ELEMENTARY ABELIAN GROUPS ACTING ON PRODUCTS OF SPHERES

A. ADEM AND D. J. BENSON

1. Introduction

One of the basic results concerning actions of finite groups is the well–known fact that every finite abelian group acting freely on a sphere must be cyclic. The extension of this theorem to the case of a product of spheres has been a problem of longstanding interest in both topology and algebra (see [1], [2], [7], [8], [9], [5]).

The first important result beyond the case of a single sphere was the theorem of Gunnar Carlsson [8], [9] which states that if an elementary abelian p-group G of rank r acts freely and cellularly on a CW -complex X ≃ (S^{n})^{k} (a product of spheres of the same dimension) with trivial action on homology, then k ≥ r. His proof yields further information— that the cellular chains on X decompose up to equivariant homotopy equivalence as a tensor product of complexes which have the homology of a single sphere. The chain complex is determined up to equivariant chain homotopy equivalence by the transgressions in H^{∗}(BG; 𝔽_{p}) of the fundamental classes of the spheres. A different proof of the same theorem was given in Benson–Carlson [5] using the machinery of varieties for modules.

Some time later, the case of homologically non–trivial actions was considered in Adem-Browder [2], where the inequality above was established (and generalized) except in the cases p = 2 and n = 1, 3, or 7. The methods used there, however, were considerably different, involving the notion of degree and rational representation theory. At the end of [2], the authors raised the question of whether or not the inequality could be strengthened to r ≤ dimH_{n}(X; 𝔽_{p})^{G}.

In this paper we establish this stronger inequality for large classes of actions: namely those which induce a permutation module on H^{n}(X; 𝔽_{p}). Geometrically these are clearly the most natural kinds of actions on a product of spheres. What is slightly less evident is the fact that for p = 2, it follows from the Hopf invariant one theorem that provided n ≠ 1, 3 or 7, the basis of H^{n}(X; 𝔽_{2}) corresponding to the k spheres is permuted by G, so that H^{n}(X; 𝔽_{2}) is always isomorphic to a permutation 𝔽_{2}G-module.

Our method, which is an extension of the method of [5], works without restriction on p and as in the case of [9] yields more information than the inequality alone. Our main result is the following.

Theorem 1.1. Let G be an elementary abelian p-group of rank r acting freely on a finite dimensional CW complex X ≃ (S^{n})^{t}, in such a way that the basis u_{1}, …, u_{t} of H^{n}(X; 𝔽_{p}) corresponding to the t spheres is permuted by G. Then the number of orbits of G on {u_{1}, …, u_{t}} is at least r. Furthermore, the mod p cellular chains on X are chain homotopy equivalent to a tensor product of complexes corresponding to the orbits. The complex corresponding to an orbit is tensor induced from a complex for the stabilizer, with the homology of a single n-sphere.

Both authors are partly supported by grants from the NSF.

2. Varieties and tensor induction

The proof of the main theorem depends on understanding varieties for modules and tensor induction. So we begin with a brief review of these topics.

We shall work in terms of the rank variety, which is defined as follows. Let G = ⟨g_{1}, …, g_{r}⟩ be an elementary abelian p-group of rank r, and k be an algebraically closed field of characteristic p. Let X_{i} be the element g_{i}−1 of the group algebra kG, so that X_{i}^{p} = 0 and X_{i} is in the Jacobson radical J(kG). Then we set V_{G}^{r}(k) equal to J(kG)/J^{2}(kG), and write x_{1}, …, x_{r} for the images of X_{1}, …, X_{r} in V_{G}^{r}(k). In fact, these elements form a basis for V_{G}^{r}(k) as a k-vector space. If

0 ≠ α = λ_{1}x_{1} + ⋯ + λ_{r}x_{r} ∈ V_{G}^{r}(K)

then we set

u_{α} = 1 + λ_{1}X_{1} + ⋯ + λ_{r}X_{r},

so that u_{α} is a unit of order p in kG. Now if M is a finitely generated kG-module, we set

V_{G}^{r}(M) = {0} ∪ {α ∈ V_{G}^{r}(k) | M ↓_{u_{α}} is not free} ⊆ V_{G}^{r}(k),

This is a closed homogeneous subvariety of V_{G}^{r}(k), which is equal to {0} if and only if M is a projective kG-module.

