MASSEY PRODUCTS IN THE HOMOLOGY OF THE LOOPSPACE OF

A p-COMPLETED CLASSIFYING SPACE: FINITE GROUPS WITH CYCLIC SYLOW p-SUBGROUPS

DAVE BENSON AND JOHN GREENLEES

Abstract. Let G be a finite group with cyclic Sylow p-subgroup, and let k be a field of characteristic p. Then H^{∗}(BG; k) and H_{∗}(ΩBGp^{∧}; k) are A_{∞} algebras whose structure we determine up to quasi-isomorphism.

1. Introduction

The general context is that we have a finite group G, and a field k of characteristic p. We are interested in the differential graded cochain algebra C^{∗}(BG; k) and the differential graded algebra C_{∗}(Ω(BGp^{∧}); k) of chains on the loop space: these two are Koszul dual to each other, and the Eilenberg-Moore and Rothenberg-Steenrod spectral sequences relate the cohomology ring H^{∗}(BG; k) to the homology ring H_{∗}(Ω(BGp^{∧}); k). Of course if G is a p-group, BG is p-complete so Ω(BGp^{∧}) ≃ G, but in general H_{∗}(Ω(BGp^{∧}); k) is infinite dimensional. Henceforth we will omit the brackets from Ω(BGp^{∧}).

We consider a simple case where the two rings are not formal, but we can identify the A_{∞} structures precisely (see Section 3 for a brief summary on A_{∞}-algebras). From now on we suppose specifically that G is a finite group with cyclic Sylow p-subgroup P, and let BG be its classifying space. Then the inclusion of the Sylow p-normaliser N_{G}(P) → G and the quotient map N_{G}(P) → N_{G}(P)/O_{p′}N_{G}(P) induce mod p cohomology equivalences

B(N_{G}(P)/O_{p′}N_{G}(P)) ← BN_{G}(P) → BG,

and hence homotopy equivalences after p-completion

B(N_{G}(P)/O_{p′}N_{G}(P))p^{∧} ⟵^{∼} BN_{G}(P)p^{∧} ⟶^{∼} BGp^{∧}.

Here, O_{p′}N_{G}(P) denotes the largest normal p′-subgroup of N_{G}(P). Thus N_{G}(P)/O_{p′}N_{G}(P) is a semidirect product ℤ/p^{n} ⋊ ℤ/q, where q is a divisor of p − 1, and ℤ/q acts faithfully as a group of automorphisms of ℤ/p^{n}. In particular, the isomorphism type of N_{G}(P)/O_{p′}N_{G}(P) only depends on |P | = p^{n} and the inertial index q = |N_{G}(P) : C_{G}(P)|, and therefore so does the homotopy type of BGp^{∧}. Our main theorem determines the multiplication maps m_{i} in the A_{∞} structure on H^{∗}(BG; k) and H_{∗}(Ω(BGp^{∧}); k) arising from C^{∗}(BG; k) and C_{∗}(Ω(BGp^{∧}); k) respectively. We will suppose from now on that p^{n} > 2, q > 1 since the case of a p-group is well understood.

The starting point is the cohomology ring

H^{∗}(BG; k) = H^{∗}(Bℤ/p^{n}; k)^{ℤ/q} = k[x] ⊗ Λ(t) with |x| = −2q, |t| = −2q + 1.

There is a preferred generator t ∈ H^{1}(Bℤ/p^{n}; k) = Hom(ℤ/p^{n}, k) and we take x to be the nth Bockstein of t.

Before stating our result we should be clear about grading and signs.

Remark 1.1. We will be discussing both homology and cohomology, so we should be explicit that everything is graded homologically, so that differentials always lower degree. Explicitly, the degree of an element of H^{i}(G; k) is −i.

Remark 1.2. Sign conventions for Massey products and A_{∞} algebras mean that a specific sign will enter repeatedly in our statements, so for brevity we write

ϵ(s) = \begin{cases}+1 & s ≡ 0, 1 mod 4 \\ −1 & s ≡ 2, 3 mod 4 \end{cases} .

Theorem 1.3. Let G be a finite group with cyclic Sylow p-subgroup P of order p^{n} and inertial index q so that

H^{∗}(BG; k) = k[x] ⊗ Λ(t) with |x| = −2q, |t| = −2q + 1 and β_{n}t = x

Up to quasi-isomorphism, the A_{∞} structure on H^{∗}(BG; k) is determined by

m_{p^{n}}(t, …, t) = ϵ(p^{n})x^{h}

where h = p^{n} − (p^{n} − 1)/q. This implies

m_{p^{n}}(x^{j_{1}}t, …, x^{j_{p^{n}}}t) = ϵ(p^{n})x^{h+j_{1}+···+j_{p^{n}}}

for all j_{1}, …, j_{p^{n}} ≥ 0. All m_{i} for i > 2 on all other i-tuples of monomials give zero. If q > 1 and p^{n} ≠ 3 then

H_{∗}(ΩBGp^{∧}; k) = k[τ] ⊗ Λ(ξ) where |τ| = 2q − 2, |ξ| = 2q − 1.

