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Homotopy types of orbit spaces and their self-equivalences for the periodic groups Z/a \rtimes (Z/b x T^\star_n) and Z/a \rtimes (Z/b x O^\star_n)

Let G be a finite group given in one of the forms listed in the title with period 2d and X(n) an n-dimensional CW-complex with the homotopy type of an n-sphere. We study the automorphism group Aut(G) to compute the number of distinct homotopy types of orbit spaces X(2dn-1)/\mu with respect to free and cellular G-actions \mu on all CW-complexes X(2dn-1). At the end, the groups E(X(2dn-1)/\mu) of self homotopy equivalences of orbit spaces X(2dn-1)/\mu associated with free and cellular G-actions \mu on X(2dn-1) are determined.

group with periodic cohomology of period 2d acts freely and cellularly on a (2d − 1)dimensional CW -complex of the homotopy type of a (2d − 1)-sphere. It is worth to mention that useful cohomological and geometric aspects associated to group actions are presented in [2] and a list of basic conjectures is provided.
Backing to the case of a (2n − 1)-dimensional CW -complex X(2n − 1) with the homotopy type of a (2n − 1)-sphere, by means of results in [11], it is shown in [12,Theorem 1.8] that the set of homotopy types of spherical space forms of all free cellular G-actions on X(2n − 1) is in one-to-one correspondence with the orbits, which contain a generator of the cyclic group H 2n (G) = Z/|G| under the action of ±Aut (G) (see [4] for another approach). This plays also a fundamental role in the calculation of the group E(X(2n − 1)/µ) of self homotopy equivalences of the orbit space X(2n − 1)/µ.
All finite periodic groups has been completely described by Suzuki-Zassenhaus and their classification can be found in the table [1, Chapter IV; Theorem 6.15]. The present paper is part of the project to describe the homotopy types of the orbit spaces and the group of self homotopy equivalences for all periodic groups. It continues the works of [4,5,6,8], where the cases corresponding to the families I and II from the table [1, Chapter IV; Theorem 6.15] with the Suzuki-Zassenhaus classification of finite periodic groups have been solved. Here we have two goals. The first one is to calculate the numbers of homotopy types of spherical spaces forms for the groups Z/a ⋊ (Z × T ⋆ n ) and Z/a ⋊ (Z × O ⋆ n ) corresponding to the families III and IV from the table mentioned above. The second one is to determine the group of homotopy classes of self-equivalences for space forms given by free actions of those both families of finite periodic groups. The results of [4,5,6,8], taking care for the groups from families I and II of that table, are essential to make crucial calculations to develop the main results stated in Theorem 2.2 and Theorem 3.2.
In order to obtain these results, we divide the paper into two parts. The first part consists of some algebraic results. The automorphism group Aut(A ⋊ α G) of a semi-direct product A ⋊ α G of some finite groups A, G leads in [6] to a splitting short exact sequence 0 → Der α (G, A) −→ Aut (A ⋊ α G) −→ Aut (A) × Aut α (G) → 1.
Section 1 makes use of this to achieve automorphisms of the groups in question. This is the approach to develop in Proposition 1.1 and Proposition 1.2 the groups Der α (G, A) and Aut (A) × Aut α (G), respectively.
Then, in the second part, we present geometric interpretations of those algebraic results in terms of G-actions. Section 2 uses the group Aut (A)×Aut α (G) established in Proposition 1.2 and Lyndon-Hochschild-Serre spectral sequence to deal with the number of homotopy types of spherical space forms for actions of the groups Z/a ⋊ (Z/b × T ⋆ n ) and Z/a ⋊ (Z/b × O ⋆ n ). The main results of this section are stated in are appropriate restrictions of γ and τ , respectively. Then: Then, Corollary 2.3 says that the number of such homotopy types of those space forms coincides with that of (4n − 1)-lens spaces studied in [4] provided the least period of the groups in question is ≤ 4.
The group of crossed homomorphisms Der α (G, A) studied in Proposition 1.1 plays a key role in Section 3 dealing with the structure of groups E(X(2dn − 1)/µ) of self homotopy equivalences for spherical space forms X(2dn − 1)/µ with respect to free and cellular Z/a ⋊ (Z/b × T ⋆ n )-and Z/a ⋊ (Z/b × O ⋆ n )-actions µ, respectively. We point out that by means of [4, Proposition 3.1] (see also [10,Theorem 1.4]), the group E(X(2k − 1)/µ) is independent of the action µ on X(2k − 1). Writing X(2k − 1)/G for the corresponding orbit space, we close the paper with ) acts freely and cellularly on a which deals with explicit formulae for those groups of self homotopy equivalences.
