{q}, or

(2) i′_{q} = i_{q} and j′_{q} < j_{q}.

Thus by Theorem 3.1, the next lemma follows.

Lemma 3.7 If 4 ≤ i_{1} < i_{2} < ⋯ < i_{q} ≤ n, then the elements B_{i_{1}j_{1}} … B_{i_{q}j_{q}} are linearly independent.

Proof of Theorem 1.5. By Corollary 3.5 and Theorem 3.1, the Poincaré series for H^{∗}(K_{n}; Z) is given by

(1 + t)(1 + 2t) ⋯ (1 + (n − 1)t)/(1 + t)(1 + (n − 1)t).

By Lemma 3.7 the elements B_{ij} generate H^{∗}(K_{n}; Z) as an algebra. The rest of Theorem 1.5 follows from the definition of the B_{ij} together with Theorem 3.1 and Lemma 3.8.

Lemma 3.8 (1) B_{i2}B_{i3} = 0 for i ≥ 4,

(2) B_{i2}B_{i4} = B_{42}[B_{i3} + B_{i4}] for i ≥ 5, and

(3) B_{ij}B_{ik} = εB_{kj}B_{ij} + ∑ ±B_{ab}B_{cd} for ε = 0, 1 and a < c with (i) d < k, or (ii) d = k and a < i.

Furthermore, all relations follow from B_{42}B_{43} = 0 together with equivariance.

Proof That B_{42}B_{43} = 0 follows from the relation A_{ij}A_{ik} = A_{kj}(A_{ik} − A_{ij}) and the fact that α^{∗} is a monomorphism. Next notice that τ_{i}B_{i2}B_{i3} = B_{i+1,2}B_{i+1,3} where τ_{i} is the transposition (i, i+1) and i ≥ 4. Thus B_{i2}B_{i3} = 0. To check that B_{i2}B_{i4} = B_{42}[B_{i3} +B_{i4}], i ≥ 5, notice that it suffices to check that the formula is correct for i = 5 by equivariance. But 0 = τ_{3}(B_{52}B_{53}) = (B_{52} − B_{42})(B_{53} − B_{54}). The result follows. Thus the relations for B_{ij}B_{ik} imply the relations for B_{i+q,j}B_{i+q,k} by equivariance and it suffices to determine the relation for B_{k+1,j}B_{k+1,k}. This follows by induction on k with j fixed and then by induction on j.

As examples, we give an explicit presentation of H^{∗}(K_{n}; Z) as an algebra over Σ_{n} for n ≤ 6. First consider the embedding of Σ_{n−q} in Σ_{n} given by the subgroup Σ_{n−q} × 1^{q} in Σ_{n}. By the above, there is a map of algebras

p : H^{∗}(K_{n}; Z) → H^{∗}(K_{n−q}; Z)

given by

p(B_{ij}) = \{ \begin{matrix}0 & if i > n − q \\ B_{ij} & if i ≤ n − q. \end{matrix}

Furthermore, p is Σ_{n−q}-equivariant. A table for the cohomology of K_{6} is given below and thus for K_{n}, n ≤ 6, by the above remark. The proof is a straightforward check using the formula for α^{∗}(B_{ij}).

Example 3.9 H^{∗}(K_{n}; Z), n ≤ 6.

(12)  (23)  (34)  (45)  (56)
B_{42} −B_{42} B_{42} + B_{43} −B_{42}  B_{52}  B_{42}
B_{43} B_{42} + B_{43} −B_{43} B_{42} + B_{43}  B_{53}  B_{43}
B_{52} −B_{52} B_{52} + B_{53} B_{52} − B_{42}  B_{42}  B_{62}
B_{53} B_{52} + B_{53} −B_{53} B_{53} + B_{54}  B_{43}  B_{63}
B_{54} B_{54} − B_{42} B_{54} + B_{53} −B_{54} \begin{matrix}B_{54} + B_{53} + B_{52} \\ −B_{43} − B_{42} \end{matrix}  B_{64}
B_{62} −B_{62} B_{62} + B_{63} B_{62} − B_{42}  B_{62}  B_{52}
B_{63} B_{62} + B_{63} −B_{63} B_{64} + B_{63}  B_{63}  B_{53}
B_{64} B_{64} − B_{42} B_{63} + B_{64} −B_{64}  B_{64} + B_{65}  B_{54}
B_{65} \begin{matrix}B_{65} − B_{52} \\ +B_{42} \end{matrix} B_{65} B_{65} + B_{64}  −B_{65}  \begin{matrix}B_{65} + B_{64} + B_{63} \\ +B_{62} − B_{54} \\ −B_{53} − B_{52} \end{matrix}

Relations:

\begin{matrix}(1) & B_{42}B_{43} = B_{52}B_{53} = B_{62}B_{63} = 0 \\ (2) & B_{52}B_{54} = B_{42}[B_{53} + B_{54}] \\ (3) & B_{53}B_{54} = −B_{42}B_{53} + B_{43}B_{54} \\ (4) & B_{62}B_{64} = B_{42}[B_{63} + B_{64}] \\ (5) & B_{62}B_{65} = −B_{42}[B_{63} + B_{64}] + B_{52}[B_{63} + B_{64} + B_{65}] \\ (6) & B_{63}B_{64} = −B_{42}B_{63} + B_{43}B_{64} \\ (7) & B_{63}B_{65} = B_{42}B_{63} − B_{43}B_{64} − B_{52}B_{63} + B_{53}B_{64} + B_{53}B_{65} \\ (8) & B_{64}B_{65} = (B_{42} + B_{43} − B_{52} − B_{53})B_{64} + B_{54}B_{65} \end{matrix}

4 Proof of Theorem 1.1

Consider the group extension 1 → K_{n} → Γ_{0,0}^{n} → Σ_{n} → 1 and recall that F_{p} denotes the field with p elements for p prime. If p > n, the E_{2}-term of the Lyndon-Hochschild-Serre spectral sequence for this extension is the module of invariants H^{∗}(K_{n}; F_{p})^{Σ_{n}}. By Corollary 3.5, α^{∗} restricts to a monomorphism on the module of invariants:

α^{∗} : H^{∗}(K_{n}; F_{p})^{Σ_{n}} → H^{∗}(P_{n}; F_{p})^{Σ_{n}}.

It was shown in Cohen-Lada-May [11, pp.276–283] that H^{∗}(P_{n}; F_{p})^{Σ_{n}} vanishes in degrees greater than one and is spanned by A = ∑_{n≥i>j≥1} A_{ij} in degree one. By Lemma 3.4,

i^{∗}(A) = \frac{n(n−1)}{2}z^{∗}+ others. Thus if p > n, H^{∗}(K_{n}; F_{p})^{Σ_{n}} is trivial in positive degrees and H̄^{∗}(Γ_{0,0}^{n}; Z) is all p-torsion for p ≤ n. Since there is an extension 1 → Z/2 → Δ_{g} → Γ_{0,0}^{2g+2} → 1, a similar proof gives the rest of Theorem 1.1.

5 The mod-5 Cohomology of Γ_{2,0}^{0} and Γ_{0,0}^{6}

We first exhibit maps α : Z/p → Γ_{0,0}^{p+1} and α : Z/p → Δ_{g} with p = 2g + 1 and p prime which induce isomorphisms on the p-primary component of homology. As the homology of all groups are torsion by Theorem 1.1, it suffices to check that α induces an isomorphism in (co-)homology with F_{p}-coefficients.

