QUASILINEAR ACTIONS ON PRODUCTS OF SPHERES

ÖZGÜN ÜNLÜ AND ERGÜN YALÇIN

Abstract. For some small values of f, we prove that if G is a group having a complex (real) representation with fixity f, then it acts freely and smoothly on a product of f + 1 spheres with trivial action on homology.

1. introduction

Some of the most interesting questions about group actions on topological spaces are about group actions on products of spheres. Historically, these questions are generalizations of questions about group actions on a sphere. In general, one is interested in whether a given finite group G can act freely on a product of k spheres for a given integer k ≥ 1. One of the long lasting conjectures in this subject states that if a group G acts freely on a product of k spheres then rk(G) ≤ k, where rk(G) is the rank of the group G defined as the largest integer s such that (Z/p)^{s} ⊂ G for some prime number p. In terms of construction of actions, it is known that every finite group can act freely on a product of spheres. But the following problem is still open.

Problem. Show that every finite group can act freely on a product of spheres with trivial action on homology.

The type of action demanded in the above problem is much harder to construct since we are not allowed to permute the spheres in the product. In this paper, we are interested in the following question, which is closely related to the above problem.

Question. Suppose that G is a finite group which has a complex representation or an oriented real representation with fixity f. Then, does it act freely and smoothly on a product of f + 1 spheres with trivial action on homology?

Let G be a finite group and F be a field. The fixity of an F -representation V of G is defined as the maximum value of dim_{F} V^{g} among all 1 ≠ g ∈ G. If G has a representation with fixity zero, then it acts freely on a sphere. Adem, Davis, and Ünlü [1] showed that if G has a complex representation V of dimension n with fixity one, then G acts freely and smoothly on S^{2n−1} × S^{4n−5}. They construct this action by taking the unit sphere of the Whitney sum double of the tangent bundle of S(V), where S(V) denotes the unit sphere of V. We observe that their argument can be extended to some higher values of fixity. In particular, for complex representations we prove the following:

Date: May. 13, 2009.

1991 Mathematics Subject Classification. Primary: 57S25; Secondary: 20C15.

The second author is partially supported by TÜBİTAK and TÜBA-GEBİP/2005-16.

Theorem 1.1. If G is a finite group which has a faithful n-dimensional complex representation with fixity 2, then G acts freely and smoothly on X = S^{2n−1} ×S^{4n−5} ×S^{q(4n−8)−1} for some q ≥ 2 with trivial action on homology.

The value of q in the above product depends on the value of n. For example, if n ≥ 4, then the last sphere can be taken as low as S^{8n−17}. This shows in particular that finite subgroups of SU(4) act freely and smoothly on X = S^{7} × S^{11} × S^{15}. We prove Theorem 1.1 using pullbacks of bundles over quaternionic Stiefel manifolds.

For real representations, we prove a similar result.

Theorem 1.2. Let G be a finite group which has a faithful representation ρ : G → SO(n) with fixity f ≤ 4. Assume that n ≥ 12 when f = 4. Then G acts freely and smoothly on a product of f + 1 spheres with trivial action on homology.

For fixity f ≤ 3, we can prove the above statement using a similar argument to the complex case, but for f = 4 this method seems to fail. So we use a slightly different argument which involves a sequence of R-algebras satisfying certain properties. The argument gives a recipe for constructing free actions for all values of f, but because of the well known dimension restriction on division algebras over R, at this point we can only make it work for f ≤ 4.

Finally, we would like to remark that, as was explained in [1], to obtain a free action of a finite group on a finite complex homotopy equivalent to a product of k spheres, it is enough to construct an action on a product of k − 1 spheres with rank 1 isotropy. Then, using a technique given in [2] one gets a free action on a finite complex having homotopy type of a product of k-spheres. This allows us to state the following results as a corollary of Theorems 1.1 and 1.2.

Corollary 1.3. Let G be a finite group. If G has a complex representation with fixity 3, then G acts freely on a finite complex X homotopy equivalent to a product of 4 spheres with trivial action on homology. If G has a real representation with fixity 5 having dimension n ≥ 12, then G acts freely on a finite complex X homotopy equivalent to a product of 6 spheres with trivial action on homology.

