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Topological realizations of groups in Alexandroff spaces

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.

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Acknowledgements

We would like to thank the referee for carefully reading our manuscript and for giving such valuable comments which substantially improved some previous versions of the paper.

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Correspondence to Pedro J. Chocano.

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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).

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Chocano, P.J., Morón, M.A. & Ruiz del Portal, F. Topological realizations of groups in Alexandroff spaces. RACSAM 115, 25 (2021). https://doi.org/10.1007/s13398-020-00964-7

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Keywords

  • Automorphisms
  • Homotopy equivalence
  • Alexandroff spaces
  • Posets

Mathematics Subject Classification

  • 55P10
  • 55P99
  • 06A06