Homotopy equivariance, strict equivariance and induction theory

Abstract

An obvious question occurs at the very start of equivariant homotopy theory. What is the relationship between maps equivariant up to homotopy and strictly equivariant maps? This question has been studied by various people, usually away from the group order ([8, 11, 22, 25, 26]). We consider the problem stably and answer it by giving a spectral sequence proceeding from homotopy equivariant to strictly equivariant information. The form of the spectral sequence is not surprising, but there are three distinctive features of our approach: (1) we show that the spectral sequence may be viewed as an Adams spectral sequence based on nonequivariant homotopy, (2) we show how to exploit the product structure, and (3) we give a treatment showing how Dress's algebra of induction theory [13] applies to give non-normal subgroups equal status. As a spinoff from (3) we also obtain spectral sequences for calculating homology and cohomology of universal spaces (3.5).