We organize the paper as follows: In Section 2, some basic lemmas and Theorem 1.1 are proved. In this section, we also discuss how Theorem 1.2 can be proved for f ≤ 3 using an argument similar to the complex case. In Section 3, we prove Theorem 1.2 for f ≤ 4 using a more general argument.

2. Proof of Theorem 1.1

Throughout this section, we will use the notation and terminology used in [5]. 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 classical group U_{F} (n) is defined as the subgroup of GL(n, F) that preserves the inner product. For F = R, C, and H, we have different notations for the group U_{F} (n). The orthogonal group O(n) is the subgroup of GL(n, R) formed by n × n real matrices A satisfying the property (Av, Aw) = (v, w) for all v, w ∈ R^{n}. Similarly, the unitary group U(n) is the subgroup of GL(n, C) which preserves the inner product, and Sp(n) is the subgroup of GL(n, H) which preserves the inner product for vector space H^{n}. We define classical groups SO(n) and SU(n) as subgroups of O(n) and U(n) formed by matrices with determinant equal to 1.

In our proofs, we will be using fibre bundles arising from Stiefel manifolds. Let V_{k}(F^{n}) denote the subspace of F^{nk} formed by the k-tuples of vectors (v_{1}, v_{2}, …, v_{k}) such that v_{i} ∈ F^{n} for all i = 1, …, k, and for every pair (i, j), we have (v_{i}, v_{j}) = 1 if i = j and zero otherwise. There is a homeomorphism between V_{k}(F^{n}) and the coset space U_{F} (n)/U_{F} (n − k) (see Theorem 1.3 in Chapter 8 of [5]).

There is a sequence of fibre bundles

V_{n}(F^{n}) → ⋯ → V_{k+1}(F^{n}) → V_{k}(F^{n}) → ⋯ → V_{2}(F^{n}) → 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 fibre 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 [5]). 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}) where (v_{1}, …, v_{k}) ∈ V_{k}(F^{n}) and (v_{i}, v_{k+1}) = 0 for all i = 1, …, k.

Lemma 2.1. The vector bundle \overline{q}_{k} : \overline{V}_{k+1}(F^{n}) → V_{k}(F^{n}) is stably trivial.

Proof. Note that a vector bundle ξ is stably trivial if there is a bundle isomorphism ξ ⊕ τ^{i} ≅ τ^{j} for some trivial bundles τ^{i} and τ^{j} of dimensions i and j. Note that in our case we can consider the bundle ξ = (\overline{q}_{k} : \overline{V}_{k+1}(F^{n}) → V_{k}(F^{n})) as a subbundle of the trivial bundle V_{k}(F^{n}) × F^{n} which is orthogonal to the bundle θ defined as follows: Let θ : E → V_{k}(F^{n}) denote the bundle with total space

E = {((v_{1}, v_{2}, …, v_{k}), w) | w ∈ ⟨v_{1}, …, v_{k}⟩, (v_{1}, …, v_{k}) ∈ V_{k}(F^{n}), w ∈ F^{n}}

and with obvious projection map. It is easy to see that ξ ⊕ θ is isomorphic to the trivial bundle V_{k}(F^{n}) × F^{n} and that θ is a trivial bundle. □

Another way to see this result is the following: Note that a vector bundle ξ over a paracompact base space B is stably trivial if its classifying map ξ̂ : B → BU_{F} (n) becomes null-homotopic when it is composed with BU_{F} (n) → BU_{F} (n + i) for some i. In the above lemma, the classifying map for ξ = (\overline{q}_{k} : \overline{V}_{k+1}(F^{n}) → V_{k}(F^{n})) is a map of the form ϕ : V_{k}(F^{n}) → BU_{F} (n − k). Note that BU_{F} (n − k) can be described as the space of (n − k)-dimensional subspaces in F^{∞}. Under this description, the classifying map ϕ takes a k-tuple (v_{1}, …, v_{k}) of vectors in F^{n} to the subspace orthogonal to subspace generated by v_{i}’s.

