Every group retraction can be realized as a topological retraction

2025/11/05 by Pedro J. Chocano, Chocano, Pedro J.
Computer Science · Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2511.03472

openalex publication_date 2025/11/05 · openalex created_date 2025/11/07 · openalex updated_date 2026/07/28

Abstract

Given a group retraction r: G → H , we construct a finite topological space X_r of height 1, together with a topological retraction r: X_r → X_r , such that the group of automorphisms Aut(X_r) (or the group of self-homotopy equivalences E(X_r) ) of X_r is isomorphic to G , and Aut(r(X_r)) (or E(r(X_r)) ) is isomorphic to H. Moreover, there is a natural map r' : Aut(X_r) → Aut(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.

Citations

• Realization of permutation modules via Alexandroff spaces
• Smallest posets with given cyclic automorphism group
• On some topological realizations of groups and homomorphisms
• Topological realizations of groups in Alexandroff spaces

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