On the realizability of group actions
Cristina Costoya, Antonio Viruel

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 is a -module for a group , we say that a simply-connected space realize this action if, for some , as a -module for the group of self-homotopy equivalences of , is isomorphic to as a -module. Which modules can be so realized? In this paper we obtain a positive answer for any faithful finitely generated -module, where 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.
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