Every group retraction can be realized as a topological retraction

Abstract

Given a group retraction $r: G \rightarrow H $, we construct a finite topological space $ X_r $ of height 1, together with a topological retraction $\overline{r}: X_r \rightarrow X_r $, such that the group of automorphisms $ \mathrm{Aut}(X_r) $ (or the group of self-homotopy equivalences $ \mathcal{E}(X_r) $) of $X_r$ is isomorphic to $ G $, and $ \mathrm{Aut}(\overline{r}(X_r)) $ (or $\mathcal{E}(\overline{r}(X_r)) $) is isomorphic to $ H$. Moreover, there is a natural map $\overline{r}' : \mathrm{Aut}(X_r) \rightarrow \mathrm{Aut}(\overline{r}(X_r)) $ that coincides with the original group retraction $ r $. As a direct consequence of this construction, we show that height 1 is the minimal height required to realize any finite group as the group of automorphisms (or the group of self-homotopy equivalences) of a finite topological space, except in the case where $ G $ is a symmetric group. In that unique case, the group can be realized by a finite topological space of height 0.