Periodic Complexes and Group Actions

Abstract

Abstract. In this paper we show that the cohomology of a connected CW -complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on R n × S m ; we construct non-standard free actions of rank two simple groups on finite complexes Y ≃ S n × S m ; and we prove that a finite p-group P acts freely on such a complex if and only if it does not contain a subgroup isomorphic to (Z/p) 3 .

All 𝑝-local finite groups of rank two for odd prime 𝑝

“…The B(3, r; 0, γ, 0)-conjugation class of E i consists of the subgroups ζ, ζ , sa , where a ∈ γ 1 , sa has order 3 and a = s a 1 1 s a 2 2 with a 1 ≡ i modulo 3. By taking square powers it is straightforward that these subgroups are exactly the same as the subgroups ζ, ζ , s 2 Therefore we are left with the case of r = 2k when γ 2 is the only F-Alperin proper subgroup. Because F is a saturated fusion system and γ 2 is fully normalized, then, by Definition 2.2, Aut B(3,r;1,0,0) (γ 2 ) is a Sylow of Aut F (γ 2 ).…”

“…As γ 1 is abelian and P must contain its centralizer as it is centric, P must equal γ 1 . Recalling the orders of s 1 and s 2 we check that only the case r odd is allowed. Suppose now that P is not contained in γ 1 .…”

“…possible p-local structures, is meaningful when working in other research areas such as transformation groups (e.g. when constructing actions on spheres [2,1]) or modular representation theory (e.g. the study of the p-local structure of a group is a first step when checking conjectures like those of Alperin [3] or Dade [17]).…”

Constructing homologically trivial actions on products of spheres

“…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.…”

“…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].…”