On the realizability of group actions

Abstract

Let the pair (G,M)R denote a group G acting on an R-module M, R a unitary ring. We ask for the existence of an R-local space X such that (E(X),πk(X))R is equivalent, in a natural way, to (G,M)R, for some k≥2, where E(X) denotes the group of homotopy classes of self-homotopy equivalences of X. If such an X exists, we say that X realizes the group action (G,M)R. We prove that if G is finite and acts faithfully on a finitely generated Q-module M, there exist infinitely many rational spaces realizing (G,M)Q. Our proof relies on providing a positive answer to Kahn's realizability problem for a large class of orthogonal groups that strictly contains finite ones. As a matter of fact, we enlarge the class of groups that is known to be realizable in the classical Kahn's sense.