CONSTRUCTING HOMOLOGICALLY TRIVIAL ACTIONS ON PRODUCTS OF SPHERES

ÖZGÜN ÜNLÜ AND ERGÜN YALÇIN

Abstract. We prove that if a finite group G has a representation with fixity f, then it acts freely and homologically trivially on a finite CW-complex homotopy equivalent to a product of f + 1 spheres. This shows, in particular, that every finite group acts freely and homologically trivially on some finite CW-complex homotopy equivalent to a product of spheres.

1. Introduction

It is known that every finite group acts freely on a product of spheres 𝕊^{n_{1}} × ⋯ × 𝕊^{n_{k}} for some n_{1}, n_{2}, …, n_{k}. This follows from a construction given in [15, pg. 547] which is attributed to J. Tornehave by B. Oliver. The construction is based on a simple idea that one can permute the spheres in a product to get smaller isotropy. More specifically, for a finite group G, one defines the G-space X as a product of G-spaces Map_{⟨g⟩}(G, 𝕊(ρ_{g})) over all elements g ∈ G, where 𝕊(ρ_{g}) denotes the unit sphere of a nontrivial one-dimensional complex representation ρ_{g} : ⟨g⟩ → ℂ. Note that because of the way X is constructed, G acts freely on X, but the induced action of G on the homology of X is not a trivial action in general. It is interesting to ask if there exists a similar construction so that the induced action on homology is trivial. The following was stated as a problem by J. Davis in the Problem Session of Banff 2005 conference on Homotopy Theory and Group Actions:

Question 1.1. Does every finite group act freely on some product of spheres with trivial action on homology?

To find a free, homologically trivial action of a finite group G on a product of spheres, one may try to take a family of G-spheres 𝕊^{n_{i}} for i = 1, …, k and let G act on the product 𝕊^{n_{1}} × ⋯ × 𝕊^{n_{k}} by diagonal action. An action which is obtained in this way is called a product action in general and a linear action if each G-sphere in the product is a unit sphere of a representation. By construction, product actions are homologically trivial if the action on each sphere is homologically trivial but not every finite group has a free product action on a product of spheres. For example, it is known that the alternating group A_{4} cannot act freely on a product of spheres by a product action. This follows from a result of Oliver [15, Theorem 1] which says that A_{4} does not act freely on any product of equal dimensional spheres with trivial action on homology. On the other hand, it can be shown that A_{4} acts freely on 𝕊^{2} × 𝕊^{3} with trivial action on homology. So, one cannot answer the above question affirmatively by considering only the product actions.

Date: April 17, 2013.

2010 Mathematics Subject Classification. Primary: 57S25; Secondary: 55R91.

Both of the authors are partially supported by TÜBİTAK-TBAG/110T712.

The situation with A_{4} is not an exceptional case. For example, if one searches through the characters of a finite group G and tries to see when G has a family of characters χ_{1}, …, χ_{k} such that G acts freely on 𝕊(χ_{1}) × ⋯ × 𝕊(χ_{k}), then one notices that not many groups have such a family of characters. In fact, Urmie Ray [16] showed that finite groups which have free linear actions are very rare: If a finite group G has a free linear action on a product of spheres, then all nonabelian simple sections of G are isomorphic to A_{5} or A_{6}.

In this paper we attack the problem stated above using some more recent construction methods that were developed to study the rank conjecture. In particular, we use some ideas from our earlier papers [19] and [20] where we constructed free actions on products of spheres for finite groups which have representations with small fixity and for p-groups with small rank. Fixity of a representation V of a finite group G over a field F is defined as the maximum of dimensions dim_{F} V^{g} over all elements g ∈ G − {1}. If ρ : G → U(n) is a faithful complex representation with fixity f, then G acts freely on the space X = U(n)/U(n − f − 1). For small values of f, one can modify the space X to obtain a free action on a product of f + 1 spheres (see [1] and [19]). In this paper we improve this result to all values of fixity for actions on finite CW-complexes homotopy equivalent to product of spheres.

Theorem 1.2. Let G be a finite group. If G has a faithful complex representation with fixity f, then G acts freely on a finite complex X homotopy equivalent to a product of f + 1 spheres with trivial action on homology.

We prove this theorem using a recursive method for constructing free actions. In fact in the construction we use symplectic representations instead of complex representations and obtain free actions on a products of spheres with dimensions n_{i} ≥ 3. The recursive method involves the construction of a G-equivariant spherical fibration p : E → X over a given finite G-CW-complex X in each step. We require that the total space E is also homotopy equivalent to a finite G-CW-complex and that the G-action on E has smaller isotropy than the G-action on X. Once they are constructed, by taking fiber joins these fibrations can be replaced by G-fibrations which are non-equivariantly homotopy equivalent to trivial fibrations. This gives a G-action on a finite CW-complex Y homotopy equivalent to X × 𝕊^{N} for some N > 0. Using this method, after some steps, one gets a free action on a product of spheres. This method was first developed by Connolly and Prassidis [4] and later used also in [2] and [18].

Since every finite group has a faithful complex representation, every finite group has a complex representation with fixity f for some positive integer f. Hence, as a corollary of Theorem 1.2 we obtain an affirmative answer to Question 1.1 in homotopy category.

Corollary 1.3. Every finite group acts freely and homologically trivially on some finite CW-complex X homotopy equivalent to a product of spheres

The paper is organized as follows: In Section 2, we introduce G-fibrations and discuss the effects of taking fiber joins of G-fibrations. In Section 3, we discuss the equivariant Federer spectral sequence introduced by Møller [14] and using it, we give another proof for a theorem by M. Klaus [12] (see Theorem 3.1). In Section 4, we introduce our main construction method, and finally in Section 5, we prove Theorem 1.2.

2. G-fibrations

In this section, we first give some preliminaries on G-fibrations and then prove some lemmas on fiber joins of G-fibrations. For more details on this material we refer the reader to [13] and [22]. Some of this material also appears in [4], [9], [12], and [18].

Definition 2.1. A G-fibration is a G-map p : E → B which satisfies the following homotopy lifting property for every G-space X: Given a commuting diagram of G-maps

\xymatrix{X × \{0\} \ar[d] \ar[r]^{h} & E \ar[d]^{p} \\ X × I \ar[r]^{H} & B,}

there exists a G-map H̃ : X × I → E such that H̃|_{X×{0}} = h and p ∘ H̃ = H.

Given a G-fibration p : E → B over B, the isotropy subgroup G_{b} ≤ G of a point b ∈ B acts on the fiber space F_{b} := p^{−1}(b). So, F_{b} is a G_{b}-space. Let us denote the set of isotropy subgroups of the G-action on B by Iso(B).

