Relation between finite topological spaces and finitely presentable groups

TL;DR

This paper presents a method to construct finite topological spaces with a fundamental group isomorphic to any given finitely presentable group, including all finite groups, linking algebraic and topological structures.

Contribution

It introduces a construction technique for finite topological spaces corresponding to finitely presentable groups, expanding the understanding of their relationship.

Findings

Constructed finite spaces for any finitely presentable group

Applicable to a broad class of groups, including all finite groups

The construction may not produce minimal-sized spaces

Abstract

In this paper it is shown how to construct a finite topological space $X$ for a given finitely presentable group $G$ such that $\pi_1(X)\cong G$. Our construction is not optimal in the sense that the cardinality of the space $X$ might not be the smallest possible. Our main result applies to a large class of interesting groups, including all finite groups.