The dimension of V_{G}^{r}(M) is an invariant called the complexity of M as a kG-module. It measures the polynomial rate of growth of the minimal resolution of M. The properties of rank varieties and complexity that we shall be using are summarized in the following proposition.

Proposition 2.1. Let M and N be finitely generated kG-modules. Then

(i) V_{G}^{r}(M ⊕ N) = V_{G}^{r}(M) ∪ V_{G}^{r}(N) (ii) V_{G}^{r}(M ⊗_{k} N) = V_{G}^{r}(M) ∩ V_{G}^{r}(N) (iii) V_{G}^{r}(M) = {0} if and only if M is projective. (iv) Let ζ ∈ H^{n}(G, k) be a nonzero element of cohomology, represented by a cocycle ζ̂ : Ω^{n}k → k. Then the kernel L_{ζ} of ζ̂ is a kG-module of complexity r−1 if ζ is non-nilpotent, and complexity r if ζ is nilpotent.

(v) If 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence of kG-modules, then for {i, j, k} = {1, 2, 3} we have V_{G}^{r}(M_{i}) ⊆ V_{G}^{r}(M_{j}) ∪ V_{G}^{r}(M_{j}). In particular, the two largest complexities among M_{1}, M_{2} and M_{3} are equal.

If H is a subgroup of a finite group G and M is a kH-module, then the usual process of induction can be written as M ↑^{G}= ⨁_{g} g ⊗ M, where the elements g run over a set of left coset representatives of H in G, and G acts as follows. If g_{0} ∈ G then for each coset representative g, g_{0}g can be written as g′h where g′ is another coset representative and h ∈ H. Then the action is defined by g_{0}(g ⊗ m) = g′ ⊗ hm.

The tensor induced module M ⊗^{↑}^{G} is defined in a similar way, but using the tensor product of the g ⊗M instead of the direct sum, with the same definition for the action. Details can be found in Section 3.15 of [3]. There is also a corresponding process of tensor induction for chain complexes of modules, which is described in detail in Section 4.1 of [4].

Lemma 2.2. Let G ≥ G′ be elementary abelian p-groups. Then

V_{G}^{r}((kG′)⊗^{↑}^{G}) = ⋃_{G=G′×G′′} V_{G′′}^{r}(k).

Proof. By Proposition 3.15.2 (iv) of [3], kG′⊗^{↑}^{G} is a direct sum of modules induced up from subgroups G′′ satisfying G′∩G′′ = {1}. The restriction of kG′⊗^{↑}^{G} to any such G′′ has trivial direct summands, so V_{G′′}^{r}(k) ⊆ V_{G}^{r}(kG′⊗^{↑}^{G}). The maximal such G′′ satisfy G = G′×G′′.

Lemma 2.3. Let G ≥ G′ be elementary abelian p-groups, with G of p-rank r. Let ζ be nonzero element of H^{n}(G′, k) for some n > 0.

(i) If ζ is non-nilpotent, then the complexity of L_{ζ} ⊗^{↑}^{G} is equal to r − 1. (ii) If ζ is nilpotent, then the complexity of L_{ζ} ⊗^{↑}^{G} is equal to r.

Proof. We prove statement (i) by induction on r. The proof of statement (ii) is similar. Consider the complex 0 → L_{ζ} → Ω^{n}(k) → 0 of kG′-modules, whose homology is k in degree zero, and zero elsewhere. Tensor induce this complex up to G. This gives a complex of the form