We also have a fibration of the form

V_{k}(F^{n}) → BU_{F} (n − k) → BU_{F} (n)

which comes from the homeomorphism V_{k}(F^{n}) ≃ U_{F} (n)/U_{F} (n − k). The projection map BU_{F} (n − k) → BU_{F} (n) is given by inclusion of an (n − k)-dimensional subspace in F^{∞} to an n-dimensional subspace in F^{∞}. The inclusion of a fibre over a subspace of dimension n is exactly the map described above. So, the map V_{k}(F^{n}) → BU_{F} (n − k) in the above fibration coincides with the classifying map ϕ. So, composing the classifying map ϕ with BU_{F} (n − k) → BU_{F} (n) gives a null-homotopic map. This shows that the bundle ξ = (\overline{q}_{k} : \overline{V}_{k+1}(F^{n}) → V_{k}(F^{n})) is stably trivial

Another important ingredient in our proofs is a theorem about stably trivial bundles. Let ξ be a vector bundle E → B over an n-dimensional CW-complex B. The following is a special case of a theorem given in [5].

Lemma 2.2 ([5], Theorem 1.5, Chapter 9). Let ξ be a k-dimensional F vector bundle with dim_{R} F = c. Suppose that the base space B is an n-dimensional CW-complex. If ξ is stably trivial and n ≤ c(k + 1) − 2, then ξ is trivial.

Proof. Let f : B → BU_{F} (k) denote the classifying map for ξ. Assume that the composition of f with BU_{F} (k) → BU_{F} (k + j) is null-homotopic for some j. Then f lifts to a map f : B → V_{j}(F^{k+j}). Using the fibration sequence for Stiefel manifolds, one can easily show that π_{i}(V_{j}(F^{k+j})) = 0 for i ≤ c(k + 1) − 2. So, the result follows from the so called compression lemma in homotopy theory (see Lemma 4.6 in [4] or Corollary 7.13 in [3]). □

Lemma 2.2, in particular, shows that given a stably trivial vector bundle over a finite dimensional complex, we can obtain a trivial bundle by taking Whitney sum multiples of it sufficiently many times.

Now, we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. Let V be a faithful complex representation of G with dim V = n and fixity 2. Since V is faithful, we must have n ≥ 3. Consider the pull-back diagram

\xymatrix{E_{1} \ar[r] \ar[d] & V_{2}(H^{n}) \ar[d]^{q_{1}} \\ V_{1}(C^{n}) \ar[r]^{f} & V_{1}(H^{n})}

where f is the map induced from the inclusion of C into H defined by a + ib → a + ib + j0 + k0. Since the map f : V_{1}(C^{n}) = S^{2n−1} → V_{1}(H^{n}) = S^{4n−1} is null-homotopic, the bundle E_{1} → V_{1}(C^{n}) is a trivial bundle with fibre S^{4n−5}. So E_{1} ≃ S^{2n−1} × S^{4n−5}.

Now, consider the pull back bundle

\xymatrix{E_{2} \ar[r] \ar[d]^{ξ} & \overline{V}_{3}(H^{n}) \ar[d]^{q_{2}} \\ E_{1} \ar[r]^{f̃} & V_{2}(H^{n}) .}

The bundle ξ is stably trivial since it is a pullback of a stably trivial bundle. Taking a q-fold Whitney sum ξ ⊕ ⋯ ⊕ ξ of ξ, we obtain a bundle ξ^{⊕q} : E_{2}(q) → E_{1}. By Lemma 2.2, the bundle ξ^{⊕q} is a trivial bundle when

dim E_{1} ≤ 4(q(n − 2) + 1) − 2 = 4q(n − 2) + 2 .

Note that since n ≥ 3, we can always find a q which makes this inequality hold. In fact, since dim E_{1} = 6n − 6, the inequality holds even for q = 2 when n ≥ 4.