Approaching of homotopy types of spherical space forms and their self homotopy equivalences for the rest of the groups from the table in [1, Chapter IV; Theorem 6.15], or more precisely for the family of VI of this table, is in progress.
1. Algebraic backgrounds. Let a finite group G be given by an extension where the orders of groups G 1 and G 2 are relatively prime. We recall that by [7] any automorphism of G leaves the subgroup G 1 invariant and consequently, there is a map ψ : Aut (G) → Aut (G 1 ) × Aut (G 2 ) of automorphism groups. Given Then, an isomorphism follows. we can choose the action α given above. Then, we consider the semi-direct product the binary tetrahedral group. More generally, for n ≥ 1 consider the action α n : Z/3 n → Aut (Q 8 ) as the composition of the quotient map Z/3 n → Z/3 with the action α : Z/3 → Aut (Q 8 ). Then, for the group by means of [13, p. 198], it holds in virtue of generators and relations. In particular, the cyclic group Z/3 n is the abelianization of T ⋆ n for any n ≥ 1 and the center Z(T ⋆ n ) = Z/2 ⊕ Z/3 n−1 . The symmetric group S 3 has two distinct extensions by Q 8 , with respect to the outer action α : This follows from the facts that Z(Q 8 ) = Z/2 and H 2 (S 3 , Z/2) = Z/2. These extensions are the semi-direct product Q 8 ⋊ S 3 and where O ⋆ is the binary octahedral group. Because ϕ −1 (A 3 ) = T ⋆ for the alternating subgroup A 3 ⊆ S 3 , so we achieve the extension In the light of [13, p. 198] the group O ⋆ n is given by in virtue of generators and relations. It follows that the cyclic group Z/2 is isomorphic to the abelianization of O ⋆ n and the center Z(O ⋆ n ) as well. Now, consider the periodic groups Z/a ⋊ γ (Z/b × T ⋆ n ) and Z/a ⋊ τ (Z/b × O ⋆ n ) corresponding to the families III and IV [1, Theorem 6.15] with (a, b) = (ab, 6) = 1 respectively, on the cyclic group Z/a. The group Aut (Z/a) is abelian, a fortiori the actions γ and τ are uniquely determined by their But the abelianizations of T ⋆ n and O ⋆ n are isomorphic to the groups Z/3 n and Z/2, respectively. Whence, the actions γ 2 and τ 2 are uniquely determined by γ 2 (X) and τ 2 (R), respectively, with γ 2 (X) To study the groups Aut (Z/a ⋊ γ (Z/b × T ⋆ n )) and Aut (Z/a ⋊ γ (Z/b × O ⋆ n )), we need, in the light of [6, Proposition 1.3], to describe the groups Der γ (Z/b × T ⋆ n , Z/a), , respectively. First, let a = p k with p = 2, 3 prime and k ≥ 0. Because the actions γ 2 and τ 2 factor through the abelianizations of T ⋆ n and O ⋆ n , which are isomorphic to the groups Z/3 n and Z/2, respectively, whence Ker γ 2 is trivial or Ker γ 2 = Q 8 ⋊ αn Z/3 n 0 for some n 0 ≤ n and Ker τ 2 is trivial or equals T ⋆ n (as a subgroup with index two in O ⋆ n and containing T ⋆ n ). Consequently, by [6], we achieve that Der and Z/b × Z/2 are cyclic whence, as in [6, Corollary 1.5], elements of Derτ (Z/b × Z/3 n−n 0 , Z/p n ) and Derτ (Z/b × Z/2, Z/p n ) might be described by means of some elements in Z/p n . Now, if a is a positive integer with (a, 6) = 1 and a = p k 1 1 · · · p ks s its prime factorization with k i ≥ 1 then p i = 2, 3 for all i = 1, . . . , s. Obviously, any isomorphism Z/a ∼ = → Z/p n 1 1 × · · · × Z/p ns s yields isomorphisms α : Aut (Z/a) ∼ = → Aut (Z/p n 1 1 × · · · × Z/p ns s ) and for an action γ : Z/b × T ⋆ n → Aut (Z/a), and for an action τ : Then, the well-known (see e.g. [6, Lemma and where α i is the composition of α with an appropriate projection map Aut (Z/p k 1 1 ) × · · · × Aut (Z/p ks s ) → Aut (Z/p k i i ) for i = 1, . . . , s. Thus, we may summarize the discussion above as follows. and if γ 2 and τ 2 are trivial; (2) Der γ (Z/b×T ⋆ n , Z/p k ) ∼ = Derγ (Z/b×Z/3 n−n 0 , Z/p k ) provided Ker γ 2 = Q 8 ⋊ αn Z/3 n 0 , whereγ : If γ : Z/b × T ⋆ n → Aut (Z/a) and τ : Z/b × O ⋆ n → Aut (Z/a) are actions with (a, b) = (ab, 6) = 1 and a = p k 1 1 · · · p ks s is the prime factorization of a with k i ≥ 1 for i = 1, . . . , s then and where α i is the composition of an isomorphism α : Aut (Z/a) Now, move to the groups Aut γ (Z/b×T ⋆ n ) and Aut τ (Z/b×O ⋆ n ), where γ = (γ 1 , γ 2 ) : . Furthermore, the groups Aut (T ⋆ n ) and Aut (O ⋆ n ) have been fully described in [7] for all n ≥ 1. In the light of [6, Corollary 1.4] we achieve and But ϕ ∈ Aut γ 2 (T ⋆ n ) (resp. ϕ ∈ Aut τ 2 (O ⋆ n )) if and only if γ 2 (X) = (γ 2 ϕ)(X) (resp. τ 2 (R) = (τ 2 ϕ)(R)). Now, if Ker γ 2 = Q 8 ⋊ αn Z/3 n 0 then, from the list of elements in Aut (T ⋆ n ) presented in [7], it follows that Aut γ 2 (T ⋆ n ) = {ϕ ∈ Aut (T ⋆ n ); ϕ(X) = X l(1+3 n 0 +1 ) for l = 0, . . . , 3 n−n 0 −1 }.