Up to quasi-isomorphism, the A_{∞} structure is determined by

m_{h}(ξ, …, ξ) = ϵ(h)τ^{p^{n}}.

This implies

m_{h}(τ^{j_{1}}ξ, …, τ^{j_{h}}ξ) = ϵ(h)τ^{p^{n}+j_{1}+···+j_{h}}

for all j_{1}, …, j_{p^{n}} ≥ 0. All m_{i} for i > 2 on all other i-tuples of monomials give zero. If q > 1 and p^{n} = 3 then q = 2 and

H_{∗}(ΩBGp^{∧}; k) = k[τ, ξ]/(ξ^{2} + τ^{3}),

and all m_{i} are zero for i > 2.

2. The group algebra and its cohomology

We assume from now on, without loss of generality, that G has a normal cyclic Sylow p-subgroup P = C_{G}(P), with inertial index q = |G : P |. We shall assume that q > 1, which then forces p to be odd. For notation, let

G = ⟨g, s | g^{p^{n}} = 1, s^{q} = 1, sgs^{−1} = h^{γ}⟩ ≅ ℤ/p^{n} ⋊ ℤ/q,

where γ is a primitive qth root of unity modulo p^{n}. Let P = ⟨g⟩ and H = ⟨s⟩ as subgroups of G.

Let k be a field of characteristic p. The action of H on kP by conjugation preserves the radical series, and since |H| is not divisible by p, there are invariant complements. Thus we may choose an element U ∈ J(kP) such that U spans an H-invariant complement of J^{2}(kP) in J(kP). It can be checked that

U = ∑_{\begin{matrix}1⩽j⩽p^{n}−1, \\ (p,j)=1 \end{matrix}} g^{j}/j

is such an element, and that sUs^{−1} = γU. This gives us the following presentation for kG:

kG = k⟨s, U | U^{p^{n}} = 0, s^{q} = 1, sU = γUs⟩.

We shall regard kG as a ℤ[\frac{1}{q}]-graded algebra with |s| = 0 and |U| = 1/q. Then the bar resolution is doubly graded, and taking homomorphisms into k, the cochains C^{∗}(BG; k) inherit a double grading. The differential decreases the homological grading and preserves the internal grading. Thus the cohomology H^{∗}(G, k) = H^{∗}(BG; k) is doubly graded:

H^{∗}(BG; k) = k[x] ⊗ Λ(t)

where |x| = (−2q, p^{n}), |t| = (−2q + 1, h), and h = p^{n} − (p^{n} − 1)/q. Here, the first degree is homological, the second internal. The Massey product ⟨t, t, …, t⟩ (p^{n} repetitions) is equal to −x^{h}. This may easily be determined by restriction to P, where it is well known that the p^{n}-fold Massey product of the degree one exterior generator is a non-zero degree two element. The usual convention is to make the constant −1, because this Massey product is minus the nth Bockstein of t [5, Theorem 14].

3. A_{∞}-algebras

An A_{∞}-algebra over a field is a ℤ-graded vector space A with graded maps m_{n} : A^{⊗n} → A of degree n − 2 for n ≥ 1 satisfying

∑_{r+s+t=n}(−1)^{r+st}m_{r+1+t}(id^{⊗r} ⊗ m_{s} ⊗ id^{⊗t}) = 0

for n ≥ 1. The map m_{1} is therefore a differential, and the map m_{2} induces a product on H_{∗}(A).

A theorem of Kadeishvili [2] (see also Keller [3, 4] or Merkulov [8]) may be stated as follows. Suppose that we are given a differential graded algebra A, over a field k. Let Z^{∗}(A) be the cocycles, B^{∗}(A) be the coboundaries, and H^{∗}(A) = Z^{∗}(A)/B^{∗}(A). Choose a vector space splitting f_{1} : H^{∗}(A) → Z^{∗}(A) ⊆ A of the quotient. Then this gives by an inductive procedure an A_{∞} structure on H^{∗}(A) so that the map f_{1} is the degree one part of a quasi-isomorphism of A_{∞}-algebras.

If A happens to carry auxiliary gradings respected by the product structure and preserved by the differential, then it is easy to check from the inductive procedure that the maps in the construction may be chosen so that they also respect these gradings. It then follows that the structure maps m_{i} of the A_{∞} structure on H^{∗}(A) also respect these gradings.

Let us apply this to H^{∗}(BG; k). We examine the elements m_{i}(t, …, t). By definition, we have m_{1}(t) = 0 and m_{2}(t, t) = 0. The degree of m_{i}(t, …, t) is i times the degree of t, increased in the homological direction by i − 2. This gives

|m_{i}(t, …, t)| = i(−2q + 1, h) + (i − 2, 0) = (−2iq + 2i − 2, ih).