0 → L_{ζ} ⊗^{↑}^{G} → U_{t−1} → U_{t−2} → ⋯ → U_{1} → Ω^{n}k⊗^{↑}^{G} → 0,

with t = |G : G′|, whose homology is again k in degree zero. The action of G on this degree zero homology is trivial, because for a p-group in characteristic p there is no possible nontrivial action. The augmentation map from Ω^{n}k⊗^{↑}^{G} to k represents Norm_{G′,G}(ζ) ∈ H^{nt}(G, k), and so its kernel is L_{Norm_{G′,G}(ζ)}. Each of the modules U_{i} with 1 ≤ i ≤ t − 1 is a direct sum of modules induced up from kG′′-modules with G′ ≤ G′′ < G. The kG′′- modules in question have the property that their restriction to G′ is a tensor product of L_{ζ} with some module, so their complexity is strictly less than the rank of G′′. It follows that the complexity of each U_{i} with 1 ≤ i ≤ t − 1 is at most r − 2. Since L_{Norm_{G′,G}(ζ)} has complexity exactly r − 1, the same is true of the image of U_{i} → U_{i−1} for 2 ≤ i ≤ t − 1 by induction on i (using part (v) of Proposition 2.1), and so L_{ζ} ⊗^{↑}^{G} also has complexity exactly r − 1.

3. Proof of the main theorem

This section consists of a proof of Theorem 1.1. Let X ≃ (S^{n})^{t} be a finite CW complex, and G an elementary abelian p-group of rank r acting freely on X. We may also assume that the action is cellular. Let k be an algebraically closed field of characteristic p, and let v_{1}, …, v_{t} be the fundamental classes of the t spheres in H^{n}(X; k). In other words, if ρ_{i} : X → S^{n} is the projection onto the ith sphere, then v_{i} is the image of the fundamental class of S^{n} under ρ_{i}^{∗} : H^{n}(S^{n}; k) → H^{n}(X; k). We also assume that G acts on H^{n}(X; k) in such a way as to permute the basis v_{1}, …, v_{t}. Let m be the number of orbits of this action. Our goal is to prove that r ≤ m.

Denote by C_{∗} the complex

0 → C_{s} ⟶^{∂_{s}} C_{s−1} → ⋯ ⟶^{∂_{1}} C_{0} → 0

of mod p cellular chains on X, and choose a diagonal approximation C_{∗} → C_{∗} ⊗ C_{∗} (see for example tom Dieck [10], p. 104). Also, if G′ is a subgroup of G, then thinking of X as a space with G′-action, we obtain a G-action in the usual way on

Hom_{G′}(G, X) = X × ⋯ × X (|G : G′| times).

A chain approximation to the diagonal embedding X → X ×⋯×X gives rise to a diagonal approximation C_{∗} → (C_{∗})_{G′} ⊗^{↑}^{G}, where the latter is the tensor induced complex.

Now the truncation

C_{n} → C_{n−1} → ⋯ → C_{0} → 0

of C_{∗} is a complex of free kG-modules whose homology is k in degree zero and which is exact elsewhere. The kG-module Ker(∂_{n})/Im(∂_{n+1}) ≅ H_{n}(X; k) has a free k-basis ξ_{1}, …, ξ_{t} dual to the basis v_{1}, …, v_{t} of H^{n}(X; k). So for each orbit O of G on ξ_{1}, …, ξ_{t}, say with stabilizer G′, we have a surjective map Ker(∂_{n}) → k_{G′} ↑^{G}.

By Frobenius reciprocity (sometimes called the Nakayama relations) we have

Hom_{kG}(Ker(∂_{n}), k_{G′} ↑^{G}) ≅ Hom_{kG′}(Ker(∂_{n}), k).

Denote by L_{O} the kernel of the corresponding kG′-module homomorphism Ker(∂_{n}) → k. Then we have a diagram of kG′-modules

[figure: 0; L_{O}; 0; C_{s}; ⋯; C_{n+1}; C_{n}; C_{n−1}; ⋯; C_{0}; 0; 0; C_{n}/L_{O}; C_{n−1}; ⋯; C_{0}; 0; 0]

The bottom row of this diagram is a direct sum of a complex C_{∗}^{O} of finitely generated modules and an exact complex of projectives. Note that the homology of C_{∗}^{O} is the homology of S^{n}. Then the map of kG′-complexes C_{∗} → C_{∗}^{O} gives rise to a map of kGcomplexes (C_{∗})_{G′} ⊗^{↑}^{G} → C_{∗}^{O}⊗^{↑}^{G}. Write Ĉ_{∗}^{O} for the complex C_{∗}^{O}⊗^{↑}^{G}. Then composing this map with the diagonal map C_{∗} → (C_{∗})_{G′} ⊗^{↑}^{G}, we obtain a map μ_{O} : C_{∗} → Ĉ_{∗}^{O}.

