Every group retraction can be realized as a topological retraction
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909893863145472 |
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| author | Chocano, Pedro J. |
| author_facet | Chocano, Pedro J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Every group retraction can be realized as a topological retraction Chocano, Pedro J. Algebraic Topology General Topology 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. |
| title | Every group retraction can be realized as a topological retraction |
| topic | Algebraic Topology General Topology |
| url | https://arxiv.org/abs/2511.03472 |