The homological degree is even, so if m_{i}(t, ⋯, t) is non-zero then it is a multiple of a power of x. Comparing degrees, if m_{i}(t, …, t) is a non-zero multiple of x^{α} then we have

2iq − 2i + 2 = 2αq, ih = αp^{n}.

Eliminating α, we obtain (iq − i + 1)p^{n} = ihq. Substituting h = p^{n} − (p^{n} − 1)/q, this gives i = p^{n}. Finally, since the Massey product of p^{n} copies of t is equal to −x^{h}, it follows that m_{p^{n}}(t, …, t) = ϵ(p^{n})x^{h}, where the sign is as defined in Remark 1.2 [6, Theorem 3.1]. Thus we have

m_{i}(t, …, t) = \begin{cases}ϵ(p^{n})x^{h} & i = p^{n} \\ 0 & otherwise. \end{cases}

We shall elaborate on this argument in a more general context in the next section, where we shall see that the rest of the A_{∞} structure is also determined in a similar way.

4. A_{∞} structures on a polynomial tensor exterior algebra

In this section, we shall examine the following general situation. Our goal is to establish that there are only two possible A_{∞} structures satisfying Hypothesis 4.1 below, and that the Koszul dual also satisfies the same hypothesis with the roles of a and b, and of h and ℓ reversed.

Hypothesis 4.1. A is a ℤ × ℤ-graded A_{∞}-algebra over a field k, where the operators m_{i} have degree (i − 2, 0), satisfying

(1) m_{1} = 0, so that m_{2} is strictly associative,

(2) ignoring the m_{i} with i > 2, the algebra A is k[x] ⊗ Λ(t) where |x| = (−2a, ℓ) and |t| = (−2b − 1, h), and

(3) ha − ℓb = 1.

Remarks 4.2. (i) The A_{∞}-algebra H^{∗}(BG; k) of the last section satisfies this hypothesis, with a = q, b = q − 1, h = p^{n} − (p^{n} − 1)/q, ℓ = p^{n}.

(ii) By comparing degrees, if we have m_{ℓ}(t, …, t) = ϵ(p^{n})x^{h} then (2b + 1)ℓ + 2 − ℓ = 2ah and so ha − ℓb = 1. This explains the role of part (3) of the hypothesis. The consequence is, of course, that a and b are coprime, and so are h and ℓ.

Lemma 4.3. If m_{i}(t, …, t) is non-zero, then i = ℓ > 2 and m_{ℓ}(t, …, t) is a multiple of x^{h}.

Proof. The argument is the same as in the last section. The degree of m_{i}(t, …, t) is i|t| + i − 2 = (−2ib − 2, ih). Since the homological degree is even, if m_{i}(t, …, t) is non-zero then it is a multiple of some power of x, say x^{α}. Then we have

2ib + 2 = 2αa, ih = αℓ.

Eliminating α gives (ib + 1)ℓ = iha, and so using ha − ℓb = 1 we have i = ℓ. Substituting back gives α = h. □

Elaborating on this argument gives the entire A_{∞} structure. If m_{ℓ}(t, …, t) is non-zero, then by rescaling the variables t and x if necessary we can assume that m_{ℓ}(t, …, t) = x^{h} (note that we can even do this without extending the field, since ℓ and h are coprime).

Proposition 4.4. If m_{ℓ}(t, …, t) = 0 then all m_{i} are zero for i > 2. If m_{ℓ}(t, …, t) = x^{h} then m_{ℓ}(x^{j_{1}}t, …, x^{j_{ℓ}}t) = x^{h+j_{1}+···+j_{ℓ}}, and all m_{i} for i > 2 on all other i-tuples of monomials give zero.

Proof. All monomials live in different degrees, so we do not need to consider linear combinations of monomials. Suppose that m_{i}(x^{j_{1}}t^{ε_{1}}, …, x^{j_{i}}t^{ε_{i}}) is some constant multiple of x^{j}t^{ε}, where each of ε_{1}, …, ε_{i}, ε is either zero or one. Then comparing degrees, we have

(j_{1} + ⋯ + j_{i})|x| + (ε_{1} + ⋯ + ε_{i})|t| + (i − 2, 0) = j|x| + ε|t|.

Setting

α = j_{1} + ⋯ + j_{i} − j, β = ε_{1} + ⋯ + ε_{i} − ε

we have β ⩽ i, and

α(−2a, ℓ) + β(−2b − 1, h) + (i − 2, 0) = 0.

Thus

2αa + 2βb + β + 2 − i = 0, αℓ + βh = 0.

Eliminating α, we obtain

−2βha + 2βℓb + βℓ + 2ℓ − iℓ = 0.

