Quasilinear actions on products of spheres

Abstract

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.

Fusion systems and group actions with abelian isotropy subgroups

“…However, one would need to construct free actions for certain families of groups where the surgery obstructions could be detected when they are reduced to subgroups in these families. Some general methods which can be used to do constructions for some of these families are given in [1,14,15]. Our goal in this paper is to improve the methods used in [15] and, as a result, obtain some new actions.…”

“…So, it cannot act freely on a product of linear spheres. On the other hand, it is easy to construct homologically trivial actions on products of spheres for supersolvable groups and many other groups with small rank and fixity (see [11,14]).…”

Constructing homologically trivial actions on products of spheres

“…First we introduce some notation and recall some basic facts about Stiefel manifolds. For more details, we refer the reader to [19].…”

“…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.…”