On the realizability of group actions

Abstract

We raise the question of realizability of group actions which is an extended version of the 1960's Kahn realizability problem for (abstract) groups. Namely, if $M$ is a $\mathbb ZG$-module for a group $G$, we say that a simply-connected space $X$ realize this action if, for some $k$, $\pi_k(X)$ as a $\mathbb Z \mathcal E (X) $-module for the group $\mathcal E (X)$ of self-homotopy equivalences of $X$, is isomorphic to $M$ as a $\mathbb ZG$-module. Which modules can be so realized? In this paper we obtain a positive answer for any faithful finitely generated $\mathbb Q G$-module, where $G$ is finite. Our proof relies on providing a positive answer to Kahn's problem for a large class of orthogonal groups of which, by using invariant theory, our case is shown to be a particular one.