On the realizability of group actions

Cristina Costoya; Antonio Viruel

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arXiv · 1511.06322

On the realizability of group actions

Cristina Costoya·Antonio Viruel

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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.

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Cristina Costoya, Antonio Viruel. 2015-11-19. On the realizability of group actions. https://arxiv.org/abs/1511.06322

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