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arXiv · 2308.16675

Realization of permutation modules via Alexandroff spaces

Cristina Costoya·Rafael Gomes·Antonio Viruel
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Abstract

We raise the question of the realizability of permutation modules in the context of Kahn's realizability problem for abstract groups and the $G$-Moore space problem. Specifically, given a finite group $G$, we consider a collection $\{M_i\}_{i=1}^n$ of finitely generated $\Z G$-modules that admit a submodule decomposition on which $G$ acts by permuting the summands. Then we prove the existence of connected finite spaces $X$ that realize each $M_i$ as its $i$-th homology, $G$ as its group of self-homotopy equivalences $\E(X)$, and the action of $G$ on each $M_i$ as the action of $\E(X)$ on $H_i(X; \Z)$.

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Cristina Costoya, Rafael Gomes, Antonio Viruel. 2023-08-31. Realization of permutation modules via Alexandroff spaces. https://arxiv.org/abs/2308.16675

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