Every group retraction can be realized as a topological retraction
Pedro J. Chocano
TL;DR
The paper addresses realizing a group retraction $r: G \rightarrow H$ as a topological retraction within finite topological spaces. It introduces an explicit construction of a connected finite space $X_r$ of height $1$ and a topological retraction $\overline{r}: X_r \rightarrow X_r$ that realizes $r$, with $\mathrm{Aut}(X_r) \cong G$, $\mathrm{Aut}(\overline{r}(X_r)) \cong H$, and a natural map $\overline{r}': \mathrm{Aut}(X_r) \rightarrow \mathrm{Aut}(\overline{r}(X_r))$ compatible with $r$. The main contributions include the concrete assembly of $X_r$ from height-$1$ components $C_g$ (for $g \in G$), an explicit action $T: G \to \mathrm{Aut}(X_r)$ giving $\mathrm{Aut}(X_r) \cong G$, and a proof that $\overline{r}$ is continuous with image $\mathrm{Aut}(\overline{r}(X_r)) \cong H$; moreover, $X_r$ has no beat points, so $\mathrm{Aut}(X_r)=\mathcal{E}(X_r)$, yielding a homotopical realization. Consequently, every finite group $G$ admits a height-$1$ finite space $X_G$ with $\mathrm{Aut}(X_G) \cong \mathcal{E}(X_G) \cong G$, while symmetric groups are realizable at height $0$; an open problem on minimal cardinalities for height-$1$ realizations of non-symmetric $G$ is highlighted. This work advances the understanding of the minimal height necessary to realize finite groups as automorphism or self-homotopy groups of finite topological spaces and provides explicit models linking group-theoretic and topological data.
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.