Definition 2.2. Let {F_{H}} denote a family of H-spaces over all H ∈ Iso(B). If for every b ∈ B, the fiber space F_{b} is G_{b}-homotopy equivalent to F_{G_{b}}, then p : E → B is said to have fiber type {F_{H}}.

Note that in general a G-fibration does not have to have a fiber type, i.e., for b_{1}, b_{2} ∈ B with G_{b_{1}} = G_{b_{2}} = H, it may happen that F_{b_{1}} and F_{b_{2}} are not H-homotopy equivalent. But throughout the paper we only consider G-fibrations which have a fiber type. Observe that if p : E → B is a G-fibration such that B^{H} is path connected for every H ∈ Iso(B), then p has a fiber type since for every b_{1}, b_{2} ∈ B^{H}, the fiber spaces F_{b_{1}} and F_{b_{2}} are Hhomotopy equivalent by a standard argument in homotopy theory. In our applications the G-fibrations that we construct will often satisfy this connectedness property.

If p : E → B is a G-fibration with fiber type {F_{H}} such that for all H ∈ Iso(B) the fixed point space B^{H} is connected, then the family {F_{H}} satisfies a certain compatibility condition. To see this, let H, K ∈ Iso(B) such that g^{−1}Kg ≤ H for some g ∈ G. Then, we have gB^{H} ⊆ B^{K}, so by the connectedness of B^{K}, we obtain that for every b ∈ B^{H}, the K-space gF_{b} is K-homotopy equivalent to F_{K}. This means that the K-spaces Res_{K} g^{∗}F_{H} and F_{K} are K-homotopy equivalent for all H ∈ Iso(B) where Res_{K} g^{∗}F_{H} is the space F_{H} which is considered as a K-space through the map K → H defined by k → g^{−1}kg.

Definition 2.3. Let H be a family of subgroups of G closed under conjugation. A family of H-spaces {F_{H}} over all H ∈ H is called a compatible family of H-spaces if for every H, K ∈ H with g^{−1}Kg ≤ H for some g ∈ G, the K-space Res_{K} g^{∗}F_{H} is K-homotopy equivalent to F_{K}, where Res_{K} g^{∗}F_{H} is the the space F_{H} considered as a K-space through the map K → H defined by conjugation k → g^{−1}kg.

The main aim of this section is to introduce some tools for construction of G-fibrations with fiber type {F_{H}} for a given compatible family {F_{H}}. We first introduce some more terminology:

Given two G-fibrations p_{1} : E_{1} → B and p_{2} : E_{2} → B over the same G-space B, a G-map f : E_{1} → E_{2} is called a fiber preserving map if it satisfies p_{2} ∘ f = p_{1}. Two fiber preserving G-maps f, f′ : E_{1} → E_{2} are said to be G-fiber homotopic if there is a G-map H : E_{1} × I → E_{2} which is fiber preserving at each t ∈ I such that H(x, 0) = f(x) and H(x, 1) = f′(x) for all x ∈ E_{1}. We say two G-fibrations p_{1} : E_{1} → B and p_{2} : E_{2} → B are G-fiber homotopy equivalent if there are fiber preserving G-maps f_{1} : E_{1} → E_{2} and f_{2} : E_{2} → E_{1} such that f_{1} ∘ f_{2} and f_{2} ∘ f_{1} are G-fiber homotopic to identity maps.

For an H-space F_{H}, let Aut_{H}(F_{H}) denote the topological monoid of self H-homotopy equivalences of F_{H}. Note that Aut_{H}(F_{H}) is not a connected space in general but it is easy to show that all its components have the same homotopy type. When we need to choose a component, we often take the connected component which includes the identity map. We denote this component by Aut_{H}^{I}(F_{H}).

Since Aut_{H}(F_{H}) is a monoid, the usual construction of classifying spaces for monoids applies, and we get a universal fibration E Aut_{H}(F_{H}) → B Aut_{H}(F_{H}) with fiber Aut_{H}(F_{H}). From this one also obtains a fibration

F_{H} → E_{H} → B Aut_{H}(F_{H})

where E_{H} = E Aut_{H}(F_{H}) ×_{Aut_{H}(F_{H})} F_{H}. This is actually an H-fibration with trivial Haction on the base space. It turns out that this fibration is a universal fibration for all H-equivariant fibrations with trivial action on the base space.

Theorem 2.4. Let H be a finite group, F_{H} be a finite H-CW-complex, and B be a CWcomplex with trivial H-action. Then, there is a one-to-one correspondence between H-fiber homotopy classes of H-fibrations over B with fiber F_{H} and the set of homotopy classes of maps B → B Aut_{H}(F_{H}). The correspondence is given by taking the pullback of the universal H-fibration described above via the map f : B → B Aut_{H}(F_{H}).

This theorem is proved in [9] in full detail. The proof is based on the proof of Stasheff’s theorem on the classification of Hurewicz fibrations [17]. More general versions of this theorem also appear in [8] and [22].

Also note that, as in the case of orientable vector bundle theory, we can give an orientable version of the classification of H-fibrations over a trivial H-space B. If p : E → B is an H-fibration with fiber F_{H} over a trivial H-space, then there is a natural group homomorphism χ : π_{1}(B) → π_{0}(Aut_{H}(F_{H})) = E_{H}(F_{H}) where E_{H}(F_{H}) denotes the group of homotopy classes of self H-homotopy equivalences of F_{H}. If this homomorphism is trivial, then we call the H-fibration p a homotopy orientable H-fibration. This notion of orientability is stronger than usual notion of orientable fibration where one only requires the action on the homology of F_{H} to be trivial (see [6]). Note that an H-fibration is homotopy orientable if and only if its classifying map f : B → B Aut_{H}(F_{H}) lifts to a map f̃ : B → B Aut_{H}^{I}(F_{H}). Also note that homotopy orientable fibrations are classified (as homotopy orientable fibrations) by the homotopy classes of maps [B, B Aut_{H}^{I}(F_{H})]. We will be using these facts later in the proof of Lemma 2.8.

