On some topological realizations of groups and homomorphisms

Abstract

Let f : G → 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.

Shape of compacta as extension of weak homotopy of finite spaces

“…Recent research recognizes the critical role played by the theory of finite topological spaces in several fields of mathematics such as dynamical systems [6,16,9], group theory [5,4,11] algebraic topology (see [3,18] and the references given there) and geometric topology [21,10]. It is worth pointing out that important conjectures can be stated in terms of the theory of finite topological spaces.…”