Note that the total space E_{2}(q) can be considered as the subspace of C^{n(3+2q)} formed by (3+2q)-tuples of complex vectors (v_{1,1}, v_{2,1}, v_{2,2}, v_{3,1}, v_{3,2}, …, v_{3,2q}) satisfying the property

((v_{1,1}, 0), (v_{2,1}, v_{2,2}), (v_{3,2i−1}, v_{3,2i})) ∈ \overline{V}_{3}(H^{n})

for all i = 1, …, q. Here we consider a pair of complex numbers (a + ib, c + id) in H as the quaternion y = a + ib + jc + kd.

Let V be an n-dimensional faithful complex representation of G, and let W = H ⊗_{C} V. By taking the average, we can assume the inner product on W is G-invariant. Note that if we consider elements of W as pairs of complex vectors (v_{1}, v_{2}), then the G-action on W can be written as a diagonal action g(v_{1}, v_{2}) = (gv_{1}, gv_{2}). From this it is easy to see that the G-action on \overline{V}_{3}(W) gives a G-action on E_{2}(q) for any q.

Let X denote the total space of the sphere bundle of E_{2}(q). Then X = S^{2n−1} ×S^{4n−5} × S^{q(4n−8)−1} and G acts smoothly on X. Since V has fixity 2, the quaternionic representation W also has fixity 2. This implies that if g ∈ G fixes a point

(v_{1,1}, v_{2,1}, v_{2,2}, v_{3,1}, v_{3,2}, …, v_{3,2q}) ∈ \overline{V}_{3}(W),

then v_{3,i} = 0 for all i. Thus G acts freely on X. □

For real representations the argument given in the above proof can be extended to prove that if a finite group has a faithful real representation ρ : G → SO(n) with fixity f ≤ 3, then G acts freely and smoothly on a product of f + 1 spheres with trivial action on homology. For this, we consider the diagram

\xymatrix{E_{2} \ar[r] \ar[d] & E′_{2} \ar[r] \ar[d] & V_{3}(H^{n}) \ar[d]^{q_{2}} \\ E_{1} \ar[r]^{f′} \ar[d] & V_{2}(C^{n}) \ar[r]^{h} \ar[d]^{q_{1}} & V_{2}(H^{n}) \\ V_{1}(R^{n}) \ar[r]^{f} & V_{1}(C^{n})}

where each square in the diagram is a pullback square and the maps f and h are maps induced by inclusions R ⊂ C ⊂ H. Since the map f is null-homotopic, E_{1} ≃ S^{n−1}×S^{2n−3}. The space V_{2}(H) is (4n−6)-connected. For n ≥ 2, the inequality dim E_{1} = 3n−4 ≤ 4n−6 holds, so by the compression lemma in homotopy theory, the composition h ∘ f′ is also null-homotopic. This implies that E_{2} ≃ S^{n−1} × S^{2n−3} × S^{4n−9}.

Note that if G has a real representation with fixity f = 1 or 2, then G acts freely and smoothly on E_{f} in the diagram. The G-actions are induced from the G-actions on Stiefel manifolds as in the complex case. For f = 3, we first consider the pullback square

\xymatrix{E_{3} \ar[r] \ar[d]^{ξ} & \overline{V}_{4}(H^{n}) \ar[d]^{q_{3}} \\ E_{2} \ar[r]^{h∘f′} & V_{3}(H^{n})}

and then we take a q-fold Whitney sum of ξ to make the dimension condition in Lemma 2.2 to hold. For this we need q to satisfy dim E_{2} = 7n − 13 ≤ q(4n − 12) + 2. Note that since we have n ≥ 4, there always exists a q which makes this inequality hold. If n ≥ 9, the inequality holds even for q = 2. Let X denote the total space of the sphere bundle of E_{3}(q), then X is a product of 4 spheres. If G has a faithful real representation ρ : G → SO(n) with fixity f = 3, then G acts freely and smoothly on X.