In the rest of this section we focus on the fiber join construction performed on Gfibrations. We use fiber joins to kill obstructions that occur in the construction of Gfibrations. The fiber join of two G-fibrations is defined in the following way: Let p_{1} : E_{1} → B and p_{2} : E_{2} → B be two fibrations. We define a fibration E_{1} ×_{B} E_{2} and maps E_{1} ×_{B} E_{2} → E_{i} for i = 1, 2 by the following pullback diagram

\xymatrix{E_{1} ×_{B} E_{2} \ar[r] \ar[d] & E_{1} \ar[d]^{p_{1}} \\ E_{2} \ar[r]^{p_{2}} & B.}

Then the G-space E_{1} ∗_{B} E_{2} is defined as the homotopy pushout of the following diagram

\xymatrix{E_{1} ×_{B} E_{2} \ar[r] \ar[d] & E_{1} \ar[d] \\ E_{2} \ar[r] & E_{1} ∗_{B} E_{2}.}

By the universal property of homotopy pushouts we get a G-fibration

p_{1} ∗ p_{2} : E_{1} ∗_{B} E_{2} → B

called the fiber join of p_{1} and p_{2}. To see that p_{1} ∗ p_{2} is a fibration, we can use equivariant lifting functions. Recall that a G-map p : E → B is a G-fibration if and only if there is a G-map

λ : Ω_{p} = {(e, ω) ∈ E × B^{I}| p(e) = ω(0)} → E^{I}

such that λ(e, ω)(0) = e and pλ(e, ω) = ω (see [23, pg. 30]). The G-map λ is called a G-lifting function. Let λ_{1} and λ_{2} denote the G-lifting functions for p_{1} and p_{2}. Then we can define a lifting function for the join p_{1} ∗ p_{2} as follows

λ(te_{1} + (1 − t)e_{2}, ω)(u) = tλ_{1}(e_{1}, ω)(u) + (1 − t)λ_{2}(e_{2}, ω)(u)

for all t, u in I. So, p_{1} ∗ p_{2} is a G-fibration.

Iterating this construction, we obtain a G-fibration

∗_{k} p : \underbrace{E ∗_{B} E ∗_{B} ⋯ ∗_{B} E}_{k-many} → B

which we call the k-fold fiber join of p with itself. Note that if p is a G-fibration with fiber type {F_{H}}, then the fiber type of the k-fold join ∗_{k}p is {∗_{k}F_{H}}. If B has trivial H-action, then the k-fold join is classified by a map B → B Aut_{H}(∗_{k}F_{H}). We would like to explain this map in terms of the classifying map of p. For this, observe that there is a monoid homomorphism

φ : Aut_{H}(F_{H}) × ⋯ × Aut_{H}(F_{H}) → Aut_{H}(∗_{k}F_{H})

defined by

φ(a_{1}, …, a_{k})(x_{1}t_{1}, …, x_{k}t_{k}) = (a_{1}(x_{1})t_{1}, …, a_{k}(x_{k})t_{k})

for every x_{1}, …, x_{k} ∈ F_{H} and t_{1}, …, t_{k} ∈ [0, 1] with ∑_{i} t_{i} = 1. We have the following lemma:

Lemma 2.5. Let H be a finite group, F_{H} be a finite H-CW-complex, and B be a CWcomplex with trivial H-action. If p : E → B is a H-fibration with fiber type F_{H} whose classifying map is f : B → B Aut_{H}(F_{H}), then the classifying map of the G-fibration ∗_{k} p is given by the composition

\xymatrix{B \ar[r]^{f×···×f} & B Aut_{H}(F_{H}) × ⋯ × B Aut_{H}(F_{H}) \ar[r]^{Bφ} & B Aut_{H}(∗_{k}F_{H})}

where Bφ is the map induced from the monoid homomorphism φ defined above.

Proof. Let A = ∏_{i=1}^{k} Aut_{H}(F_{H}). By standard properties of homotopy pushout diagrams, we observe that the fibration ∗_{k} p is the pullback fibration of the fibration

q : EA ×_{A} (∗_{k}F_{H}) → BA

via the map ∏_{i} f : B → BA. Note that A acts on ∗_{k}F_{H} via the monoid homomorphism φ, so the classifying map of q is Bφ. This completes the proof. □

A special case of a G-fibration is a G-fiber bundle over a G-CW-complex. More specifically, if ξ : E → B is a complex G-vector bundle over a G-CW-complex B, then the sphere bundle p : S(E) → B of this vector bundle is a spherical G-fibration. Note that for every b ∈ B, the fiber space p^{−1}(b) is a G_{b}-space which is homeomorphic to 𝕊(V_{G_{b}}) where V_{G_{b}} denotes vector space ξ^{−1}(b) with the induced G_{b}-action.

Note that when B^{H} is path connected for all H ∈ Iso(B), the family of complex representations {V_{H}} defined over all H ∈ Iso(B) is a compatible family. The compatibility of a family of representations is defined in the following way: A family of representations α_{H} : H → U(n) over H ∈ H is called a compatible family of representations if for every map c_{g} : H → K induced by conjugation with g ∈ G, there exists a γ ∈ U(n) such that the following diagram commutes

\xymatrix{K \ar[d]^{c_{g}} \ar[r]^{α_{K}} & U(n) \ar[d]^{c_{γ}} \\ H \ar[r]^{α_{H}} & U(n).}

Note that if F_{H} is an H-space which is H-homotopy equivalent to 𝕊(V_{H}) for some compatible family of complex H-representation V_{H}, then {F_{H}} is a compatible family of H-spaces. So, the sphere bundle p : S(E) → B of a G-vector bundle is a spherical G-fibration with fiber type {𝕊(V_{H})}. Note also that for every k ≥ 1, the fiber join ∗_{k}𝕊(V_{H}) is H-homotopy equivalent to the H-space 𝕊(V_{H}^{⊕k}) where V_{H}^{⊕k} denotes the k-fold direct sum of V_{H}.

In Section 4, we construct G-fibrations with fiber types of the form {𝕊(V_{H})}. The following result is used in those constructions.

Lemma 2.6. Let H be a finite group, F_{H} be an H-space which is H-homotopy equivalent to 𝕊(V_{H}) for some complex H-representation V_{H}. Let γ, γ^{1} : Aut_{H}(F_{H}) → Aut_{H}(∗_{k}F_{H}) be maps defined by γ(a) = φ(a, a, …, a) and γ^{1}(a) = φ(a, id, …, id), respectively. Then, the induced group homomorphisms γ_{∗} and γ_{∗}^{1} on homotopy π_{q}(Aut_{H}(F_{H})) → π_{q}(Aut_{H}(∗_{k}F_{H})) satisfy the relation γ_{∗} = kγ_{∗}^{1}.