Since ha − ℓb = 1, this gives β = ℓ(i − 2)/(ℓ − 2). Combining this with β ⩽ i gives i ⩽ ℓ. If i < ℓ then β is not divisible by ℓ, and so αℓ + βh = 0 cannot hold. So we have β = i = ℓ, ε_{1} = ⋯ = ε_{ℓ} = 1, ε = 0, α = −h, and j = h + j_{1} + ⋯ + j_{ℓ}. Finally, the identities satisfied by the m_{i} for an A_{∞} structure show that all the constant multiples have to be the same, hence all equal to zero or after rescaling, all equal to minus one. □

Combining the lemma with the proposition, we obtain the following.

Theorem 4.5. Under the hypothesis above, if ℓ > 2 then there are two possible A_{∞} structures on A. There is the formal one, where m_{i} equal to zero for i > 2, and the non-formal one, where after replacing x and t by suitable multiples, the only non-zero m_{i} with i > 2 is m_{ℓ}, and the only non-zero values on monomials are given by

m_{ℓ}(x^{j_{1}}t, …, x^{j_{ℓ}}t) = x^{h+j_{1}+···+j_{ℓ}}.

Theorem 4.6. Let G = ℤ/p^{n} ⋊ ℤ/q as above, and k a field of characteristic p. Then the A_{∞} structure on H^{∗}(G, k) given by Kadeishvili’s theorem may be taken to be the non-formal possibility named in the above theorem, with a = q, b = q − 1, h = p^{n} − (p^{n} − 1)/q, ℓ = p^{n}.

Proof. Since we have m_{p^{n}}(t, …, t) = ϵ(p^{n})x^{h}, the formal possibility does not hold. □

Remark 4.7. Dag Madsen’s thesis [7] has an appendix in which the A_{∞} structure is computed for the cohomology of a truncated polynomial ring, reaching similar conclusions by more direct methods.

5. Loops on BGp^{∧}

In general for a finite group G we have H^{∗}(BGp^{∧}; k) ≅ H^{∗}(BG; k) = H^{∗}(G, k) and π_{1}(BGp^{∧}) = G/O^{p}(G), the largest p-quotient of G. In our case, G = ℤ/p^{n} ⋊ ℤ/q with q > 1, we have G = O^{p}(G) and so BGp^{∧} is simply connected. So the Eilenberg–Moore spectral sequence converges to the homology of its loop space:

Tor_{∗,∗}^{H^{∗}(G,k)}(k, k) ⇒ H_{∗}(ΩBGp^{∧}; k).

The internal grading on C^{∗}(BG; k) gives this spectral sequence a third grading that is preserved by the differentials, and H_{∗}(ΩBGp^{∧}; k) is again doubly graded. Since H^{∗}(G, k) = k[x] ⊗ Λ(t) with |x| = (−2q, p^{n}) and |t| = (−2q + 1, h), it follows that the E^{2} page of this spectral sequence is k[τ] ⊗ Λ(ξ) where |ξ| = (−1, 2q, p^{n}) and |τ| = (−1, 2q − 1, h) (recall h = p^{n} − (p^{n} − 1)/q). Provided that we are not in the case h = 2, which only happens if p^{n} = 3, ungrading E^{∞} gives

H_{∗}(ΩBGp^{∧}; k) = k[τ] ⊗ Λ(ξ)

with |τ| = (2q − 2, h) and |ξ| = (2q − 1, p^{n}).

In the exceptional case h = 2, p^{n} = 3, we have q = 2, and the group G is the symmetric group Σ_{3} of degree three. An explicit computation (for example by squeezed resolutions [1]) gives

H_{∗}(Ω(BΣ_{3})3^{∧}; k) = k[τ, ξ]/(ξ^{2} + τ^{3})

with |τ| = (2, 2) and |ξ| = (3, 3), and the two gradings collapse to a single grading.

Applying Theorem 4.5, and using the fact that either formal case is Koszul dual to the other, we have the following.

Theorem 5.1. Suppose that p^{n} ≠ 3. Then the A_{∞} structure on H_{∗}(ΩBGp^{∧}; k) = k[τ] ⊗ Λ(ξ) is given by

m_{h}(τ^{j_{1}}ξ, …, τ^{j_{h}}ξ) = ϵ(h)τ^{p^{n}+j_{1}+···+j_{h}},

and for i > 2, all m_{i} on all other i-tuples of monomials give zero. □

Using [6] again, we may reinterpret this in terms of Massey products.

Corollary 5.2. In H_{∗}(ΩBGp^{∧}), the Massey products ⟨ξ, …, ξ⟩ (i times) vanish for 0 < i < h, and give −τ^{p^{n}} for i = h. □

Note that the exceptional case p^{n} = 3 also fits the corollary, if we interpret a 2-fold Massey product as an ordinary product.

References

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