The first lemma records H_{1} of various groups.

Lemma 5.1 (1) H_{1}(Γ_{0,0}^{n}; Z) ≅ Z/q(n − 1), q = g.c.d.(2, n). Thus

H_{1}(Γ_{0,0}^{n}; Z) ≅ \{ \begin{matrix}Z/(n − 1) & if n ≡ 1(2) and \\ Z/2 ⊕ Z/(n − 1) & if n ≡ 0(2). \end{matrix}

(2) H_{1}(Δ_{g}; Z) ≅ Z/q(n − 1), q = g.c.d.(4, n), n = 2g + 2.

Proof The relation σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1} forces H_{1} to be cyclic. The relation

(σ_{1} … σ_{n−1})^{n} = 1 = (σ_{1} … σ_{n−1})(σ_{n−1} … σ_{1})

forces n(n − 1)g = 0 = 2(n − 1)g where g generates H_{1}(Γ_{0,0}^{n}; Z) and (1) follows. The relations (σ_{1} … σ_{2g+1})^{2g+2} = 1 = Ω^{2} forces a similar result and (2) follows.

The following lemma is an exercise in the manipulation of the relations for B_{n}.

Lemma 5.2 [(σ_{1}σ_{2} … σ_{n})σ_{1}]^{n} = [σ_{1}σ_{2} … σ_{n}]^{n+1} in B_{n+1}.

Define α : Z/n → Γ_{0,0}^{n+1} and α : Z/(2g+1) → Δ_{g} by the equation α(1) = (σ_{1}σ_{2} … σ_{n})σ_{1} and notice that Lemma 5.2, together with the relations in Section 2, gives that α is a homomorphism. Let ϕ : G → H_{1}(G; Z) denote the Hurewicz homomorphism. By Lemma 5.1, one has ϕα(1) = (n + 1)(1). If n + 1 ≡ 1(2), then ϕα is an isomorphism by Lemma 5.1. If n + 1 ≡ 0(2), then let π denote the projection to Z/n from H_{1}(Γ_{0,0}^{n+1}; Z) ≅ Z/2 ⊕ Z/n; thus πϕα is an isomorphism. Hence one has

Lemma 5.3 There are homomorphisms α : Z/n → Γ_{0,0}^{n+1} and β : Γ_{0,0}^{n+1} → Z/n such that βα is an isomorphism.

The next step requires an explicit calculation which is postponed until Section 8.

Theorem 5.4

H^{p}(Σ_{6}; H^{q}(K_{6}; F_{5})) ≅ \{ \begin{matrix}F_{5} & if (1) q = 0 and p ≡ 0, 7 mod 8 (2) q = 1 and p ≡ 0, 1 mod 8 (3) q = 2 and p ≡ 1, 2 mod 8 (4) q = 3 and p ≡ 2, 3 mod 8, and \\ 0 & otherwise \end{matrix}

Notice that ∑_{p+q=n} H^{p}(Σ_{6}; H^{q}(K_{6}; F_{5})) is isomorphic to F_{5}. By Lemma 5.3, E_{2} = E_{∞} in the Lyndon-Hochschild-Serre spectral sequence converging to H^{∗}(Γ_{0,0}^{6}; F_{5}). Since there is a central extension 1 → Z/2 → Γ_{2,0}^{0} → Γ_{0,0}^{6} → 1, the mod-5 cohomology of Γ_{2,0}^{0} and Γ_{0,0}^{6} follow; H^{q}(Γ_{2,0}^{0}; F_{5}) and H^{q}(Γ_{0,0}^{6}; F_{5}) are isomorphic to F_{5}, and Theorem 1.2 follows.

We remark that H^{∗}(Σ_{6}; H^{∗}(K_{6}; F_{5})) splits into a direct sum decomposition indexed by the dimension of H^{∗}(K_{6}; F_{5}). This decomposition corresponds to the eigenspace decomposition of H^{∗}(K(Z/5, 1); F_{5}) given in Holzsager [15], Cohen [10] and thus corresponds to the cohomology of the stable summands of K(Z/5, 1).

6 The mod-3 Cohomology of Γ_{2,0}^{0}

In this section, we give some (meager) information about the mod-3 cohomology of Γ_{2,0}^{0}. The central extension 1 → Z/2 → Γ_{2,0}^{0} →^{π} Γ_{0,0}^{6} → 1 gives that π induces an isomorphism on mod-3 cohomology. Thus, as before, we study H^{∗}(Σ_{6}; H^{∗}(K_{6}; F_{3})). As the Sylow 3-subgroup of Σ_{6} is Z/3 × Z/3, we first require information about the embedding

H^{∗}(Σ_{6}; H^{∗}(K_{6}; F_{3})) → H^{∗}(Z/3 × Z/3; H^{∗}(K_{6}; F_{3}))

where Z/3×Z/3 is generated by the 3-cycles α = (132) and β = (465). For the convenience of the reader, a chart is provided giving the action of α and β on H^{∗}(K_{6}; F_{3}).

Table 6.1.

α  β
B_{42}  −B_{42} − B_{43}  B_{62}
B_{43}  B_{42}  B_{63}
B_{52}  −B_{52} − B_{53}  B_{42}
B_{53}  B_{52}  B_{43}
B_{54}  B_{54} + B_{53} − B_{42} − B_{43}  B_{64} + B_{63} + B_{62} − B_{43} − B_{42}
B_{62}  −B_{62} − B_{63}  B_{52}
B_{63}  B_{62}  B_{53}
B_{64}  B_{63} + B_{64} − B_{42} − B_{43}  B_{65} + B_{64} + B_{63} + B_{62} − B_{53} − B_{52}
B_{65} B_{65} − B_{53} − B_{52} + B_{43} + B_{42} −B_{65} − B_{64} − B_{63} − B_{62} + B_{54} + B_{53} + B_{52}

Lemma 6.2 H^{1}(K_{6}; F_{3}) is a free F_{3}[Z/3 × Z/3]-module and for q ≥ 0 we have

H^{q}(Σ_{6}; H^{1}(K_{6}; F_{3})) = 0.

Proof Consider the orbit of B_{42} − B_{54} under the Z/3 × Z/3-action. By a check which is omitted, the linear span of this orbit is 9-dimensional over F_{3}. Thus H^{1}(K_{6}; F_{3}) is a cyclic F_{3}[Z/3×Z/3]-module as H^{1}(K_{6}; F_{3}) is 9-dimensional over F_{3}. Hence F_{3}[Z/3×Z/3] is isomorphic to H^{1}(K_{6}; F_{3}) and is free. Thus H^{q}(Σ_{6}; H^{1}(K_{6}; F_{3})) = 0 if q > 0 and is H^{1}(K_{6}; F_{3})^{Σ_{6}}, the module of invariants, if q = 0. But a direct check gives

Lemma 6.3 The module of invariants H^{p}(K_{6}; F_{3})^{Σ_{6}} vanishes if p > 0.

