Topological realizations of groups in Alexandroff spaces

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Given a group G, we provide a constructive method to get infinitely many (non-homotopy-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.
This is a pre-print of an article published in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas . The final authenticated version is available online at:”.
1] P.S. Alexandroff. Diskrete Raume. Mathematiceskii Sbornik (N.S.) 2, 501-518, 1937. [2] M. Arkowitz, Problems on self-homotopy equivalences, in: Groups of homotopy self-equivalences and related topics, Contemp. Math. 274, 309-315, 2001. [3] M. Arkowitz, The Group of Self-Homotopy Equivalences - A Survey, Groups of Self-Equivalenes and Related Topics, Lecture Notes in Mathematics, vol. 1425, Springer Verlag, pp. 170-203, 1990. [4] J.A. Barmak. Algebraic topology Topology of Finite Topological Spaces and Applications. Lecture Notes in Mathematics 2032, Springer-Verlag, 2011. [5] J.A. Barmak. and E.G. Minian Automorphism groups of finite posets. Discrete Math., Vol 309 , Issue 10, 3424-3426, 2009. [6] G. Birkhoff On groups of automorphisms. Rev. Un. Mat. Argentina, 11, pp. 155-157, 1946. [7] C. Costoya, A. Viruel, Every finite group is the group of self homotopy equivalences of an elliptic space, Acta Mathematica 213, 49-62, 2014. [8] C. Costoya, A. Viruel, A primer on the group of self-homotopy equivalences: a rational homotopy theory approach, [9] A. Dold und R. Thom. Quasifaserungen und unendliche symmetrische Produkte, Annals of Mathematics(2),vol. 67, pp. 239-281, 1958. [10] Y. Félix. Problems on mapping spaces and related subject, in Homotopy Theory of Function Spaces and Related Topics. Contemp. Math.,519, pp. 217-30. Amer. Math. Soc., Providence, RI, 2010. [11] K. Hrbacek and T. Jech. Introduction to set theory. Third edition, revised and expanded. Marcel Dekker, Inc., 1999. [12] D.W. Kahn. Realization problems for the group of homotopy classes of self-equivalences. Math. Ann., 220, 37-46, 1976. [13] M.J. Kukiela. On homotopy types of Alexandroff spaces. Order 27, no. 1, 9-21, 2010. [14] J.P. May, Finite spaces and larger contexts. [15] M.C. McCord. Singular homology and homotopy groups of finite spaces. Duke Math. J. 33, 465-474, 1966. [16] J. Rutter, Spaces of Homotopy Self-Equivalences - A Survey, Springer-Verlag, Berlin, Lecture Notes in Mathematics, vol. 1662, 1997. [17] R.E. Stong. Finite topological spaces. Trans. Amer. Math. Soc., 123, pp. 325-340, 1966. [18] M.C. Thornton. Spaces with given homeomorphism groups. Proc. Amer. Math. Soc 33, 127-131, 1972. [19] J. H. C. Whitehead. Combinatorial homotopy I. Bull. Amer. Math. Soc., 55, 213-245, 1949.