On some topological realizations of groups and homomorphisms

Abstract

Given a homomorphism of groups f : G → H f:G\rightarrow H , we construct a topological space X f X_f such that its group of homeomorphisms A u t ( X f ) Aut(X_f) is isomorphic to G G , its group of homotopy classes of self-homotopy equivalences E ( X f ) \mathcal {E}(X_f) is isomorphic to H H and the natural map between A u t ( X f ) Aut(X_f) and E ( X f ) \mathcal {E}(X_f) is f f . In addition, we consider realization problems involving homology groups, homotopy groups and groups of automorphisms.