Define f : Z/3 → Γ_{0,0}^{6} by f(1) = (σ_{2}σ_{1})^{−1}(σ_{5}σ_{4})^{−1} and observe that the composite

Z/3 →^{f} Γ_{0,0}^{6} →^{λ} Σ_{6} sends 1 to αβ = (132)(465). Thus (λf)_{∗} sends the 4-dimensional generator of H_{4}(Z/3; F_{3}) to twice the generator of H_{4}(Σ_{6}; F_{3}) ≅ F_{3}. As the transfer gives

a stable map tr : Σ^{∞}BΣ_{6} → Σ^{∞}BΣ_{3} and the composite Σ^{∞}BZ/3 ⟶^{Σ^{∞}(λf)} Σ^{∞}BΣ_{6} →^{tr} Σ^{∞}BΣ_{3} gives an isomorphism of H_{4}( ; F_{3}), one has

Lemma 6.4 The space BΣ_{3} is a stable 3-local retract of BΓ_{0,0}^{6} (and of course BΣ_{6}). Thus H_{3+4q}(Γ_{0,0}^{6}; Z) contains a Z/3-summand.

Proof The mod-3 cohomology of BΣ_{3} is isomorphic to Λ[x_{3}] ⊗ F_{3}[x_{4}] as an algebra where x_{i} is of degree i. By a check of the Steenrod operations, any self-map of the mod-3 cohomology of BΣ_{3} as a module over the Steenrod algebra which is an isomorphism on H^{4}( ; F_{3}) is an isomorphism. The lemma follows.

Corollary 6.5 H^{q}(Γ_{0,0}^{6}; F_{3}) = 0 if q = 1, 2 and

H^{3}(Γ_{0,0}^{6}; F_{3}) = F_{3} ⊕ H^{1}(Σ_{6}; H^{2}(K_{6}; F_{3})).

Proof As H^{q}(Σ_{6}; F_{3}) = 0 if q = 1, 2 and H^{4}(Σ_{6}; F_{3}) is F_{3}, Lemma 6.4 implies that H^{4}(Σ_{6}; H^{0}(K_{6}; F_{3})) lives to E_{∞} in the Lyndon-Hochschild-Serre spectral sequence for the extension 1 → K_{6} → Γ_{0,0}^{6} → Σ_{6} → 1. Furthermore E_{2}^{0,q} = 0 if either (1) q = 1 or (2) q = 3 and p = 0 by Lemmas 6.2 and 6.3. Thus a direct inspection gives Corollary 6.5.

Recapitulating, one has

Theorem 6.6 The 3-primary component of H_{q}(Γ_{2,0}^{0}; Z) and H_{q}(Γ_{0,0}^{6}; Z) vanishes if q = 1 or 2; if q ≡ 3(4), then this group contains a F_{3}-summand induced by a homomorphism of Z/3 to Γ_{0,0}^{6}. In addition, H_{3}(Γ_{2,0}^{0}; F_{3}) is F_{3} ⊕ H_{1}(Σ_{6}; H_{2}(K_{6}; F_{3})).

This gives Theorem 1.7.

7 The mod-2 Cohomology of Γ_{2,0}^{0}; Theorems 1.3 and 1.4

By Theorem 1.1, H_{q}(Γ_{2,0}^{0}; Z) is all 2-, 3-, or 5-torsion. By Theorem 1.2, the 5-primary component of H_{2q}(Γ_{2,0}^{0}; Z) vanishes. By Corollary 6.5, the 3-primary component of H_{2}(Γ_{2,0}^{0}; Z) vanishes. Thus one has

Lemma 7.1 H_{2}(Γ_{2,0}^{0}; Z) is a finite abelian 2-group.

To obtain some non-trivial 2-torsion, consider the commutative diagram

\begin{matrix}B_{n} & ⟶^{γ} & Γ_{0,0}^{n} \\ ↓ & ↗γ̄ & ↓ \\ B̄_{n} & ⟶^{π̄} & Σ_{n} \end{matrix}

where B̄_{n} is defined to be B_{n}/center, and the center of B_{n} is isomorphic to Z with generator (σ_{1} … σ_{n−1})^{n} (Chow [7]) which has trivial image in Γ_{0,0}^{n}.

Lemma 7.2 The map γ^{∗} is an epimorphism in mod-2 cohomology.

Proof By Cohen-Lada-May [11, pp.226–227] π : B_{n} → Σ_{n} is a mod-2 homology monomorphism.

For convenience of the reader, we list H_{∗}(B_{n}; F_{2}) (Cohen-Lada-May [11, pp.347–349]; the Steenrod operations are incorrectly given on pp.347–349 of [11]). The natural inclusion of B_{2n} in B_{2n+1} is an isomorphism in mod-2 homology. A basis for H_{∗}(B_{2n}; F_{2}) is given by the submodule of the polynomial algebra F_{2}[x_{1}, x_{2}, …, x_{n}, …], with degree(x_{n}) = 2^{n} − 1, spanned by the monomials x_{1}^{a_{1}}x_{2}^{a_{2}} … x_{k}^{a_{k}} satisfying a_{1} + 2a_{2} + 4a_{3} + ⋯ + 2^{k−1}a_{k} ≤ n.

We include a relevant digression here. Let Φ_{n} denote the automorphism group of the free group F_{n}. There is a homomorphism ϕ_{n} : B_{n} → Φ_{n} given in Magnus-Karrass-Solitar [19]. Let B_{∞} and Φ_{∞} denote the direct limit of the B_{n} and Φ_{n}.

Lemma 7.3 The mod-2 homology of Φ_{∞} is a Hopf algebra which is free over H_{∗}(B_{∞}; F_{2}), this last Hopf algebra is given by F_{2}[x_{1}, x_{2}, …, x_{n}, …]. Thus the Thom spectrum obtained from the homomorphism Φ_{∞} → GL(R) given by abelianizing F_{n} is a wedge of K(Z/2)’s.

Proof See Cohen [9].

Returning to B̄_{n}, Chow [7] implies that H_{1}(B̄_{n}; Z) is isomorphic to Z/2⊕ odd torsion if n ≡ 2 or 3 mod 4. Thus by Lemma 5.1, γ̄ : B̄_{n} → Γ_{0,0}^{n} induces an isomorphism on the 2-primary component of H_{1} if n ≡ 2 mod 4.

Lemma 7.4 The maps γ̄ : B̄_{n} → Γ_{0,0}^{n} and λ : Γ_{0,0}^{n} → Σ_{n} induce isomorphisms on the 2-primary component of H_{1}( ; Z) if n ≡ 2 mod 4.

Next observe that the central extension 1 → Z → B_{n} → B̄_{n} → 1 is obtained by applying the long exact sequence in homotopy to the fibration

K(B_{n}, 1) →^{θ} K(B̄_{n}, 1) → K(Z, 2).

As θ^{∗} is an epimorphism in mod-2 cohomology, the Serre spectral sequence in mod-2 cohomology for this last fibration collapses. Thus one has the following equation for the module of indecomposables:

QH^{∗}(B̄_{n}; F_{2}) ≅ QH^{∗}(B_{n}; F_{2}) ⊕ (possibly F_{2} concentrated in degree 2).

But then γ̄^{∗} induces an epimorphism on the module of indecomposables by commutation with Sq^{1}. Theorems 1.3 and 1.4 follow.

Remark. If n ≡ 2(4), we actually proved that the map π̄ : B̄_{n} → Σ_{n} is onto in mod-2 cohomology. Dually, the elements (Q_{1}[1])^{2} and Q_{2}[1] are both in the image of H_{2}(B̄_{n}; F_{2}) as Sq_{∗}^{1}Q_{2}[1] = Q_{1}[1] and Sq_{∗}^{1}(Q_{1}[1])^{2} = 0. See Cohen-Lada-May [11].

