Realization of Permutation Modules via Alexandroff Spaces

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 \(\{M_i\}_{i=1}^n\) ${\{M_i\}}_{i=1}^n$ of finitely generated \(\mathbb {Z}G\) $\mathbb{Z}G$ -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 \(M_i\) $M_i$ as its i-th homology, G as its group of self-homotopy equivalences \(\mathcal {E}(X)\) $\mathcal{E}(X)$ , and the action of G on each \(M_i\) $M_i$ as the action of \(\mathcal {E}(X)\) $\mathcal{E}(X)$ on \(H_i(X; \mathbb {Z})\) $H_i(X;\mathbb{Z})$ .