Finite groups as homotopy self-equivalences of finite spaces

TL;DR

The paper addresses the realization problem of finite groups as the group of homotopy classes of self-homotopy equivalences of finite spaces. It connects this problem to automorphisms of Hasse diagrams and Frucht-type graphs, and introduces an asymmetric-family construction $F_k$ along with a block-replacement scheme to encode group structure via colored Cayley graphs. The main result shows that for any finite group $G$ and generating set $S$, the finite space $X(G,S)$ satisfies that the group of homotopy classes of self-homotopy equivalences is isomorphic to $G$, i.e., Aut$(X(G,S)) cong G$. This provides a constructive solution to the realization problem for finite spaces and links graph automorphisms with homotopy self-equivalences through a explicit finite-space construction.

Abstract

We study the realization problem of finite groups as the group of homotopy classes of self-homotopy equivalences of finite spaces. Let $G$ be a finite group. Using an infinite family of pairwise non weakly homotopic asymmetric spaces we present a new construction of a finite space whose group of homotopy classes of self-homotopy equivalences is isomorphic to $G$.

Figures (5)

• Figure 1: The asymmetric minimal finite space $F_3$ with 14 points
• Figure 2: $C_{\mathbb{Z}/3\mathbb{Z},\{x\}}$
• Figure 3: $X(\mathbb{Z}/3\mathbb{Z},\{x\})$
• Figure 4: $C_{D_6,\{\tau,\sigma\}}$
• Figure 5: $X(D_6,\{\tau,\sigma\})$

Theorems & Definitions (22)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Definition 1.5
• Definition 1.6
• Theorem 1.7
• proof
• Corollary 1.8
• Remark 2.1