Topological realizations of groups in Alexandroff spaces

Abstract

Given a group G, we provide a constructive method to get infinitely many (nonhomotopy-equivalent) Alexandroff spaces, such that the group of autohomeomorphisms, the group of homotopy classes of self-homotopy equivalences and the pointed version are isomorphic to G. As a result, any group G can be realized as the group of homotopy classes of self-homotopy equivalences of a topological space X, for which there exists a CW complex K(X) and a weak homotopy equivalence from K(X) to X.

On some topological realizations of groups and homomorphisms

“…X G H has the same homotopy type of X * H . Therefore, repeating same arguments used in [7], we can obtain the desired result.…”

“…f is a homeomorphism, so f |W h is also a homeomorphism. Then, we have that h = t, otherwise, we would get a contradiction since W h is homeomorphic to W t if and only if h = t. Using Proposition 2.9, it is easy to check that f fixes S (h,0) for every h ∈ H. On the other hand, [7,Remark 4.2] says that if a homeomorphism g : X * H \ {W h |h ∈ H} → X * H \ {W h |h ∈ H} coincides in one point with the identity map, then g is the identity map. Then, it can be obtained that f is the identity map, so Aut(X * H ) is the trivial group.…”

“…). Now, we want to construct a topological space X * H satisfying that Aut(X * H ) is trivial and E(X * H ) is isomorphic to H. Again, by [7], there exists a finite T 0 topological space X H with E(X H ) Aut(X H ) H. In figure 1, the Hasse diagram of X H corresponds to the red and black parts of the diagram on the right. We need to modify X H in order to reduce the number of self-homeomorphisms without changing the number of self-homotopy equivalences.…”

On the triviality of flows in Alexandroff spaces

“…In particular, the fundamental group of such spaces can be chosen to be isomorphic to any prefixed group. On the other hand every group can be realized as the group of autohomeomorphisms of an Alexandroff space, see [4].…”

Coincidence theorems for finite topological spaces

“…Finite topological spaces are becoming a significant part of topology and a good tool so as to model and face problems of different nature. For instance, they can be used to reconstruct compact metric spaces [20] or to solve realization problems of groups in topological categories [8]. From an algebraic point of view, they are interesting since they have the same homotopy and singular homology groups of simplicial complexes [18].…”