Note that in the above construction the space E_{1} can be considered as the space pairs of n-dimensional real vectors (v_{1,1}, v_{2,1}, v_{2,2}) such that v_{1,1} + i0 is orthogonal to v_{2,1} + iv_{2,2} as complex vectors. This is equivalent to saying that v_{1,1} ⊥ v_{2,1} and v_{1,1} ⊥ v_{2,2} as real vectors. It is clear from this that if G acts orthogonally on R^{n}, then it will act on E_{1} with diagonal action on coordinates. In a similar way, the space E_{2} can be considered as the space of tuples of the form

(v_{1,1}, v_{2,1}, v_{2,2}, v_{3,1}, v_{3,2}, v_{3,3}, v_{3,4})

where v_{1,1} ⊥ v_{k,l} for all k, l and v_{2,1} + iv_{2,2} is perpendicular to v_{3,1} + iv_{3,2} and v_{3,3} + iv_{3,4} as complex vectors. This is equivalent to saying that as pairs of real vectors, (v_{2,1}, v_{2,2}) is perpendicular to (v_{3,1}, v_{3,2}) and (v_{3,3}, v_{3,4}), and that (−v_{2,2}, v_{2,1}) is perpendicular to (v_{3,1}, v_{3,2}) and (v_{3,3}, v_{3,4}). Note that this second relation makes it possible to conclude that the G-action on E_{2} is free for a representation with fixity f = 2. If we had just taken all the tuples

(v_{1,1}, v_{2,1}, v_{2,2}, v_{3,1}, v_{3,2}, v_{3,3}, v_{3,4})

with v_{1,1} ⊥ v_{k,l} for all k, l and (v_{2,1}, v_{2,2}) is perpendicular to (v_{3,1}, v_{3,2}) and (v_{3,3}, v_{3,4}) we would still have a product of spheres with a G-action but the action would no longer be free for a real representation with fixity 2. In the next section, we elaborate on this idea and give a more general construction of “quasilinear” actions.

3. Proof of Theorem 1.2

Let V be a real representation of G. Suppose that we are given a sequence of R-algebras A_{1} ⊂ A_{2} ⊂ ⋯ ⊂ A_{s} which are possibly non-commutative and non-associative. For each i, define V_{i} = V ⊗_{R} A_{i}. Then, V_{i} is a real representation with the G-action given by left multiplication g(v ⊗ a) = gv ⊗ a. Note that V_{i} is also a module over A_{i} where the the action of a′ ∈ A on v ⊗ a is given by (v ⊗ a)a′ = v ⊗ (aa′). In the non-associative case, this does not give a module structure in the usual sense by it still gives a R-bilinear map V_{i} × A_{i} → V_{i} which is what we need.

If there is an inner product on V, we can define an inner product on V_{i} by (v⊗a, v′⊗a′) = (v, v′)(a, a′) where the inner products on A_{i}’s can be taken in such a way that A_{i} ⊂ A_{i+1} is a subspace as an inner product space. This gives us a sequence of real representations of G

V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{s}

such that each V_{i} is a module over A_{i}.

Form now on we assume that V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{s} is a sequence of real representations such that each V_{i} is a module over A_{i}, but not necessarily of the form V_{i} = V ⊗_{R} A_{i} for a fixed real representation V. We choose an inner product on each V_{i} such that V_{i} ⊂ V_{i+1} is an inclusion of inner product spaces. For each u ∈ V_{i}, let ⟨u⟩_{A_{i}} denote the vector space {ua | a ∈ A_{i}}. We denote the norm of a vector by |u|. We define

W_{k}({V_{i}}_{i=1}^{s}) = {(v_{1}, v_{2}, …, v_{k}) | v_{i} ∈ V_{i} with |v_{i}| = 1, and v_{j} ⊥ ⟨v_{i}⟩_{A_{j}} for all i < j}.

To simplify notation, we write W_{k}(V) for W_{k}(\{V_{i}\}_{i=1}^{s}). The subspace \overline{W}_{k}(V) is defined as the space of k-tuples as above but we no longer require the last vector v_{k} to be a unit vector.