8 H^{∗}(Σ_{6}; H^{∗}(K_{6}; F_{5})); Theorem 5.4

Let π denote the subgroup of Σ_{6} generated by the 5-cycle σ = (15432). As π is the 5-Sylow subgroup of Σ_{6}, the restriction

i^{∗} : H^{∗}(Σ_{6}; H^{∗}(K_{6}; F_{5})) → H^{∗}(π; H^{∗}(K_{6}; F_{5}))

is a monomorphism. The normalizer of π in Σ_{6} acts on H^{∗}(π; H^{∗}(K_{6}; F_{5})) with the invariants equal to the image of i^{∗}. Thus our computation breaks down into two steps, the first of which is to prove

Theorem 8.1 If p > 0 and q = 0, 1, 2, 3, then H^{p}(π; H^{q}(K_{6}; F_{5})) is F_{5}.

To do this last computation, we first record the action of σ on H^{∗}(K_{6}; F_{5}) in the following table obtained from Example 3.9.

σ
B_{42}  −B_{53}
B_{43}  −B_{52}
B_{52}  −B_{53} − B_{54}
B_{53}  B_{42} − B_{52}
B_{54}  B_{43} − B_{53}
B_{62} B_{52} − B_{62} − B_{63} − B_{64} − B_{65}
B_{63}  −B_{52} + B_{62}
B_{64}  −B_{53} + B_{63}
B_{65}  −B_{53} − B_{54} + B_{64}

Observe that the vector subspace M_{1} spanned by B_{ij}, i = 4, 5, is a free F_{5}π-submodule of H^{1}(K_{6}; F_{5}). The quotient H^{1}(K_{6}; F_{5})/M_{1} = N_{1} is spanned by the cosets B̄_{6j}, 2 ≤ j ≤ 5. The action of π on N_{1} is specified in the following table.

σ
B̄_{62} −B̄_{62} − B̄_{63} − B̄_{64} − B̄_{65}
B̄_{63}  B̄_{62}
B̄_{64}  B̄_{63}
B̄_{65}  B̄_{64}

By direct calculation from the minimal resolution of F_{5} as a trivial F_{5}π-module and Example 3.9, one has

Lemma 8.2 If q > 0, H^{q}(π; N_{1}) is F_{5} and so H^{q}(π; H^{1}(K_{6}; F_{5})) is F_{5}. Furthermore, the module of invariants H^{1}(K_{6}; F_{5})^{Σ_{6}} is one-dimensional spanned by

ξ = 2B_{42} + B_{43} + 3B_{52} + 2B_{53} + B_{54} + 4B_{62} + 3B_{63} + 2B_{64} + B_{65}

and a generator for H^{2q}(π; N_{1}) is given by the class of ξ in N_{1}.

Proof The proof is a direct computation from the minimal resolution for the cyclic group. Here N^{∗} = (1 + σ + σ^{2} + σ^{3} + σ^{4})^{∗} is the zero map and (1 − σ)^{∗} has a one-dimensional cokernel over F_{5}. Notice that H^{1}(N_{1}; F_{5})^{π} is one-dimensional spanned by y = 4B̄_{62} + 3B̄_{63} + 2B̄_{64} + B̄_{65}. The lemma follows as ξ is fixed by Σ_{6} and projects to y in N_{1}.

Let M_{2} denote the vector subspace of H^{2}(K_{6}; F_{5}) spanned by B_{4i}B_{5j}, 2 ≤ i ≤ 3, 2 ≤ j ≤ 4. M_{2} is an F_{5}π-submodule of H^{2}(K_{6}; F_{5}) by inspection of the relations in Example 3.9. Let N_{2} = H^{2}(K_{6}; F_{5})/M_{2} and let R_{2} denote the vector subspace of M_{2} generated by σ^{k}(B_{42}B_{52}), 0 ≤ k ≤ 4.

Lemma 8.3 The modules R_{2} and N_{2} are free as F_{5}π-modules. Thus H^{q}(π; H^{2}(K_{6}; F_{5})) is isomorphic to H^{q}(π; M_{2}/R_{2}) if q > 0, and M_{2}/R_{2} is a trivial π-module of dimension one over F_{5}; H^{q}(π; H^{2}(K_{6}; F_{5})) is F_{5} if q > 0. A generator of H^{2q}(π; H^{2}(K_{6}; F_{5})), q > 0, is represented by any element of the form B_{42}B_{53} + others where “others” is fixed by π.

Proof The module R_{2} has basis over F_{5} given by

{B_{42}B_{52}, B_{42}B_{53} − B_{43}B_{54}, B_{42}B_{53} − B_{43}B_{52}, B_{42}B_{54}, B_{43}B_{53}}

and is thus a free F_{5}π-module. Next notice that N_{2} is 20-dimensional over F_{5} and that the F_{5}π-submodule of N_{2} generated by the cosets of B_{42}B_{65}, B_{43}B_{65}, B_{53}B_{65} and B_{54}B_{65} is thus all of N_{2} by inspection. Hence N_{2} is free over F_{5}π.

To finish the proof of Lemma 8.3, observe that the coset of B_{42}B_{53} in M_{2}/R_{2} is nonzero and that σ fixes B_{42}B_{53}. Thus M_{2}/R_{2} is a trivial π-module. The module of fixed points M_{2}^{π} is spanned by α = B_{42}[B_{52}−2B_{53}+B_{54}]+B_{43}[B_{52}+B_{53}+B_{54}] and β = B_{42}B_{53}, and β projects to a generator of H^{2q}(π; M_{2}/R_{2}). The lemma follows.

Let M_{3} denote the vector subspace of H^{3}(K_{6}; F_{5}) spanned by [σ^{i}(B_{42}B_{52})]B_{6j}, 0 ≤ i ≤ 4, 2 ≤ j ≤ 5. By inspection, M_{3} is a free F_{5}π-submodule of H^{3}(K_{6}; F_{5}) and the quotient H^{3}(K_{6}; F_{5})/M_{3} = N_{3} is spanned by the cosets \overline{B_{42}B_{53}B_{6j}}, 2 ≤ j ≤ 5.

Lemma 8.4 The F_{5}π-modules N_{1} and N_{3} are isomorphic. Thus H^{q}(π; H^{3}(K_{6}; F_{5})) ≅ F_{5} if q > 0. Furthermore, ξ.B_{42}B_{53} represents a generator of H^{2q}(π; H^{3}(K_{6}; F_{5})) (where ξ appears in Lemma 8.2).

Proof First notice that π fixes B_{42}B_{53} and so it fixes ξ.B_{42}B_{53} ≠ 0. Next notice that N_{1} and N_{3} are isomorphic F_{5}π-modules with the isomorphism given by sending B̄_{6j} to \overline{B_{42}B_{53}B_{6j}}. As the class of ξ generates H^{2q}(π; N_{1}), the coset of ξ.B_{42}B_{53} which is non-zero generates H^{2q}(π; N_{3}). The lemma follows.

The proof of Theorem 8.1 is a quotation of Lemmas 8.2 to 8.4.

To complete the computation of H^{∗}(Σ_{6}; H^{∗}(K_{6}; F_{5})), first observe that the 4-cycle τ = (1254) normalizes π as τ^{−1}στ = σ^{2} for σ = (15432). We use this to compute the elements in H^{p}(π; H^{q}(K_{6}; F_{5})) invariant under τ.