Proof. Let γ^{i} : Aut_{H}(F_{H}) → Aut_{H}(∗_{k}F_{H}) be the map defined by

γ^{i}(a) = φ(id, …, a, …, id)

where a is on the i-th coordinate. We have γ = γ^{1}γ^{2} ⋯ γ^{k} under the product induced by the product in the monoid A. Since the group operation on π_{q}(Aut_{H}(∗_{k}F_{H})) coming from the monoid structure on Aut_{H}(∗_{k}F_{H}) coincides with the usual group structure on homotopy groups, we have γ_{∗} = γ_{∗}^{1} + ⋯ + γ_{∗}^{k}. So, to complete the proof, it is enough to show that γ^{i} and γ^{j} are homotopic for every i, j ∈ {1, …, k}. Since F_{H} is H-homotopy equivalent to 𝕊(V_{H}), it is enough to prove this for 𝕊(V_{H}). Note that in this case we have γ^{i} = T (i, j)γ^{j} where T (i, j) : V_{H}^{⊕k} → V_{H}^{⊕k} is a linear transformation which swaps the j-th summand with the i-th summand. Since U(n) is connected, there is a path between T (i, j) and the identity. Using this path, we can define a homotopy between γ^{i} and γ^{j}. □

Remark 2.7. For more general H-spaces F_{H}, there exists a swap map

S(i, j) : ∗_{k}F_{H} → ∗_{k}F_{H},

which swaps the i-th and j-th coordinates, similar to the linear transformation T (i, j) in the proof of Lemma 2.6. If F_{H} is a free H-space homotopy equivalent to an odd dimensional sphere, then S(i, j) will be homotopy equivalent to the identity map. If the H-action on F_{H} is not free, then the swap map S(i, j) is not homotopy equivalent to the identity in general even when F_{H} is homotopy equivalent to an odd dimensional sphere. On the other hand, if F_{H} is a homotopy representation with the property that all fixed point spheres are odd dimensional, then under certain conditions on H or on the dimension function of F_{H}, one can prove that S(i, j) is homotopy equivalent to the identity (see Proposition 20.12 in [13]).

We end this section with the following observation.

Lemma 2.8. Let H be a finite group and p : E → 𝕊^{n} be an H-fibration over the trivial H-space 𝕊^{n} where n ≥ 2. Suppose that the fiber type F_{H} of p is H-homotopy equivalent to 𝕊(V_{H}) for some complex H-representation V_{H}. If π_{n−1}(Aut_{H}(F_{H})) is a finite group of order N, then ∗_{N}p is H-fiber homotopy equivalent to the trivial fibration.

Proof. By Theorem 2.4, the H-fibration p is classified by the homotopy class of a map f : 𝕊^{n} → B Aut_{H}(F_{H}). Since n ≥ 2, this map lifts to map f̃: S^{n} → B Aut_{H}^{I}(F_{H}), so we can assume that the H-fibration p is an homotopy orientable fibration . Since B Aut_{H}^{I}(F_{H}) is simply connected, we have

[S^{n}, B Aut_{H}^{I}(F_{H})] ≅ π_{n}(B Aut_{H}^{I}(F_{H})) ≅ π_{n−1}(Aut_{H}(F_{H})).

So, p is classified by a homotopy class α ∈ π_{n−1}(Aut_{H}(F_{H})). By a slightly modified version of Lemma 2.5, it is easy to see that the fiber join ∗_{N}p is classified by γ_{∗}(α) where γ : Aut_{H}(F_{H}) → Aut_{H}(∗_{H}F_{H}) is the map defined by γ(a) = φ(a, …, a). By Lemma 2.6, we have

γ_{∗}(α) = Nγ_{∗}^{1}(α) = γ_{∗}^{1}(Nα) = 0.

So, ∗_{N}p is H-fiber homotopy equivalent to the trivial fibration. □

3. Equivariant Federer spectral sequence

The main purpose of this section is to prove the following theorem which is due to M. Klaus [12]. We give a different proof here using the equivariant Federer spectral sequence which was introduced by Møller in [14].

Theorem 3.1 (Klaus [12]). Let G be a finite group and V be a complex representation of G. Then, for every n > 0, there is an m ≥ 1 such that π_{n}(Aut_{G}(𝕊(V^{⊕k})) is finite for all k ≥ m.

Before the proof, we first introduce Møller’s theorem (Theorem 5.3 in [14]) and discuss how the Bredon cohomology with coefficients in a G-equivariant local coefficients system can be calculated for the case we are interested in. For more details on Bredon cohomology, we refer the reader to [3] and [13].

Let (X, A) be a finite G-CW-complex, p : E → B be a G-fibration, and u : X → E be a G-equivariant map. Assume that for all H ≤ G and for all b ∈ B, the space p^{−1}(b)^{H} is a path connected simple space, and let π_{q}(F) be the G-equivariant local coefficient system on X, induced from the local coefficient system on B associated to the G-fibration p (see Example 2.2 in [14]). Let F_{u}(X, A; E, B)^{G} denote the space of all equivariant maps v : X → E such that v|_{A} = u|_{A} and p ∘ v = p ∘ u with compact open topology.

Theorem 3.2 (Møller [14]). There is a spectral sequence with E^{2}-term

E_{pq}^{2} = H_{G}^{−p}(X, A; π_{q}(F))

for p+q ≥ 0 and E_{pq}^{2} = 0 otherwise, converging to π_{p+q}(F_{u}(X, A; E, B)^{G}, u) when p+q > 0.

The spectral sequence above is called the equivariant Federer spectral sequence since it is the equivariant version of a spectral sequence introduced by Federer [7]. We will be using this spectral sequence for the following special case: Let X be a finite G-CW-complex such that X^{H} is a path connected simple space for all H ≤ G. Take A = ∅, E = X, B = ∗, p : E → B to be the constant map, and u : X → E to be the identity map. Then F_{id}(X, ∅; X, ∗) will be homotopy equivalent to the identity component of Aut_{G}(X). Since all the components of Aut_{G}(X) have the same homotopy type, we have

π_{n}(Aut_{G}(X)) ≅ π_{n}(F_{id}(A, ∅; X, ∗))

for all n > 0. So, we can use the equivariant Federer spectral sequence to calculate the homotopy groups of Aut_{G}(X).

Also note that in the situation we consider, the local coefficient system is constant on orbits. In this case Bredon cohomology with local coefficients has an alternative description which involves modules over the orbit category. We define this category now.

Definition 3.3. Let G be a finite group and Or_{G} denote the orbit category of G whose objects are orbits G/H where H is a subgroup of G. The morphisms of Or_{G} from G/H to G/K are G-maps f : G/H → G/K, where we consider the left cosets G/H and G/K as left G-sets. We denote the morphism from G/H to G/K which sends H to aK by â.

