Homotopy types of locally linear representation forms

Abstract

The authors of [6] investigated certain locally linear actions of a cyclic groupG of odd order on homotopy spheres, the so-calledG-representation forms [16]. In particular, several conditions on a dimension function were described that made sure that it can be realized as the dimension function of aG-representation form. It remained unclear, whether all homotopy types with those dimension functions would support a locally linear structure. It is the aim of this note to show that this is not the case, i.e., to give examples of homotopy representations [17] with the same dimension functions some of which support a locally linear structure with stably trivial tangent bundle and others do not. The main tools are formulated as general splitting principles for fixed point and restriction functors that may have some interest in their own right, too.