Recall that if p is odd, then H^{∗}(Z/p; F_{p}) is isomorphic to Λ[x_{1}] ⊗ F_{p}[x_{2}] with the degree of x_{i} equal to i. Thus if p = 5, the invariants under τ in H^{∗}(π; H^{0}(K_{6}; F_{5})) are Λ[x_{1}x_{2}^{3}] ⊗ F_{5}[x_{2}^{4}] as τ^{∗}x_{2} = 2x_{2}. Furthermore if M is an F_{5}π-module, then H^{∗}(π; M) is periodic of period 2 with periodicity operator given by cup-product with x_{2} [6]. Since τ induces an isomorphism of algebras in cohomology, the above together with Lemmas 8.2 to 8.4 gives a calculation of the invariants in H^{2p}(π; H^{q}(K_{6}; F_{5})): let ξ ∈ H^{1}(K_{6}; F_{5}) be as in Lemma 8.2. As ξ is fixed by all of Σ_{6}, ξ.x_{2}^{4i} is fixed by τ and ξ.x_{2}^{j} is not if j ≢ 0(4). Thus H^{8j}(π; H^{1}(K_{6}; F_{5})) is the module of even-dimensional invariants on the 1-line. Next recall that H^{2q}(π; H^{i}(K_{6}; F_{5})), i = 2, 3, is spanned by B_{42}B_{53}.x_{2}^{q} and ξ.B_{42}B_{53}.x_{2}^{q} by Lemmas 8.3 and 8.4.

Recall that

α = B_{42}[B_{52} − 2B_{53} + B_{54}] + B_{43}[B_{52} + B_{53} + B_{54}]

and

β = B_{42}B_{53}.

A direct computation from Example 3.9 gives the equation

τ^{∗}(β) = α + 3β.

As α + 3β represents 3 times the generator of H^{2q}(π; H^{2}(K_{6}; F_{5})), q > 0, the invariants in H^{2q}(π; H^{2}(K_{6}; F_{5})) under the action of τ are spanned by the class of β.x_{2}^{4q+1}. Thus

H^{8j+2}(π; H^{2}(K_{6}; F_{5}))

is the module of even-dimensional invariants on the 2-line.

As ξ is fixed by Σ_{6}, τ^{∗}(ξ.β) = ξα + 3(ξ.β). Thus

H^{8j+2}(π; H^{3}(K_{6}; F_{5}))

is the module of even-dimensional invariants on the 3-line.

Since ∑_{p+q=n} H^{p}(Σ_{6}; H^{q}(K_{6}; F_{5})) must contain a copy of F_{5} by Lemma 5.3, it follows

that H^{1}(Σ_{6}; H^{2}(K_{6}; F_{5})) and H^{3}(Σ_{6}; H^{4}(K_{6}; F_{5})) are both isomorphic to F_{5}.

A “picture” of E_{2} = H^{∗}(Σ_{6}; H^{∗}(K_{6}; F_{5})) is given below where dots represent a copy of F_{5} and other groups are zero:

[figure: 1; 2; 3; 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19]

As a by-product of the computation of H^{∗}(Σ_{6}; H^{∗}(K_{6}; F_{5})), one also obtains the cohomology H^{∗}(Σ_{5}; H^{∗}(K_{5}; F_{5})) as follows. The module M_{1} is isomorphic to H^{1}(K_{5}; F_{5}) as a Σ_{5}-module. Furthermore, M_{2} is isomorphic to H^{2}(K_{5}; F_{5}) as a Σ_{5}-module. The differentials in the Lyndon-Hochschild-Serre spectral sequence converging to H^{∗}(Γ_{0,0}^{5}; F_{5}) vanish by degree reasons and thus the following theorem is immediate.

Theorem 8.5

H^{q}(Γ_{0,0}^{5}; F_{5}) ≅ \{ \begin{matrix}F_{5} & if q ≡ 0, 3 mod 4 \\ 0 & otherwise. \end{matrix}

It would seem likely that the classifying space BΓ_{0,0}^{5} is a 5-local stable retract of BZ/5.

9 Theorem 1.6

The proof of Theorem 1.6 was outlined in Cohen [8]; details are given here. By Example 3.9, H^{1}(K_{4}; Z) is spanned by B_{42} and B_{43}; the action of Σ_{4} is specified in that example and the action of the 3-cycle σ = (132) is recorded in the following chart.

σ
B_{42} −B_{42} − B_{43}
B_{43}  B_{42}

A direct computation using the minimal resolution of F_{3} as a trivial F_{3}π-module where π is generated by σ gives

Lemma 9.1 If q > 0, H^{q}(π; H^{1}(K_{4}; F_{3})) is F_{3}. Furthermore, B_{42} − B_{43} is a generator of H^{2q}(π; H^{1}(K_{4}; F_{3})).

Notice that τ = (12) normalizes π in Σ_{4} and τ^{∗}(B_{42} − B_{43}) = B_{42} − B_{43}. Thus an argument like that in Section 8 gives

Lemma 9.2

H^{p}(Σ_{4}; H^{q}(K_{4}; F_{3})) ≅ \{ \begin{matrix}F_{3} & if (1) q = 0 and p ≡ 3, 0 mod 4, (2) q = 1 and p ≡ 0, 1 mod 4, and \\ 0 & otherwise. \end{matrix}

Since the group Γ_{0,0}^{4} has Z/3 as a retract, E_{2} = E_{∞} in the Lyndon-Hochschild-Serre spectral sequence converging to H^{∗}(Γ_{0,0}^{4}; F_{3}), and one has

Proposition 9.3 There is a group homomorphism α : Z/3 → Γ_{0,0}^{4} which induces an isomorphism in mod-3 cohomology.

To compute the mod-2 cohomology of K_{4}, consider the 2-Sylow subgroup of Σ_{4}, Σ_{2}≀Σ_{2}, together with the elements σ^{i}, 0 ≤ i ≤ 2, a complete set of right coset representatives for Σ_{2} ≀ Σ_{2} in Σ_{4}. The left action of Σ_{4} on the vector space F_{2}[Σ_{4}/Σ_{2} ≀ Σ_{2}] is given by the following chart.

(12) (23) (34)
σ̄^{0} σ̄^{0}  σ̄^{1}  σ̄^{0}
σ̄^{1} σ̄^{2}  σ̄^{0}  σ̄^{2}
σ̄^{2} σ̄^{1}  σ̄^{2}  σ̄^{1}

Thus F_{2}[Σ_{4}/Σ_{2} ≀ Σ_{2}] splits as F_{2} ⊕ B as a left F_{2}Σ_{4}-module where F_{2} is spanned by σ̄^{0} + σ̄^{1} + σ̄^{2} and B is spanned by σ̄^{0} + σ̄^{1} = α and σ̄^{1} + σ̄^{2} = β. The left action of Σ_{4} on B is given by the chart

(12) (23) (34)
α α + β α α + β
β β α + β β

It is mildly easier to use homology here. Observe that B is isomorphic to H_{1}(K_{4}; F_{2}) as a left Σ_{4}-module with the isomorphism given by sending α to B_{42∗} and β to B_{43∗} (where B_{ij∗} is the dual basis in homology). Furthermore, there are isomorphisms

f : F_{2}[Σ_{4}/Σ_{2} ≀ Σ_{2}] → F_{2}[Σ_{4}] ⊗_{Σ_{2}≀Σ_{2}} F_{2}

and

g : H_{∗}(Σ_{4}; F_{2}[Σ_{4}/Σ_{2} ≀ Σ_{2}]) → H_{∗}(Σ_{2} ≀ Σ_{2}; F_{2}).

