Realization of permutation modules via Alexandroff spaces

TL;DR

This paper demonstrates how certain permutation modules over finite groups can be realized as homology groups of finite topological spaces with a group action, linking algebraic modules to topological realizations.

Contribution

It establishes the existence of finite connected spaces that realize given permutation modules and group actions as homology and self-homotopy equivalences.

Findings

Finite spaces can realize permutation modules as homology groups.

Group actions on modules correspond to self-homotopy equivalences of spaces.

Constructs connect algebraic module properties with topological realizations.

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 $H_i(X; \Z)$.