Let Γ denote the orbit category Or_{G}. A functor from Or_{G} to the category of abelian groups Ab is called an ℤΓ-module. Morphisms between ℤΓ-modules are given by natural transformations. Given a G-CW-complex X, we define a chain complex of ℤΓ-modules by taking C_{n}(X^{?}) as the functor Or_{G} → Ab which sends G/H to the n-th cellular chains C_{n}(X^{H}) and sends â : G/H → G/K to the group homomorphism a_{∗} : C_{n}(X^{K}) → C_{n}(X^{H}). The boundary maps at each H are compatible with the maps a_{∗} for every â : G/H → G/K, so they combine together to give a ℤΓ-module map ∂ : C_{n}(X^{?}) → C_{n−1}(X^{?}) for every n.

Given a ℤΓ-module M, we define C^{∗}(G, M) as the cochain complex of abelian groups

C^{n}(X, M) = Hom_{ℤΓ}(C_{n}(X^{?}), M)

with coboundary maps coming from the boundary maps of C_{n}(X^{?}). Note also that by the standard properties of free ℤΓ-modules (see [13, Sec. 9]), we have

C^{n}(G, M) ≅ ⨁_{[σ]∈I_{n}} M(G/G_{σ})

where I_{n} is the set of G-orbits of n-cells in X.

Let \underline{L} be a G-equivariant local coefficient system on X (see [14, def. 2.1]). Suppose that there exists a ℤΓ-module M such that L(G/H)(x) = M(G/H) and L(G/H)(γ) = id_{M(G/H)} for all x ∈ X^{H} and all paths γ in X^{H}. Then we have

H_{G}^{n}(X; \underline{L}) ≅ H^{n}(C^{∗}(X, M))

where H^{n}(G, \underline{L}) denotes the Bredon cohomology with local coefficients and C^{∗}(G, M) denotes the cochain complex introduced above (see Definition 2.3 and Example 2.4 in [14]). In our situation, this gives an isomorphism

H_{G}^{n}(X; π_{q}(F)) ≅ H_{G}^{n}(X; π_{q}(X^{?}))

where π_{q}(X^{?}) is the ℤΓ-module π_{q}(X^{?}) : Or_{G} → Ab which sends G/H to π_{q}(X^{H}) and sends â : G/H → G/K to a_{∗} : π_{q}(X^{K}) → π_{q}(X^{H}). Also note that on the cochain level, we have

C^{n}(X, π_{q}(X^{?})) = Hom_{ℤΓ}(C_{n}(X^{?}), π_{q}(X^{?})) ≅ ⨁_{[σ]∈I_{n}} π_{q}(X^{G_{σ}}).

So, we have an explicit description of the E_{pq}^{2}-terms of the equivariant Federer spectral sequence. Now we are ready to prove the main theorem of this section.

Proof of Theorem 3.1. Let G be a finite group and X be a finite G-CW-complex which is G-homotopy equivalent to 𝕊(V) for some complex representation V of G. In fact, we only need X to be a G-homotopy representation with odd dimensional fixed point spheres for our arguments to work (see [13, pg. 392] for a definition of homotopy representation). Let n be a fixed positive integer. We want to show that there is an m ≥ 1 such that π_{n}(Aut_{G}(∗_{k}X)) is finite for all k ≥ m. Let X_{k} = ∗_{k}X denote the k-fold join of X. By Theorem 3.2, there is a spectral sequence with

E_{pq}^{2} = H_{G}^{−p}(X_{k}; π_{q}(X_{k}^{?}))

for p + q ≥ 0 and E_{pq}^{2} = 0 otherwise, converging to π_{p+q}(Aut_{G}(X_{k})) when p + q > 0. Since X_{k} is finite dimensional, to show that π_{n}(Aut_{G}(X_{k})) is finite it is enough to show that H_{G}^{−p}(X_{k}; π_{q}(X_{k}^{?})) is finite for every pair (p, q) with p + q = n. Note that for this we need to show that there is an m ≥ 1 such that for all k ≥ m, the cohomology group H_{G}^{q−n}(X_{k}; π_{q}(X_{k}^{?}) ⊗ ℚ) is zero for all q ≥ n.

Let {n_{1}, n_{2}, …, n_{s}} be the set of all distinct dimensions of fixed subspaces V^{H} over all subgroups H ≤ G. Assume that n_{1} < n_{2} < ⋯ < n_{s}. Note that the fixed point spheres X_{k}^{H} have dimensions {kn_{i} − 1 | i = 1, …, s}. Since homotopy groups π_{i}(S^{2j−1}) of an odd dimensional sphere are all finite except when i = 2j − 1, we have π_{q}(X_{k}^{?}) ⊗ ℚ = 0 for all q which is not equal to kn_{i} − 1 for some i. If q = kn_{i} − 1 for some i, then we have

H_{G}^{q−n}(X_{k}; π_{q}(X_{k}^{?}) ⊗ ℚ) = H_{G}^{kn_{i}−n−1}(X_{k}; M_{i})

where M_{i} is the ℤΓ-module such that M_{i}(H) = ℚ for all subgroups H ≤ G satisfying dim V^{H} = n_{i}. To complete the proof we need to show that this cohomology group is zero for all i ∈ {1, …, s}.

Note that there is a well-known first quadrant spectral sequence with E_{2}-term

E_{2}^{pq} = Ext_{ℤΓ}^{p}(H_{q}(X_{k}^{?}), M_{i})

which converges to H_{G}^{p+q}(X_{k}; M_{i}) (see [20, Prop. 3.3]). Since the coefficient module M_{i} takes only the values ℚ, we can replace H_{p}(X_{k}^{?}) with H_{p}(X_{k}^{?}; ℚ) and take the ext-groups over ℚΓ. Note that the ℚΓ-module H_{p}(X_{k}^{?}; ℚ) is zero at all dimensions except when p = kn_{i} − 1 for some i. Let N_{i} denote the ℚΓ-module H_{kn_{i}−1}(X_{k}^{?}; ℚ) for all i = 1, …, s. To prove that H_{G}^{kn_{i}−n−1}(X_{k}; M_{i}) = 0 for all i, it is enough to show that the ext-group

Ext_{ℚΓ}^{k(n_{i}−n_{j})−n}(N_{j}, M_{i})

is zero for all j ≤ i − 1. Let l_{j} denote the length of the ℚΓ-module N_{j} for every j (see [13, pg. 325] for a definition). Then, by [13, prop. 17.31], the above ext-group is zero if k(n_{i} − n_{j}) − n ≥ l_{j}. Let l = max_{j}{l_{j}}. Then if k ≥ n + l, then the above inequality will hold for every j ≤ i − 1. This completes the proof. □

4. Construction of spherical G-fibrations

We start with proving a proposition which is an important tool for constructing Gfibrations. In different forms, this proposition also appears in [4], [12], and [18]. Here we give a proof of it for completeness since it is the main ingredient in the proof of Theorem 1.2.

