Homotopy rigidity of linear actions: characters tell all

Counting characters in linear group actions

Let G be a finite group and V be a finite G–module. We present upper bounds for the cardinalities of certain subsets of Irr(GV ), such as the set of those χ ∈ Irr(GV ) such that, for a fixed v ∈ V , the restriction of χ to 〈v〉 is not a multiple of the regular character of 〈v〉. These results might be useful in attacking the non–coprime k(GV )–problem.

Homotopy Rigidity for Grassmannians

Two n-dimensional unitary representations which differ by complex conjugation or tensoring with a character induce topologicalry equivalent actions on the Grassmann manifold of complex m-planes in «-space. This paper shows under modest dimension hypotheses that only such projectively equivalent linear representations of compact connected Lie groups can give topologicalry conjugate actions. A li...

Rigidity of Multiparameter Actions

In this survey I would like to expose recent developments and applications of the study of the rigidity properties of natural algebraic actions of multidimensional abelian groups. This study was initiated by Hillel Furstenberg in his landmark paper [Fur67]. The survey is an outgrowth of notes I gave to a series of lectures in the II Workshop on Dynamics and Randomness in December 2002; a relate...

Homotopy Actions by Topological Actions . Ii

A homotopy action of a group G on a space X is a homomorphism from G to the group HAUT(X) of homotopy classes of homotopy equivalences of X. George Cooke developed an 'obstruction theory to determine if a homotopy action is equivalent up to homotopy to a topological action. The question studied in this paper is: Given a diagram of spaces with homotopy actions of G and maps between them that are...

Characters in Global Equivariant Homotopy Theory