On the realizability of group actions

TL;DR

This paper investigates which modules can be realized as homotopy groups of simply-connected spaces with group actions, providing positive results for faithful finitely generated modules over finite groups.

Contribution

It extends the classical Kahn realizability problem by establishing realizability for a broad class of modules over finite groups using invariant theory.

Findings

Positive realizability for any faithful finitely generated QG-module with finite G

Utilizes invariant theory to connect orthogonal groups to the realizability problem

Provides a constructive approach for realizing modules as homotopy groups

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.