Now iterating the diagonal approximation C_{∗} → C_{∗} ⊗ C_{∗} enough times, we obtain a map C_{∗} → C_{∗} ⊗ ⋯ ⊗ C_{∗} (m copies). Then we compose with the maps μ_{O} to obtain a map

μ : C_{∗} → C_{∗} ⊗ ⋯ ⊗ C_{∗} \underrightarrow{⨂_{O} μ_{O}} ⨂_{O} Ĉ_{∗}^{O}.

This map is an isomorphism on degree n homology. Since both complexes have homology which comultiplies like a product of spheres, this is enough to show that μ is a homology isomorphism.

Now consider the modules in the complex Ĉ_{∗}^{O}. If the orbit O has length l, then this complex has length nl. The terms in this complex are direct sums of modules of the form M_{G′} ⊗^{↑}^{H} ↑^{G}, where G′ is the orbit stabilizer, G′ ≤ H ≤ G, and M is a tensor product of projective kG′-modules and copies of L_{O}. Except in degree nl, this tensor product involves at least one copy of kG′ and so M is a projective kG′-module. So by Lemma 2.2, for i < nl, V_{G}^{r}(Ĉ_{i}^{O}) ⊆ ⋃_{G′′} V_{G′′}^{r}(k), where G′′ runs over the set of subgroups satisfying G′ ∩ G′′ = {1}. The module Ĉ_{nl}^{O} ≅ L_{O}⊗^{↑}^{G} has complexity either r − 1 or r by Lemma 2.3.

Now tensoring together the complexes Ĉ_{∗}^{O} as O ranges over the orbits, we see that in degrees less than nt, by Proposition 2.1 (i) and (ii) we obtain modules whose variety is contained in ⋂_{O} ⋃_{G′′} V_{G′′}^{r}(k), where G′′ runs over the subgroups whose intersection with the relevant orbit stabilizer G′ is trivial.

Lemma 3.1. The union over the orbits O of the orbit stabilizers G′ ≤ G is equal to the whole of G.

Proof. If the union is not the whole of G, then there is an element g of order p in G generating a copy of ℤ/p for which H^{n}(X; k) is a free kℤ/p-module.

Choose a basepoint in Eℤ/p, and let i : X → Eℤ/p ×_{ℤ/p} X, j : X^{p} → Eℤ/p ×_{ℤ/p} X^{p} be the maps induced by the inclusion of this basepoint. Let

P : H^{n}(X; k) → H^{np}(Eℤ/p ×_{ℤ/p} X^{p}; k)

be the Steenrod power map. Finally, let h : X → X^{p} be given by

h : x ↦ (x, g(x), …, g^{p−1}(x)),

and Id × h : Eℤ/p ×_{ℤ/p} X → Eℤ/p ×_{ℤ/p} X^{p} be the induced map.

[figure: X; h; i; X^{p}; j; Eℤ/p ×_{ℤ/p} X; Id×h; Eℤ/p ×_{ℤ/p} X^{p}]

If z ∈ H^{n}(X; k) then by the construction of P we have j^{∗}P z = z × ⋯ × z ∈ H^{np}(X^{p}; k). Thus

i^{∗}(Id × h)^{∗}P z = h^{∗}j^{∗}P z = h^{∗}(z × ⋯ × z) = z.g(z) … g^{p−1}z.

It follows that z.g(z) … g^{p−1}(z) is represented by an element of H^{0}(Bℤ/p; H^{np}(X; k)) which is a universal cycle in the spectral sequence of the Borel construction

H^{∗}(Bℤ/p; H^{∗}(X; k)) ⇒ H^{∗}(Eℤ/p ×_{ℤ/p} X; k) ≅ H^{∗}(X/⟨g⟩; k).

Taking the product of these elements as z runs over an kℤ/p-basis for H^{n}(X; k), we obtain a universal cycle in the top degree on the fiber, H^{0}(Bℤ/p; H^{nm}(X; k)). It follows that the product of this element with any element on the base is a universal cycle, which contradicts the fact that this spectral sequence has to converge to the cohomology of a finite complex.

