Topological realizations of groups in Alexandroff spaces
- PDF / 651,247 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 32 Downloads / 234 Views
Topological realizations of groups in Alexandroff spaces Pedro J. Chocano1
· Manuel A. Morón2
· Francisco Ruiz del Portal2
Received: 22 July 2019 / Accepted: 7 November 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract We prove that every group can be realized as the homeomorphism group and as the group of (pointed) homotopy classes of (pointed) self-homotopy equivalences of infinitely many non-homotopy-equivalent Alexandroff spaces. Keywords Automorphisms · Homotopy equivalence · Alexandroff spaces · Posets Mathematics Subject Classification 55P10 · 55P99 · 06A06
1 Introduction Growing out of foundational papers on the subject, such as [17,22], the theory of finite topological spaces has seen a substantial development in the last two decades. This is mainly due to the interest of researchers in applying computational methods to, for instance, dynamical systems [7,19] or approximation of spaces [2,18]. Two main monographs on the issue of the algebraic aspects of the topology of finite spaces are [16] and [5]. The first one, due to J.P. May, is the result of some REU programs developed by the author at the University of Chicago. The other one is, essentially, the Ph. D. thesis of J.A. Barmak (under the supervision of E.G. Minian). Finite topological spaces are a particular case of more general topological spaces that were introduced by Alexandroff [1]. An Alexandroff space is a topological space with the property
This research is partially supported by Grants MTM2015-63612-P, PGC2018-098321-B-100 and BES-2016-076669 from Ministerio de Ciencia, Innovación y Universidades (Spain).
B
Pedro J. Chocano [email protected] https://sites.google.com/view/pedrojchocanofeito/home?authuser=0 Manuel A. Morón [email protected] Francisco Ruiz del Portal [email protected]
1
Departamento de Álgebra, Geometría y Topología, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain
2
Departamento de Álgebra, Geometría y Topología, Instituto de Matematica Interdisciplinar, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain 0123456789().: V,-vol
123
25
Page 2 of 20
P. J. Chocano et al.
that the arbitrary intersection of open sets is open. Some of the results given by McCord [17] can be rephrased in the following way: Given a polyhedron X , there exists an Alexandroff space X (X ) such that the homotopy groups of X and X (X ) are the same. Then, it can be deduced that every group can be realized as the fundamental group of an Alexandroff space. In addition, a similar statement can be obtained for abelian groups and higher homotopy groups. Alexandroff spaces can also be used to realize finite groups as homeomorphism groups, e.g., [6,8,23]. The problem of realizing a group as the group of homeomorphisms of a topological space, i.e., the realizability problem for the topological category (T op), has been widely studied. We have only cited some references related to finite topological spaces. In particular, in [6], Barmak and Minian focused on giving, for any finite group
Data Loading...
