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Topological realizations of groups in Alexandroff spaces. (English) Zbl 1461.55006

In this article, the authors construct, for every group \(G\), Alexandroff spaces \(X_G\), \(\overline{X}_G\) and \(\overline{X}^*_G\) such that the group of automorphisms of \(X_G\), the group of self homotopy equivalences of \(\overline{X}_G\) and the group of pointed self homotopy equivalences of \({\overline{X}}^*_G\) are all isomorphic to \(G\). This is related to work of J. A. Barmak and E. G. Minian [Discrete Math. 309, No. 10, 3424–3426 (2009; Zbl 1169.06001)] where they construct, for every finite group \(G\), a finite space \(X\) such that the group of automorphisms of \(X\) is isomorphic to \(G\). In Barmak and Minian’s work, the emphasis is put on improving previous results in terms of the cardinality of the space \(X\).
In the article under consideration, the construction of the Alexandroff space \(X_G\) is analogous to the construction given by Barmak and Minian. On the other hand, the main idea behind the construction of the spaces \(\overline{X}_G\) (respectively, \(\overline{X}^*_G\)) is to modify the space \(X_G\) in order to obtain a space with the same automorphism group as \(X_G\) which satisfies that the group of self homotopy equivalences (respectively, pointed self homotopy equivalences) of this space is isomorphic to its automorphism group. In the context of finite topological spaces, it is well-known that a minimal finite space satisfies this property [R. Stong, Trans. Am. Math. Soc. 123, 325–340 (1966; Zbl 0151.29502); J.A. Barmak, Algebraic topology of finite topological spaces and applications. Lecture Notes in Mathematics 2032. Berlin: Springer (2011; Zbl 1235.55001)], while the corresponding result for Alexandroff spaces that is used in this work to obtain the desired result is given by M. J. Kukieła [Order 27, No. 1, 9–21 (2010; Zbl 1187.55005)].

MSC:

55P10 Homotopy equivalences in algebraic topology
55P99 Homotopy theory
06A06 Partial orders, general
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References:

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