We will prove that under certain conditions, the map q_{k} : \overline{W}_{k+1}(V) → W_{k}(V) defined by q_{k}(v_{1}, …, v_{k+1}) = (v_{1}, …, v_{k}) is a projection map of a stably trivial vector bundle. To find these conditions, we consider some subbundles of the trivial bundle W_{k}(V) × V_{k+1} → W_{k}(V) which are defined in the following way: For each i ∈ {1, …, k}, let

E_{i} = {((v_{1}, …, v_{k}), w) | w ∈ ⟨v_{i}⟩_{A_{k+1}}, (v_{1}, …, v_{k}) ∈ W_{k}(V)},

and let q_{k,i} : E_{i} → W_{k}(V) be the obvious projection map ((v_{1}, …, v_{k}), w) → (v_{1}, …, v_{k}).

Note that for each u ∈ V_{i}, the vector space ⟨u⟩_{A_{k+1}} is isomorphic to A_{k+1}/ Ann_{A_{k+1}}(u) where Ann_{A_{k+1}}(u) = {a ∈ A_{k+1} | ua = 0}. Suppose that Ann_{A_{k+1}}(u) is equal to a fixed subspace B_{k+1,i} ⊂ A_{k+1} for every u ∈ V_{i}. Then, we can choose a basis {a_{1}, …, a_{m}} for A_{k+1} such that {a_{1}, …, a_{l}} is a basis for B_{k+1,i}. Using this basis, we can easily express w ∈ ⟨v_{i}⟩_{A_{k+1}} uniquely as ∑_{t=l+1}^{m} γ_{t}(v_{i}a_{t}) where γ_{t} ∈ R. We conclude the following:

Lemma 3.1. Suppose that Ann_{A_{k+1}}(u) is equal to a fixed subspace B_{k+1,i} ⊂ A_{k+1} for every nonzero u ∈ V_{i}. Then, the map q_{k,i} : E_{i} → W_{k}(V) defined above is the projection map of a fibre bundle θ_{k,i} which is a trivial bundle with dimension dim_{R}(A_{k+1}/B_{k+1,i}).

If a filtration {V_{i}}_{i=1}^{s} satisfies the condition given in the above lemma for all i, k with 1 ≤ i ≤ k ≤ s−1, then we say it is a uniform filtration. In our applications, this condition holds often with B_{k+1,i} = 0 for all i, k. Another condition that we can impose on our filtration is the following:

Definition 3.2. We say a sequence {V_{i}}_{i=1}^{s} is separable at k if for every (v_{1}, …, v_{k}) ∈ W_{k}(V), we have

⟨v_{j}⟩_{A_{k+1}} ∩ ⟨v_{i}⟩_{A_{k+1}} = 0

for every i, j with 1 ≤ i < j ≤ k. Then, we say the sequence {V_{i}}_{i=1}^{s} is a separable sequence if it is separable at every k with 1 ≤ k ≤ s − 1.

For uniform, separable sequence of real representations, the following is true.

Lemma 3.3. Let {V_{i}}_{i=1}^{s} be a uniform, separable sequence. Then, the space

E = {((v_{1}, …, v_{k}), w) | w ∈ ∑_{i=1}^{k}⟨v_{i}⟩_{A_{k+1}}, (v_{1}, …, v_{k}) ∈ W_{k}(V)},

together with the obvious projection map E → W_{k}(V) defines a vector bundle isomorphic to θ_{k,1} ⊕ θ_{k,2} ⊕ ⋯ ⊕ θ_{k,k}.

Let ξ_{k} denote the orthogonal complement of the bundle θ_{k,1} ⊕ ⋯ ⊕ θ_{k,k} in the trivial bundle τ_{k} : W_{k}(V) × V_{k+1} → W_{k}(V). Then, it is not very difficult to see that ξ_{k} is the bundle with the projection map q_{k} : \overline{W}_{k}(V) → W_{k}(V) described above. So, we conclude the following:

Proposition 3.4. Let {V_{i}}_{i=1}^{s} be a uniform, separable sequence of real representations. Then, the vector bundle ξ_{k} : \overline{W}_{k+1}(V) → W_{k}(V) is a stably trivial vector bundle of dimension dim_{R} V_{k+1} − ∑_{i=1}^{k} dim_{R}(A_{k+1}/B_{k+1,i}).