Proposition 4.1. Let G be a finite group, B be a G-CW-complex, and let {V_{H}} be a compatible family of complex representations over all H ∈ Iso(B). Let q_{n} : E_{n} → B^{(n)}, n ≥ 2, be a G-fibration with fiber type {𝕊(V_{H})} where B^{(n)} denotes the n-skeleton of B. Then there is an integer k ≥ 1 and a G-fibration q_{n+1} : E_{n+1} → B^{(n+1)} such that the restriction of q_{n+1} to B^{(n)} is G-fiber homotopy equivalent to ∗_{k}q_{n}. In particular, the fiber type of q_{n+1} is {𝕊(V_{H}^{⊕k})}.

Proof. By the definition of G-CW-complexes, there exists a pushout diagram

(1)  \xymatrix{∐_{i∈I_{n+1}} G/H_{i} × 𝕊^{n} \ar[r]^{∐ f_{i}} \ar[d] & B^{(n)} \ar[d] \\ ∐_{i∈I_{n+1}} G/H_{i} × 𝔻^{n+1} \ar[r]^{∐ g_{i}} & B^{(n+1)}}

where I_{n+1} is an indexing set of orbits of (n + 1)-cells in B. For each i ∈ I_{n+1}, let q_{n,i} denote the G-fibration obtained by the following pullback diagram

\xymatrix{E_{n,i} \ar[r] \ar[d]^{q_{n,i}} & E_{n} \ar[d] \ar[d]^{q_{n}} \\ G/H_{i} × 𝕊^{n} \ar[r]^{f_{i}} & B^{(n)} .}

Restricting q_{n,i} to the sphere 𝕊^{n} in G/H_{i} × 𝕊^{n} which is fixed by H_{i}, we obtain an H_{i}- fibration q_{n,i}|_{𝕊^{n}} : q_{n,i}^{−1}(𝕊^{n}) → 𝕊^{n} such that the H_{i}-action on the base space is trivial. By Theorem 2.4 and by the argument in the proof of Lemma 2.8, such a fibration is classified by a homotopy class α_{i} ∈ π_{n−1}(Aut_{H_{i}}(𝕊(V_{H_{i}}))).

By Theorem 3.1, for each H ∈ Iso(B), there is an m_{H} ≥ 1 such that π_{n−1}(Aut_{H}(𝕊(V_{H}^{⊕k}))) is finite for all k ≥ m_{H}. Let m = max{m_{H} | H ∈ Iso(B)}. Then the group

π_{n−1}(Aut_{H}(𝕊(V_{H}^{⊕m})))

has finite order, say d_{H}, for all H ∈ Iso(B). Let d = ∏_{H} d_{H}. By Lemma 2.8, the H_{i}-fibration ∗_{dm}(q_{n,i}|_{𝕊^{n}}) is H_{i}-fiber homotopy equivalent to the trivial fibration for all i ∈ I_{n+1}. This implies that the G-fibration p obtained by the following pullback diagram

\xymatrix{W \ar[r]^{f} \ar[d]^{p} & ∗_{dm}E_{n} \ar[d]^{∗_{dm} q_{n}} \\ ∐_{i∈I_{n+1}} G/H_{i} × 𝕊^{n} \ar[r]^{∐ f_{i}} & B^{(n)}}

is G-homotopy equivalent to the trivial fibration. Let

φ : ∐_{i∈I_{n+1}} G ×_{H_{i}} 𝕊(V_{H_{i}}^{⊕dm}) × 𝕊^{n} → W

be a G-fiber homotopy equivalence between the trivial fibration and p. We can use φ to glue the cone of the trivial fibration and obtain a G-quasifibration

\xymatrix{q′_{n+1} : ( ∐_{i∈I_{n+1}} G ×_{H_{i}} 𝕊(V_{H_{i}}^{⊕dm}) × 𝔻^{n+1}) ∪_{f∘φ} ( ∗_{dm}E_{n}) \ar[r] & B^{(n+1)}} .

A quasifibration p : E → B is a continuous map such that the induced map

p_{∗} : π_{i}(E, p^{−1}(x), y) → π_{i}(B, x)

is an isomorphism for all x ∈ B, y ∈ p^{−l}(x), and i ≥ 0 (see [5, def. 1.1]). An Gquasifibration is defined in a analogous way so that the corresponding modules over the orbit category are isomorphic (see [21, def. 2.1]). The fact that q′_{n+1} is a G-quasifibration follows from an equivariant version of Lemma 1.4 in [5] (see also [21, lem. 2.9]). There is a construction called gammafication that converts a quasifibration to a fibration and this construction also works for G-quasifibrations (see [21, Not. 2.8]). Applying gammafication to q′_{n+1}, we obtain a G-equivariant spherical fibration q_{n+1} : E_{n+1} → B^{(n+1)} whose fiber type is {𝕊(V_{H}^{⊕dm})}. □

Remark 4.2. Another possible way of completing the final step of the above construction is to attach trivial G-fibrations over (n + 1)-cells with the space ∗_{dm}E_{n} using G-tubes (see [10, Theorem 3.1]). When these G-tubes are used one does not need the gammafication construction since one directly gets G-fibrations. This method is explained in detail in [9] and [10].

As a corollary of Proposition 4.1, we obtain the following which is also proved in [12] as Proposition 2.7.

Proposition 4.3 (Proposition 2.7, [12]). Let G be a finite group and B be a finite dimensional G-CW-complex. Let {V_{H}} be a compatible family of complex representations over all H ∈ Iso(B). Then there exists an integer k ≥ 1 and a G-equivariant spherical fibration q : E → B such that the fiber type of q is {𝕊(V_{H}^{⊕k})}.

Proof. Let

A = (ρ_{H}) ∈ lim_{⟵_{H∈H}} Rep(H, U(n))

where H = Iso(B) and ρ_{H} is a representation for V_{H} for every H ∈ H. Let q : E_{H}(G, A) → B_{H}(G, A) denote the universal G-equivariant vector bundle with fiber type A (see [20, def. 2.4]). Since B_{H}(G, A)^{H} = BC_{U(n)}(ρ_{H}) is simply connected for all H ∈ H, by standard obstruction theory there is a G-map B^{(2)} → B_{H}(G, A) (see the proof of Theorem 4.3 in [20] for details). Pulling back the universal G-equivariant bundle via this map, we obtain a G-equivariant vector bundle over B^{(2)}. The sphere bundle of this bundle is spherical G-fibration over B^{(2)} with fiber type {𝕊(V_{H})}. Now the result follows from the repeated application of Proposition 4.1. □

We often want the total space of a G-fibration to be G-homotopy equivalent to a finite G-CW-complex. The following theorem gives a very useful criteria for this condition:

Proposition 4.4. Let G be a finite group, B be a finite G-CW-complex, and p : E → B be a G-fibration with fiber type {F_{H}}. If F_{H} is H-homotopy equivalent to a finite H-CW-complex for every H ∈ Iso(B), then E is G-homotopy equivalent to a finite G-CW-complex.