Consider the stable cofibration M(2) → B(Σ_{2} ≀ Σ_{2}) → BΣ_{4}. As the 2-Sylow subgroup of Σ_{4} is Σ_{2} ≀ Σ_{2}, there is a splitting of H_{∗}(Σ_{2} ≀ Σ_{2}; F_{2}) as H_{∗}(Σ_{4}; F_{2}) ⊕ H_{∗}(M(2); F_{2}). Thus H_{q}(Σ_{4}; H_{1}(K_{4}; F_{2})) is isomorphic to H_{q}(M(2); F_{2}).

Notice that Σ_{2} ≀ Σ_{2} is isomorphic to a subgroup of Γ_{0,0}^{4} which projects to Σ_{2} ≀ Σ_{2} in Σ_{4}, as can be seen from the picture below where x_{i} are the puncture points.

The group Σ_{2} ≀ Σ_{2} is generated by (12), (34), and (13)(24). Send (12) to rotation along the x_{3}x_{4} ray through π radians, (34) to rotation along the x_{1}x_{2} ray through π radians, and send (13)(24) to rotation along l through π radians.

Since the inclusion Σ_{2} ≀ Σ_{2} in Σ_{4} induces an epimorphism in mod-2 homology, the Lyndon-Hochschild-Serre spectral sequence converging to H_{∗}(Γ_{0,0}^{4}; F_{2}) collapses.

Corollary 9.4 H_{q}(Γ_{0,0}^{4}; F_{2}) is isomorphic to H_{q}(Σ_{4}; F_{2}) ⊕ H_{q}(ΣM(2); F_{2}).

By work of Mitchell and Priddy [20], ΣM(2) is stably the homotopy direct limit of e_{2} : Σ(RP^{∞} ∧ RP^{∞}) → Σ(RP^{∞} ∧ RP^{∞}) where e_{2} is the space level analogue of the second Steinberg idempotent (see Section 1). We shall exhibit a homomorphism ρ : Z × Σ_{2} × Σ_{2} → Γ_{0,0}^{4} such that the summand of H_{∗}(Z × Σ_{2} × Σ_{2}; F_{2}) corresponding to H_{∗}(ΣM(2); F_{2}) maps isomorphically via ρ to the summand of H_{∗}(Γ_{0,0}^{4}; F_{2}) corresponding to H_{∗}(ΣM(2); F_{2}). Furthermore Bρ induces a stable map from ΣM(2) to BΓ_{0,0}^{4} realizing this isomorphism. Since the transfer gives a stable map from BΣ_{4} to BΣ_{2} ≀ Σ_{2} with the composite H_{∗}(Σ_{4}; F_{2}) → H_{∗}(Σ_{2} ≀ Σ_{2}; F_{2}) → H_{∗}(Σ_{4}; F_{2}) an isomorphism, the next proposition follows.

Proposition 9.5 There is a stable map from BΣ_{4} ∨ ΣM(2) to BΓ_{0,0}^{4} which is a 2-local equivalence.

Thus we must check the claim concerning ρ. Let G denote the subgroup of Γ_{0,0}^{4} generated by x = σ_{2}σ_{1}^{2}σ_{2}^{−1}, y = σ_{1}σ_{3}^{−1}, and z = σ_{1}σ_{2}σ_{1}σ_{2}^{−1}σ_{3}^{−1}σ_{2}^{−1}. By a choice of the relations for Γ_{0,0}^{4} in Section 2, we see that G is abelian with y^{2} = z^{2} = 1. Next, notice that x lifts through P_{4} and K_{4}. The class of x in H_{1}(P_{4}; Z) is ϕ(Ā_{31}) by definition of ϕ(Ā_{ij}) (in Section 3). As ϕ(Ā_{31}) projects to an infinite cyclic element of H_{1}(K_{4}; Z), it is of infinite order in Γ_{0,0}^{4} and so one has Lemma 9.6 G is isomorphic to Z × Σ_{2} × Σ_{2} and there is a commutative diagram

Z  →^{i} K_{4}
↓  ↓
Z × Σ_{2} × Σ_{2} → Γ_{0,0}^{4}
↓  ↓
Σ_{2} × Σ_{2} → Σ_{4}.

Identify H_{1}(K_{4}; F_{2}) with the F_{2}Σ_{4}-submodule of F_{2}[Σ_{4}/Σ_{2} ≀Σ_{2}] spanned by the cosets σ̄^{0} + σ̄^{1} and σ̄^{1} + σ̄^{2} where B_{42∗} is identified with σ̄^{0} + σ̄^{1} and B_{43∗} is identified with σ̄^{1} + σ̄^{2}. There is a further isomorphism

f : F_{2}[Σ_{4}/Σ_{2} ≀ Σ_{2}] → F_{2}[Σ_{4}] ⊗_{Σ_{2}≀Σ_{2}} F_{2}

given by f(σ̄^{i}) = σ^{i} ⊗ 1. Let W be a free right F_{2}[Σ_{4}]-resolution of F_{2} and recall the map i : Z → K_{4} of Lemma 9.6. Consider the induced composite θ

W ⊗_{Σ_{2}≀Σ_{2}} H_{1}(Z; F_{2}) ⟶^{1⊗i_{∗}} W ⊗_{Σ_{4}} H_{1}(K_{4}; F_{2})

⊂ W ⊗_{Σ_{4}} F_{2}[Σ_{4}/Σ_{2} ≀ Σ_{2}] ⟶^{1⊗f} W ⊗_{Σ_{2}≀Σ_{2}} F_{2}.

Since i_{∗}(generator) = B_{42∗} + B_{43∗} (by dualizing Lemma 3.6), and B_{4i∗} is identified with σ̄^{i−2} + σ̄^{i−1}, it follows that

θ(w ⊗ 1) = wσ^{0} ⊗ 1 + wσ^{1} ⊗ 1 + wσ^{1} ⊗ 1 + wσ^{2} ⊗ 1.

Let Σ_{2} × Σ_{2} be the subgroup of Σ_{2} ≀ Σ_{2} generated by (13)(24) and (12)(34). The transfer T : W ⊗_{Σ_{2}≀Σ_{2}} F_{2} → W ⊗_{Σ_{2}×Σ_{2}} F_{2} is given by T (w ⊗ 1) = w ⊗ 1 + wγ ⊗ 1 for γ = (12). Thus

T θ(w ⊗ 1) = w ⊗ 1 + w(12) ⊗ 1 + w(123) ⊗ 1 + w(13) ⊗ 1.

Via the isomorphism of Σ_{3} with GL_{2}(F_{2}) which sends (12) to ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) and (123) to ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ), this gives

Lemma 9.7 T θ(w ⊗ 1) = w ⊗ 1 + w( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ⊗ 1 + w( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ) ⊗ 1 + w( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ⊗ 1

which is e_{2}(w ⊗ 1) with the right GL_{2}(F_{2}) action on W.

Theorem 1.6 follows.