Let m_{1} = dim_{R} V_{1}, and for each 1 ≤ k ≤ s − 1, define

m_{k+1} = dim_{R} V_{k+1} − ∑_{i=1}^{k} dim_{R}(A_{k+1}/B_{k+1,i}).

Note that a CW -structure can be given for the space W_{k}(V), and as CW -complex the dimension of W_{k}(V) is ∑_{i=1}^{k}(m_{i} − 1). By Lemma 2.2, we have the following:

Proposition 3.5. Let {V_{i}}_{i=1}^{s} be a uniform, separable sequence of representation. Suppose that

∑_{j=1}^{k}(m_{j} − 1) ≤ m_{k+1} − 1

holds for all 1 ≤ k ≤ s − 1. Then, the vector bundles ξ_{1}, … ξ_{s} associated to this sequence are all trivial vector bundles. In particular, for each k ≤ s, we have W_{k}(V) ≃ S^{m_{1}−1} × S^{m_{2}−1} × ⋯ × S^{m_{k}−1}.

Now, we are ready to prove Theorem 1.2.

Proof of Theorem 1.2. Let G be a finite group and V be a faithful n-dimensional oriented real representation of G with fixity 4. We assume that n ≥ 12. Consider the sequence of R-algebras

A_{1} = R ⊂ A_{2} = C ⊂ A_{3} = H ⊂ A_{4} = O ⊂ A_{5} = S

where O denotes the octanions, and S denotes the sedenions. These algebras are the first five steps of the Cayley-Dickson construction, where A_{i} is defined as the pairs of elements (a, b) with a, b ∈ A_{i−1} and the multiplication is given by

(a, b)(c, d) = (ac − d^{∗}b, da + bc^{∗})

and the conjugation for a pair is defined by (a, b)^{∗} = (a^{∗}, −b).

One particular property of the above sequence is that for every a_{i} ∈ A_{i} and a_{j} ∈ A_{j} with 1 ≤ i < j ≤ 5, if a_{i}a_{j} = 0, then a_{i} = 0 or a_{j} = 0. This is because the algebras R, C, H, and O are division algebras. The sedenions S is not a division algebra but we have (a, 0)(c, d) = 0 ⇒ ac = 0, da = 0, so the above property still holds since O is a division algebra. Because of this property, we can conclude that the sequence {V_{i}}_{i=1}^{5} with V_{i} = V ⊗_{R} A_{i} is a uniform sequence with B_{k+1,i} = 0 for all 1 ≤ i ≤ k ≤ 4.

We claim that the sequence {V_{i}}_{i=1}^{5} is also separable. To show this, we will need to show that for every k ≤ 4, the following holds: If v_{i} ∈ V_{i} and v_{j} ∈ V_{j} with 1 ≤ i < j ≤ k, then ⟨v_{i}⟩_{A_{k+1}} ∩ ⟨v_{j}⟩_{A_{k+1}} = 0. Note that for every u, t with 1 ≤ u < t ≤ 5, the algebra A_{t} is a free module over A_{u}. So, to show that ⟨v_{i}⟩_{A_{k+1}} ∩ ⟨v_{j}⟩_{A_{k+1}} = 0, it is enough to show that ⟨v_{i}⟩_{A_{j}} ∩ ⟨v_{j}⟩_{A_{j}} = 0. By choosing a free basis for A_{j} as an A_{i}-module, we can express v_{i} in V_{j} as a tuple (v_{i}, 0, …, 0) and v_{j} as a tuple (v_{j,1}, …, v_{j,j}). If ⟨v_{i}⟩_{A_{j}} ∩ ⟨v_{j}⟩_{A_{j}} ≠ 0, there exist a, b ∈ A_{i} such that v_{i}a = v_{j,l}b for some l with 1 ≤ l ≤ j. Since i < 4, the algebra A_{i} is an associative division algebra, so we can assume that b = 1.