Homotopy types of space forms. Given a group G, write H k (G) for its
kth cohomology group with constant coefficients in the integers Z for k ≥ 0. Then, any automorphism ϕ ∈ Aut (G) yields the induced automorphism ϕ * ∈ Aut (H n (G)) and we write η : Aut (G) → Aut (H k (G)) for the corresponding anti-homomorphism.
By a period of a group G we mean an integer d such that H k (G) = H k+d (G) for all k > 0, and a group G with this property is called periodic. Among all periods of a group G there is the least one; and all others are multiple of that one. That least one period we call the period of the group and by [3, Section 11] the period of any periodic group is even.
Throughout the rest of the paper, X(k) denotes a k-dimensional CW -complex with the homotopy type of a k-sphere and the group Aut (Z/a) is identified with the unit group (Z/a) ⋆ of the mod a ring Z/a. Given a free cellular action µ of a finite group G with order |G| on a CW -complex X(2k −1) write X(2k −1)/µ for the corresponding orbit space called a (2k−1)-spherical space form or a Swan (2k−1)-complex (see e.g., [2]). Then, the group G is periodic with period 2d dividing 2k and by / ≃ for its cardinality, respectively. By means of [11], it is shown in [12,Theorem 1.8] that elements of the set K 2k−1 G / ≃ are in one-to-one correspondence with the orbits, which contain a generator of H 2k (G) = Z/|G| under an action of ±Aut (G) (see also [4] for another approach). But generators of the group Z/|G| are given by the unit group (Z/|G|) ⋆ of the ring Z/|G|. Thus, those homotopy types are in one-to-one correspondence with the quotient (Z/|G|) ⋆ /{±ϕ * ; ϕ ∈ Aut (G)}, where ϕ * is the induced automorphism on the cohomology H 2k (G) = Z/|G| by ϕ ∈ Aut (G). Now, let G 1 and G 2 be finite groups with relatively prime orders |G 1 | and |G 2 |, respectively. If G 1 and G 2 are also periodic with periods 2d 1 and 2d 2 , respectively then by [1] the least common multiple [2d 1 , 2d 2 ] of 2d 1 and 2d 2 is the least period of the product G 1 × G 2 . Furthermore, given a finite group G with an action α : G → (Z/a) ⋆ write |α(g)| for the order of α(g) with g ∈ G. Let ℓ(α) = [|α(g)|; for g ∈ G] be the least common multiple of those orders. Then, for a semi-direct product Z/a⋊ α G, we have shown in [6], by means of the Lyndon-Hochschild-Serre spectral sequence, the following result. Thus, we are in a position to investigate the periodic groups Z/a ⋊ γ (Z/b × T ⋆ n ) and Z/a ⋊ τ (Z/b × O ⋆ n ). First, we find the least periods of the groups T ⋆ n and O ⋆ n . Because T ⋆ n = Q 8 ⋊ α Z/3 n whence the Lyndon-Hochschild-Serre spectral sequence applied to the short one 0, if q = 0 and p odd; Z/3 n , if q = 0 and p even with = 0; Using the cohomology with l ≥ 0, we can easily get with l ≥ 0 and consequently, 4 is the least period of the group T ⋆ n . Whence, by Proposition 2.1, the number 2[ℓ(γ), 2] is the least period of the group Z/a ⋊ γ (Z/b × T ⋆ n ).