On some topological realizations of groups and homomorphisms

TL;DR

This paper constructs a topological space that encodes a given group homomorphism through its homeomorphism and homotopy groups, linking algebraic and topological structures in a novel way.

Contribution

It introduces a method to realize group homomorphisms as maps between topological spaces with prescribed automorphism and homotopy groups.

Findings

Constructed space $X_f$ with specified automorphism and homotopy groups

Realization of homomorphisms via topological spaces

Addresses realization problems in homology and homotopy groups

Abstract

Let $f:G\rightarrow H$ be a homomorphism of groups, we construct a topological space $X_f$ such that its group of homeomorphisms is isomorphic to $G$, its group of homotopy classes of self-homotopy equivalences is isomorphic to $H$ and the natural map between the group of homeomorphisms of $X_f$ and the group of homotopy classes of self-homotopy equivalences of $X_f$ is precisely $f$. In addition, realization problems involving homology, homotopy groups and groups of automorphisms are considered.