Note that we have v_{j} ⊥ ⟨v_{i}⟩_{A_{j}}, so it gives that v_{i}a ⊥ v_{j,l} for all a ∈ A_{i} and l ∈ {1, …, k}. This because the above decomposition of V_{j} to a direct sum of V_{i}’s respects the inner product on V_{i}. Since we can not have v_{i}a = v_{j,l} and v_{i}a ⊥ v_{j,l} at the same time, we have a contradiction. Therefore, the sequence {V_{i}}_{i=1}^{5} is a separable sequence.

Since B_{k+1,i} = 0 for all k, i with 1 ≤ i ≤ k ≤ 4, we have m_{1} = dim_{R} V_{1} = n and

m_{k+1} = dim_{R} V_{k+1} − ∑_{i=1}^{k} dim_{R}(A_{k+1}/B_{k+1,i}) = n2^{k} − ∑_{i=1}^{k} 2^{k} = (n − k)2^{k}.

So, to apply Proposition 3.5, we need the following inequalities to hold:

\begin{aligned}n − 1 & ≤ 2n − 3 \\ 3n − 4 & ≤ 4n − 9 \\ 7n − 13 & ≤ 8n − 25 \\ 15n − 38 & ≤ 16n − 65 \end{aligned} ; (3.5)

Note that the first three inequalities are satisfied when n ≥ 12. The last inequality needs a higher bound for n, but we can always use the Whitney sum on the last bundle, so we really don’t need the last inequality to hold. To see this, consider the bundle \overline{W}_{5}(V) → W_{4}(V). Since the sequence {V_{i}}_{i=1}^{4} is splitting, we get W_{4}(V) ≃ S^{n−1} × S^{2n−3} × S^{4n−9} × S^{8n−25}. Taking a q-fold Whitney sum of this bundle, we get a real vector bundle, say E(q), with dimension q(16n − 64) − 1. For this bundle to be trivial, we need the inequality

15n − 38 ≤ q(16n − 64) − 1

to hold. Since we have already assumed n ≥ 12, the inequality holds even for q = 2.

Finally, we need to show that G acts freely on the total space of the sphere bundle for E(q). Let us denote this space by X. As before, we consider an element in E(q) as a tuple of real vectors (v_{1,1}, v_{2,1}, v_{2,2}, …, v_{5,1}, …, v_{5,16q}) in V which satisfies certain orthogonality conditions. Note that the inner product on V_{5} respects the decomposition of it as a direct sum of smaller V_{i}’s, so all these orthogonality conditions can be expressed in terms of these real coordinate vectors. If we choose the inner product on V as a G-invariant inner product, then G will act on E(q) and hence on X. Thus G acts smoothly on X.

Finally, we will show that the action is free when the fixity f of V is 4. Suppose that we have an element (v_{1,1}, v_{2,1}, v_{2,2}, …, v_{5,1}, …, v_{5,16q}) fixed by 1 ≠ g ∈ G. Then, the vector (v_{5,1}, …, v_{5,16}) will lie in V_{5}^{g} and will be orthogonal to subspaces ⟨v_{i}⟩_{A_{5}} of V_{i}^{g} for every v_{i} ∈ V_{i}. Here v_{1} = v_{1,1}, v_{2} = (v_{2,1}, v_{2,2}), v_{3} = (v_{3,1}, …, v_{3,4}), and v_{4} = (v_{4,1}, …, v_{4,8}). So, (v_{5,1}, …, v_{5,16}) lies in a vector space of dimension 16f − 16 · 4 which is zero when f = 4. So, G acts freely on X. □

References

[1] A. Adem, J. F. Davis, and Ö. Ünlü, Fixity and free group actions on products of spheres, Comment. Math. Helv. 79 (2004), 758–778.

[2] A. Adem and J. H. Smith, Periodic complexes and group actions, Ann. of Math. (2) 154 (2001), 407–435.

[3] J. F. Davis, P. Kirk, Lecture notes in algebraic topology, Graduate Studies in Mathematics 35, American Mathematical Society, Providence, RI, 2001.

[4] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.

[5] D. Husemoller, Fibre bundles, Third edition, Graduate Texts in Mathematics 20, Springer-Verlag, New York, 1994.

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

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