(This lemma is based on the methods developed by Bredon (see [6], page 405). This approach was used in [1] to show that if G = ℤ/p acts freely on X = (S^{n})^{k}, then M = H^{n}(X; ℤ) splits off a trivial direct summand as a ℤG module. More generally it can be extended to show that if a p-group P acts on such an X with M ⊗ 𝔽_{p} a free 𝔽_{p}P module, then X^{P} ≠ ∅ [11].)

It follows from this lemma and Proposition 2.1 (iii) that the modules in degree less than nt in the complex ⨂_{O} Ĉ_{∗}^{O} are projective, because ⋂_{O} ⋃_{G′′} V_{G′′}^{r}(k) = {0}. The module in degree nt is L = ⨂_{O} L_{O}⊗^{↑}^{G}, which, by Proposition 2.1 (ii), has complexity at least equal to r − m, where m is the number of orbits O. Our claim is that L must also be projective. To prove this, consider the map of chain complexes C_{∗} → ⨂_{O} Ĉ_{∗}^{O}. Add a finite length exact chain complex F_{∗} of free kG-modules to C_{∗} to make the map C_{∗} ⊕ F_{∗} → ⨂_{O} Ĉ_{∗}^{O} surjective. The long exact sequence in homology shows that the kernel is a finite length exact complex all of whose terms, except possibly the ntth, are projective. This implies that the ntth term is also projective, because kG is a self-injective algebra. But the ntth term is the kernel of the surjective map C_{nt} ⊕ F_{nt} → L, and C_{nt} ⊕ F_{nt} is also projective, so it follows that L is projective. Since L has complexity at least equal to r − m, we apply Proposition 2.1 (iii) to deduce that r ≤ m, and the main theorem is proved.

Given a permutation module M over kG, it is clear that M^{G} is a vector space of rank equal to the number of orbits for the action on a suitable basis. From this we obtain

Corollary 3.2. If G = (ℤ/p)^{r} acts freely on X ≃ (S^{n})^{k}, permuting a basis for H_{n}(X; 𝔽_{p}), then r ≤ dim H_{n}(X; 𝔽_{p})^{G}.

Using the fact that for n ≠ 1, 3, 7 there are no maps S^{n}×S^{n} → S^{n} of bidegree (odd, odd) (this follows from the Hopf invariant one theorem, see [6], page 419), we obtain

Corollary 3.3. If G = (ℤ/2)^{r} acts freely on X ≃ (S^{n})^{k} and n ≠ 1, 3, or 7, then r ≤ dim H_{n}(X; 𝔽_{2})^{G}.

This represents a substantial affirmative answer to question 7.1 in [2]. The general case seems rather more complicated and remains open.

References

[1] A. Adem. ℤ/p actions on (S^{n})^{k}. Trans. Amer. Math. Soc. 300 (1987), 791–809.

[2] A. Adem. and W. Browder. The free rank of symmetry of (S^{n})^{k}. Invent. Math. 92 (1988), 431–440.

[3] D. J. Benson. Representations and cohomology of finite groups, Vol. I. Cambridge University Press, 1991.

[4] D. J. Benson. Representations and cohomology of finite groups, Vol. II. Cambridge University Press, 1991.

[5] D. J. Benson and J. F. Carlson. Complexity and multiple complexes. Math. Zeit. 195 (1987), 221–238.

[6] G. Bredon. Introduction to compact transformation groups. Academic Press, 1972.

[7] W. Browder. Cohomology and group actions. Invent. Math. 71 (1983), 599–607.

[8] G. Carlsson. On the nonexistence of free actions of elementary abelian groups on products of spheres. Amer. J. Math. 102 (1980), 1147–1157.

[9] G. Carlsson. On the rank of abelian groups acting freely on (S^{n})^{k}. Invent. Math. 69 (1982), 393–400.

[10] T. tom Dieck. Transformation groups. De Gruyter Series in Mathematics 8, Walter de Gruyter, Berlin/New York 1987.

[11] E. Yalcin Ph.D. Thesis, University of Wisconsin. In preparation.

Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA

   E-mail address: adem@math.wisc.edu

Department of Mathematics, University of Georgia, Athens GA 30602, USA

   E-mail address: djb@byrd.math.uga.edu