10 Facts about B_{n} and Lemmas 3.2 and 3.3

Some well-known manipulations with B_{n} are given here; we prove Lemmas 3.2 and 3.3 as consequences. Recall the elements z = (σ_{1} … σ_{n−1})^{n} and

τ_{q} = (σ_{1} … σ_{n−q−1})^{n−q}(σ_{n−q+1} … σ_{n−1})^{−q}

(with τ_{1} = (σ_{1} … σ_{n−2})^{n−1} and τ_{n−1} = (σ_{2} … σ_{n−1})^{−n+1}) given in Section 3.

Lemma 10.1 (1) σ_{1}τ_{n−1}σ_{1}^{−1} = τ_{n−2}τ_{n−1}^{−1}z^{−1},

(2) σ_{k}τ_{n−k}σ_{k}^{−1} = τ_{n−k−1}τ_{n−k}^{−1}τ_{n−k+1}, 1 < k < n − 1,

(3) σ_{n−1}τ_{1}σ_{n−1}^{−1} = zτ_{1}^{−1}τ_{2},

(4) σ_{j}τ_{k}σ_{j}^{−1} = τ_{k} if j ≠ n − k, and

(5) σ_{j}zσ_{j}^{−1} = z.

Since z, τ_{1}, …, τ_{n−1} generate the kernel of α : B_{n} → Γ_{0,0}^{n}, Lemma 10.1 identifies the local coefficient system associated to the Lyndon-Hochschild-Serre spectral sequence for α. Recall that ϕ : G → H_{1}(G; Z) is the Hurewicz homomorphism for the discrete group G and ( )^{σ} denotes conjugation by σ.

Corollary 10.2

ϕ(τ_{q})^{σ_{j}} = \{ \begin{matrix}τ_{q} & if j ≠ n − q, \\ ϕ(τ_{n−2}) − ϕ(τ_{n−1}) − ϕ(z) & if j = 1 and q = n − 1, \\ ϕ(τ_{n−k−1}) − ϕ(τ_{n−k}) + ϕ(τ_{n−k+1}) & if j = n − k and q = k, \\ ϕ(τ_{2}) − ϕ(τ_{1}) + ϕ(z) & if j = n − 1 and q = 1, \end{matrix}

and ϕ(z)^{σ_{i}} = ϕ(z). Thus σ_{i}^{2} acts trivially on H^{∗}(Z × F_{n−1}; Z).

Proof This follows by abelianizing the right-hand side of the equations in Lemma 10.1, together with the fact that the cohomology of Z × F_{n−1} is generated by one-dimensional classes as an algebra.

Proof of Lemma 3.2 : The second assertion in Corollary 10.2 concerning the triviality of the action of σ_{i}^{2} implies that the action of Γ_{0,0}^{n} on H^{∗}(Z × F_{n−1}; Z) factors through Σ_{n} by the presentation given in Section 2. Thus Lemma 3.2 follows.

Lemma 10.3 (1) (σ_{1} … σ_{n−1})^{k}σ_{1} = σ_{k+1}(σ_{1} … σ_{n−1})^{k} if 1 ≤ k < n − 1,

(2) (σ_{1} … σ_{n−1})^{n−1}(σ_{2} … σ_{n−1})^{1−n} = σ_{n−1} … σ_{1}, and

(3) (σ_{2} … σ_{n−1})^{n−2}(σ_{3} … σ_{n−1})^{2−n} = σ_{n−1} … σ_{2}.

Proof Equation (1) follows by induction on k. To prove (2), check that

(σ_{1} … σ_{n−1})^{n−1}(σ_{2} … σ_{n−1})^{1−n} = (σ_{n−1} … σ_{n−k})(σ_{1} … σ_{n−1})^{n−1−k}
(σ_{2} … σ_{n−1})^{1−n+k}

by induction on k: if k = 1,

(σ_{1} … σ_{n−1})^{n−1}(σ_{2} … σ_{n−1})^{1−n} = (σ_{1} … σ_{n−1})^{n−2}(σ_{1} … σ_{n−1})
(σ_{2} … σ_{n−1})^{−1}(σ_{2} … σ_{n−1})^{2−n}
= (σ_{1} … σ_{n−1})^{n−2}σ_{1}(σ_{2} … σ_{n−1})^{2−n}
= σ_{n−1}(σ_{1} … σ_{n−1})^{n−2}(σ_{2} … σ_{n−1})^{2−n}.

The inductive step is similar. Let k = n − 1 to obtain equation (2). The proof of equation (3) is analogous to that of equation (2).

Lemma 10.4 σ_{1}τ_{n−1}σ_{1}^{−1} = τ_{n−2}τ_{n−1}^{−1}z^{−1} in B_{n}.

Proof Recall that τ_{n−1} = (σ_{2} … σ_{n−1})^{−n+1}, τ_{n−2} = σ_{1}^{2}(σ_{3} … σ_{n−1})^{2−n}, and z = (σ_{1} … σ_{n−1})^{n}. Thus Lemma 10.4 is equivalent to the equation σ_{1}τ_{n−1}^{−1}σ_{1}^{−1} = zτ_{n−1}τ_{n−2}^{−1} given by

(i) σ_{1}(σ_{2} … σ_{n−1})^{n−1}σ_{1}^{−1} = (σ_{1} … σ_{n−1})^{n}(σ_{2} … σ_{n−1})^{1−n}(σ_{3} … σ_{n−1})^{n−2}σ_{1}^{−2}. This last equation is equivalent to (ii) (σ_{2} … σ_{n−1})^{n−2} = (σ_{1} … σ_{n−1})^{n−1}(σ_{2} … σ_{n−1})^{1−n}(σ_{3} … σ_{n−1})^{n−2}σ_{1}^{−1}. By Lemma 10.4(2), equation (ii) is equivalent to (iii) (σ_{2} … σ_{n−1})^{n−2} = (σ_{n−1} … σ_{1})(σ_{3} … σ_{n−1})^{n−2}σ_{1}^{−1}. Since σ_{1} commutes with σ_{i}, i ≥ 3, equation (iii) is equivalent to (iv) (σ_{2} … σ_{n−1})^{n−2}(σ_{3} … σ_{n−1})^{2−n} = (σ_{n−1} … σ_{2}). But this last equation is given by Lemma 10.3(3) and Lemma 10.4 follows.

Next consider the homomorphism θ_{n} : B_{n} → B_{n}, reversing the order of the generators: θ_{n}(σ_{i}) = σ_{n−i}.

Lemma 10.5 (1) θ_{n}(τ_{k}) = τ_{n−k}^{−1} and θ_{n}(z) = z in B_{n},

(2) σ_{k}(σ_{1} … σ_{k−1})^{k}σ_{k}^{−1} = (σ_{1} … σ_{k−1})^{−k}(σ_{1} … σ_{k−2})^{k−1}σ_{k}^{−2}(σ_{1} … σ_{k})^{k+1}, and

(3) σ_{k}(σ_{k+1} … σ_{n−1})^{k−n}σ_{k}^{−1} = σ_{k}^{2}(σ_{k+2} … σ_{n−1})^{1+k−n}(σ_{k+1} … σ_{n−1})^{k−n} .(σ_{k} … σ_{n−1})^{k−n−1}.

Proof Since (σ_{1+a} … σ_{j+a})^{1+j} = (σ_{j+a} … σ_{1+a})^{1+j}, Lemma 10.5(1) follows from the definitions.