Proof. We will prove this lemma by induction over the skeletons of B. We already know that p^{−1}(B^{(0)}) is G-homotopy equivalent to a finite G-CW-complex. Now assume that p^{−1}(B^{(n)}) is G-homotopy equivalent to a finite G-CW-complex Z for some n ≥ 0. We want to show that p^{−1}(B^{(n+1)}) is G-homotopy equivalent to a finite G-CW-complex.

The pushout diagram given in (1) induces a diagram of G-spaces

(2)  \xymatrix{f^{∗}(E_{n}) \ar[r]^{\overline{f}} \ar[d]^{\overline{j}} & E_{n} \ar[d]^{\overline{J}} \\ g^{∗}(E_{n+1}) \ar[r]^{\overline{g}} & E_{n+1}}

where the spaces in the diagram are the total spaces of the fibrations obtained by taking pullbacks of the fibration q_{n+1} : E_{n+1} → B^{(n+1)} via the maps f, g, j, and J. Here f = ∐ f_{i}, g = ∐ g_{i},

j : ∐_{i∈I_{n+1}} G/H_{i} × 𝕊^{n} → ∐_{i∈I_{n+1}} G/H_{i} × 𝔻^{n+1}

is the disjoint union of inclusion maps, and J : B^{(n)} → B^{(n+1)} is the inclusion map. Since the inclusion map 𝕊^{n} → 𝔻^{n+1} is a cofibration map, the G-map j is a G-cofibration. So, by [13, Lemma 1.26], the diagram (2) is a pushout diagram and \overline{j} is a G-cofibration.

Since 𝔻^{n+1} is contractible, there is a G-fiber homotopy equivalence

∐_{i∈I_{n+1}} G ×_{H_{i}} F_{H_{i}} × 𝔻^{n+1} ⟶^{γ} g^{∗}(E_{n+1}).

This gives a commutative diagram of the following form

\xymatrix{E_{n} & f^{∗}(E_{n}) \ar[l]_{\overline{f}} \ar[r]^{\overline{j}} & g^{∗}(E_{n+1}) \\ E_{n} \ar[u]_{id} & ∐_{i∈I_{n+1}} G ×_{H_{i}} F_{H_{i}} × 𝕊^{n} \ar[u]_{γ′} \ar[r]^{id×j} \ar[l]_{\overline{f}∘γ′} & ∐_{i∈I_{n+1}} G ×_{H_{i}} F_{H_{i}} × 𝔻^{n+1} \ar[u]_{γ}}

where γ′ is the restriction of γ to the boundary spheres. Such a restriction makes sense since γ is a fiber homotopy equivalence. Now, since both id × j and \overline{j} are G-cofibrations, by [13, Lemma 2.13], the G-space E_{n+1}, which is the pushout of the diagram in the first line, is G-homotopy equivalent to the pushout of the diagram in the second line.

To find a further homotopy equivalence, note that by induction assumption E_{n} is Ghomotopy equivalent to a finite G-CW-complex Z. So, using a similar diagram as above, we can conclude that E_{n+1} is G-homotopy equivalent to the pushout of a diagram of the following form

\xymatrix{Z & ∐_{i∈I_{n+1}} G ×_{H_{i}} F_{H_{i}} × 𝕊^{n} \ar[r]^{id×j} \ar[l]_{φ} & ∐_{i∈I_{n+1}} G ×_{H_{i}} F_{H_{i}} × 𝔻^{n+1}} .

Now, we can replace the map φ with a cellular one (up to homotopy) and conclude that E_{n+1} is G-homotopy equivalent to a finite G-CW-complex since the spaces Z and F_{H_{i}} for all i ∈ I_{n+1} are finite G-CW-complexes. This completes the n-th stage of our induction.

Since B is a finite G-CW-complex, the induction will stop in finite steps. So, the proof is complete. □

5. Proof of the main theorem

Now, we are ready to prove the main theorem of the paper. First we introduce some notation and recall some basic facts about Stiefel manifolds. For more details, we refer the reader to [19].

Let F denote the field of real numbers ℝ, complex numbers ℂ, or quaternions ℍ. For a real number the conjugation is defined by \overline{x} = x, for a complex number x = a + ib by \overline{x} = a − ib, and for a quaternion x = a + ib + jc + kd by \overline{x} = a − ib − jc − kd. On the vector space F^{n}, we can define an inner product (v, w) by taking

(v, w) = v_{1}\overline{w}_{1} + v_{2}\overline{w}_{2} + ⋯ + v_{n}\overline{w}_{n}.

The Stiefel manifold V_{k}(F^{n}) is defined as the subspace of F^{nk} formed by the k-tuples of vectors (v_{1}, v_{2}, …, v_{k}) such that v_{i} ∈ F^{n} and for every pair (i, j), we have (v_{i}, v_{j}) = 1 if i = j and zero otherwise.

There is a sequence of fiber bundles

V_{n}(F^{n}) → ⋯ → V_{k+1}(F^{n}) ⟶^{q_{k}} V_{k}(F^{n}) → ⋯ → V_{2}(F^{n}) ⟶^{q_{1}} V_{1}(F^{n})

where the map q_{k} : V_{k+1}(F^{n}) → V_{k}(F^{n}) is defined by q_{k}(v_{1}, …, v_{k+1}) = (v_{1}, …, v_{k}) and the fiber of q_{k} is V_{1}(F^{n−k}) = 𝕊^{c(n−k)−1} where c = dim_{ℝ} F (see Theorem 3.8 and Corollary 3.9 in Chapter 8 of [11]). Note that the sphere bundle q_{k} : V_{k+1}(F^{n}) → V_{k}(F^{n}) is the sphere bundle of the vector bundle \overline{q}_{k} : \overline{V}_{k+1}(F^{n}) → V_{k}(F^{n}) where \overline{V}_{k+1}(F^{n}) is the space formed by (k + 1)-tuples (v_{1}, …, v_{k+1}) satisfying (v_{1}, …, v_{k}) ∈ V_{k}(F^{n}) and (v_{i}, v_{k+1}) = 0 for all i = 1, …, k.

Note that if a finite group G has a representation W over a field F, then the inner product above can be replaced by a G-invariant one and the Stiefel manifolds have natural G-actions. Moreover the sphere bundles given above become G-equivariant bundles. If the representation W has fixity f, then we have dim_{F} W^{g} ≤ f for all g ∈ G − {1}. This means if W is a faithful representation, i.e., f ≤ dim_{F} W − 1, then the G-action on V_{f+1}(W) is free. We will be using this observation in the proof of the main theorem.

