Massey products in the homology of the loopspace of a p-completed classifying space: finite groups with cyclic Sylow p-subgroups

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_*(\Omega BG\phat;k) are A_{\infty} algebras whose structure we determine up to quasi-isomorphism.


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(BG ∧ p ); 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(BG ∧ p ); k). Of course if G is a p-group, BG is p-complete so Ω(BG ∧ p ) ≃ G, but in general H * (Ω(BG ∧ p ); k) is infinite dimensional. Henceforth we will omit the brackets from Ω(BG ∧ p ).
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 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 BG ∧ p . Our main theorem determines the multiplication maps m i in the A ∞ structure on H * (BG; k) and H * (Ω(BG ∧ p ); k) arising from C * (BG; k) and C * (Ω(BG ∧ p ); 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 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 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 Up to quasi-isomorphism, the A ∞ structure on H * (BG; k) is determined by . . , x j p n t) = ǫ(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 This implies m h (τ j 1 ξ, . . . , τ j h ξ) = ǫ(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 , and all m i are zero for i > 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 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 is such an element, and that sUs −1 = γU. This gives us the following presentation for kG: We shall regard kG as a Z[ 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: 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].

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 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.
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) = ǫ(p n )x h , where the sign is as defined in Remark 1.2 [6, Theorem 3.1]. Thus we have 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.

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.   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.
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). 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|.
Combining the lemma with the proposition, we obtain the following. 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) = ǫ(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.
Applying Theorem 4.5, and using the fact that either formal case is Koszul dual to the other, we have the following.
Using [6] again, we may reinterpret this in terms of Massey products.
Note that the exceptional case p n = 3 also fits the corollary, if we interpret a 2-fold Massey product as an ordinary product.