By Lemma 10.4, σ_{1}τ_{k}σ_{1}^{−1} = τ_{k−1}τ_{k}^{−1}z^{−1} in B_{k+1}. Thus

σ_{1}(σ_{2} … σ_{k})^{−k}σ_{1}^{−1} = σ_{1}^{2}(σ_{3} … σ_{k})^{1−k}(σ_{2} … σ_{k})^{−k}(σ_{1} … σ_{k})^{−k−1}.

Apply θ_{k+1} to get

σ_{k}(σ_{k−1} … σ_{1})^{−k}σ_{k}^{−1} = σ_{k}^{2}(σ_{k−2} … σ_{1})^{1−k}(σ_{k−1} … σ_{1})^{−k}(σ_{k} … σ_{1})^{−k−1},

and so Lemma 10.5(2) follows.

To prove Lemma 10.5(3), apply θ_{n} to Lemma 10.5(2) with n − k in place of k to get

σ_{k}(σ_{n−1} … σ_{k+1})^{n−k}σ_{k}^{−1} = (σ_{n−1} … σ_{k+1})^{k−n}(σ_{n−1} … σ_{k+2})^{n−k−1}
.σ_{k}^{−2}(σ_{n−1} … σ_{k+1})^{n−k}.

Thus Lemma 10.5(3) follows.

Proof of Lemma 10.1 : Lemma 10.1(1) was proven in Lemma 10.4. To check equation 10.1(2) observe that

σ_{k}τ_{n−k}σ_{k}^{−1} = σ_{k}(σ_{1} … σ_{k−1})^{k}σ_{k}^{−1}σ_{k}(σ_{k+1} … σ_{n−1})σ_{k−n}σ_{k}^{−1}.

Expanding by Lemma 10.5(2,3), one gets

σ_{k}τ_{n−k}σ_{k}^{−1} = (σ_{1} … σ_{k−1})^{−k}(σ_{1} … σ_{k−2})^{k−1}σ_{k}^{−2}(σ_{1} … σ_{k})^{k+1}σ_{k}^{2}
.(σ_{k+2} … σ_{n−1})^{1+k−n}(σ_{k+1} … σ_{n−1})^{k−n}(σ_{k} … σ_{n−1})^{k−n−1}
= (σ_{1} … σ_{k−1})^{−k}(σ_{1} … σ_{k−2})^{k−1}(σ_{1} … σ_{k})^{k+1}
.(σ_{k+2} … σ_{n−1})^{1+k−n}(σ_{k+1} … σ_{n−1})^{k−n}(σ_{k} … σ_{n−1})^{k−n−1}
= (σ_{1} … σ_{k})^{k+1}(σ_{k+2} … σ_{n−1})^{1+k−n}(σ_{1} … σ_{k−1})^{−k}
.(σ_{k+1} … σ_{n−1})^{k−n}(σ_{1} … σ_{k−2})^{k−1}(σ_{k} … σ_{n−1})^{k−n−1}
= τ_{n−k−1}τ_{n−k}^{−1}τ_{n−k+1}.

To do Lemma 10.1(3), notice that θ_{n}(σ_{1}τ_{n−1}^{−1}σ_{1}^{−1}) = σ_{n−1}τ_{1}σ_{n−1}^{−1} by Lemma 10.5(1). But θ_{n}(σ_{1}τ_{n−1}^{−1}σ_{1}^{−1}) = θ_{n}(τ_{n−2}τ_{n−1}^{−1}z^{−1}) = zτ_{1}τ_{2}^{−1} by Lemmas 10.4 and 10.5(1).

Since (σ_{1} … σ_{k−1})^{k} is in the center of B_{k} and σ_{i}σ_{j} = σ_{j}σ_{i} if |i−j| ≥ 2, Lemmas 10.1(4) and 10.1(5) follow.

The group P_{n} is generated by Ā_{ij} = (σ_{i−1}σ_{i−2} … σ_{j+1})σ_{j}^{2}(σ_{j+1}^{−1} … σ_{i−1}^{−1}), n ≥ i > j ≥ 1. A picture of Ā_{ij} is given by

Lemma 10.6 (1) (σ_{1} … σ_{k−1})^{k} = Ā_{21}(Ā_{31}Ā_{32}) … (Ā_{k1}Ā_{k2} … Ā_{k,k−1}),

(2) θ_{n}(Ā_{ij}) = C^{−1}Ā_{n−j+1,n−i+1}C for C = Ā_{n−j+1,n−i+2}Ā_{n−j+1,n−i+3} … Ā_{n−j+1,n−j},

(3) (σ_{k+1} … σ_{n−1})^{k−n} = θ_{n}[Ā_{21}(Ā_{31}Ā_{32}) … (Ā_{n−k,1} … Ā_{n−k,n−k−1})]^{−1}.

Proof Equation (1) is a direct calculation which is also given in Chow [7] (where Ā_{ij} = A_{ij}). Next observe that

θ_{n}(Ā_{ij}) = (σ_{n−i+1} … σ_{n−j−1})σ_{n−j}^{2}(σ_{n−j−1}^{−1} … σ_{n−i+1}^{−1}).

The relation σ_{i}σ_{i+1}σ_{i}^{−1} = σ_{i+1}^{−1}σ_{i}σ_{i+1} implies that

θ_{n}(Ā_{ij}) = (σ_{n−j}^{−1}σ_{n−j−1}^{−1} … σ_{n−i}^{−1})(σ_{n−j+1}^{2})(σ_{n−i} … σ_{n−j})
= C^{−1}A_{n−j+1,n−i+1}C

where

C = Ā_{n−j+1,n−i+2}Ā_{n−j+1,n−i+3} … Ā_{n−j+1,n−j}.

Finally, (σ_{k+1} … σ_{n−1})^{k−n} = (σ_{n−1} … σ_{k+1})^{k−n} = θ_{n}(σ_{1}σ_{2} … σ_{n−k−1})^{k−n}. But the righthand side of this equation is given by

θ_{n}[Ā_{21}(Ā_{31}Ā_{32}) … (Ā_{n−k,1} … Ā_{n−k,n−k−1})]^{−1}

by Lemma 10.6(1), and the lemma follows.

Lemma 10.7 (1) ϕ(σ_{1} … σ_{k−1})^{k} = ∑_{k≥i>j≥1} ϕ(Ā_{ij}), and

(2) ϕ(σ_{k+1} … σ_{n−1})^{k−n} = − ∑_{n−k≥i>j≥1} ϕ(Ā_{k+i,k+j}).

Proof Equation (1) follows from Lemma 10.6(1). By Lemma 10.6(3)

ϕ(σ_{k+1} … σ_{n−1})^{k−n} = ϕθ_{n}[Ā_{21}(Ā_{31}Ā_{32}) … (Ā_{n−k,1} … Ā_{n−k,n−k−1})]^{−1}

which, by Lemma 10.6(2), equals − ∑_{n−k≥i>j≥1} ϕ(Ā_{k+i,k+j}).

Proof of Lemma 3.3 :

i_{∗}ϕ(τ_{L}) = i_{∗}ϕ[(σ_{1} … σ_{n−L−1})^{n−L}(σ_{n−L+1} … σ_{n−1})^{−L}].

But by Lemma 10.7, this is just

i_{∗} ∑_{n−L≥i>j≥1} ϕ(Ā_{ij}) − i_{∗} ∑_{L≥i>j≥1} ϕ(Ā_{n−L+i,n−L+j}).

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