Also observe that if G has a complex representation with fixity f, then by tensoring it with ℍ over ℂ, we obtain a symplectic representation with the same fixity. So to prove Theorem 1.2, it is enough to prove the same statement for symplectic representations. On the other hand with symplectic representations unit spheres of fixed point subspaces are simply connected, so it makes the proof easier. So, we prove the following:

Theorem 5.1. Let ρ : G → Sp(n) be a faithful symplectic representation with fixity f. Then there exists a finite G-CW-complex X homotopy equivalent to a product of f + 1 spheres such that the G action the homology of X is trivial.

Proof. Let W denote the ℍ-space corresponding to the representation ρ. Define X_{1} = V_{1}(W). We will construct finite G-CW-complexes X_{2}, X_{3}, …, X_{f+1} recursively. For all i, the G-CW-complex X_{i} will be homotopy equivalent to a product of i spheres and will satisfy the following property: If H ∈ Iso(X_{i}), then V_{i}(W)^{H} ≠ ∅ where Iso(X_{i}) denotes the set of isotropy subgroups of X_{i}.

Assume that X_{i} is constructed for some i ≥ 1. Note that for every H ≤ G, the fixed point set V_{i}(W)^{H} is either empty or simply connected. Since for every H ∈ Iso(X_{i}) we have V (W_{i})^{H} ≠ ∅, by standard equivariant obstruction theory there exists a G-map f : X_{i}^{(2)} → V_{i}(W). By pulling back the G-bundle q_{i} : V_{i+1}(W) → V_{i}(W) via f, we obtain a G-equivariant vector bundle over X_{i}^{(2)} with fiber type {S(V_{H})}. Note that this is a compatible family defined over all H ∈ Iso(X_{i}) where V_{H} is the H-space (\overline{q}_{i})^{−1}(b) for some b ∈ V_{i}(W)^{H}.

Now, applying Proposition 4.1 repeatedly to this G-fibration, we obtain a spherical G-fibration E_{i} → X_{i} with fiber type {S(V_{H}^{⊕k})} for some k ≥ 1. By taking further fiber joins, we can assume that E_{n} is a trivial fibration non-equivariantly and the action on the homology of the total space is trivial. This is shown below in Lemma 5.2. Now, by Proposition 4.4, E_{i} is G-homotopy equivalent to a finite G-CW-complex Y. Hence we can take X_{i+1} as Y and continue the induction until we reach X_{f+1}. At this stage, we have (V_{f+1}(W))^{H} ≠ ∅ implies H = {1}, so we can conclude that the G-action on X_{f+1} is free. □

Lemma 5.2. Let p : E → B be a G-fibration over a finite G-CW-complex B and n be a positive integer. Suppose that p has fiber type {F_{H}} such that F_{H} is homotopy equivalent to the sphere 𝕊^{n} for all H ∈ Iso(B). Then, there is an integer k ≥ 1 such that ∗_{k}p : ∗_{k}E → B is non-equivariantly homotopy equivalent to the trivial fibration. Moreover, if the G-action on the cohomology of B is trivial, then there is an integer m > 0 such that for every even number k > m, the G-action on the cohomology of ∗_{k}E is also trivial.

Proof. The first part of the lemma is well-known and it follows from the fact that the homotopy groups of Aut(S^{n}) is finite. For the second part, observe that since the resulting fibration is homotopy equivalent to the trivial fibration, it is in particular an orientable fibration, i.e., π_{1}(B) action on the homology of F is trivial. So, there exists a consistent choice of generators for H^{n}(F_{b}) ≅ ℤ for all b ∈ B. Note that this gives a G-action on the cohomology of the fibers H^{n}(F_{b}) which is defined by

g^{∗} : H^{n}(F_{b}) → H^{n}(F_{gb}) ≅ H^{n}(F_{b})

where the isomorphism on the right comes from identifications of generators that we have chosen. Observe that this action in general can be nontrivial since a generator u can go to −u but if we take the fiber join of p with itself, then we can assume that this action is trivial for all b ∈ B. So for every even number k, the fibration ∗_{k}p will have this property.

For an orientable spherical fibration there is a Serre spectral sequence

E_{2}^{p,q} = H^{p}(B, H^{q}(F)) ⟹ H^{p+q}(E).

Note that for a G-fibration, all the terms in this spectral sequence will be ℤG-modules and the differentials will be ℤG-module homomorphisms since every g ∈ G induces a fiber preserving continuous map g : E → E. In our case, we have a two line spectral sequence and it is easy to see that by choosing k large enough, we can assume that N = dim(∗_{k}𝕊^{n}) ≥ dim B, and hence we can conclude that H^{i}(E) ≅ H^{i}(B) for i < N and H^{i}(E) ≅ H^{i−N}(B, H^{N}(F)) for i ≥ N. Since G acts trivially on H^{∗}(B), we have trivial action on H^{∗}(E) if the G-action on H^{i−N}(B, H^{N}(F)) is trivial for all i ≥ N. Note that the Serre spectral sequence has a product structure, so the action on H^{∗}(B, H^{N}(F)) is trivial if the G-action on H^{0}(B, H^{N}(F)) is trivial. Note that H^{0}(B, H^{n}(F)) is the kernel of the map

d^{1} : Hom_{ℤ}(C_{0}(B), H^{N}(F)) → Hom_{ℤ}(C_{1}(B), H^{N}(F))

and as a ℤG-module

Hom_{ℤ}(C_{i}(B), H^{N}(F)) ≅ ⨁_{σ∈I_{i}} Hom_{ℤ}(Ind_{G_{σ}}^{G} ℤ_{σ}, H^{N}(F_{σ}))

for i = 0, 1 where the G-action on H^{N}(F_{σ}) is the one described above. So if k is a large even number, then this action is trivial, and we can conclude that H^{0}(B, H^{N}(F)) ≅ H^{0}(B) as ℤG-modules, and hence H^{0}(B, H^{N}(F)) is a trivial ℤG-module. This completes the proof of the lemma.

□

Acknowledgement. We thank Natalia Castellana for directing us towards Møller’s work on equivariant Federer spectral sequence for calculating the homotopy groups of the monoid Aut_{H}(F_{H}). We also thank the referee for careful reading of the paper and for corrections and remarks on the first version of the paper.

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Department of Mathematics, Bilkent University, Ankara, 06800, Turkey.

  E-mail address: unluo@fen.bilkent.edu.tr, yalcine@fen.bilkent.edu.tr