Publication:
Topological realizations of groups in Alexandroff spaces

dc.contributor.authorChocano Feito, Pedro José
dc.contributor.authorMorón, Manuel A.
dc.contributor.authorRomero Ruiz del Portal, Francisco
dc.date.accessioned2023-06-17T08:56:07Z
dc.date.available2023-06-17T08:56:07Z
dc.date.issued2020-11-23
dc.descriptionThis 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: https://doi.org/10.1007/s13398-020-00964-7”.
dc.description.abstractGiven 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.
dc.description.departmentDepto. de Álgebra, Geometría y Topología
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.facultyInstituto de Matemática Interdisciplinar (IMI)
dc.description.refereedFALSE
dc.description.sponsorshipMinisterio de Economía y Competitividad (MINECO)
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/63263
dc.identifier.citation1] 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, https://www.algtop.net/geto16/docs/material/ViruelNotes.pdf. [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. https://math.uchicago.edu/may/FINITE/FINITEBOOK/FINITEBOOKCollatedDraft.pdf [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.
dc.identifier.doi10.1007/s13398-020-00964-7
dc.identifier.issn1578-7303
dc.identifier.officialurlhttps://doi.org/10.1007/s13398-020-00964-7
dc.identifier.relatedurlhttps://link.springer.com/article/10.1007/s13398-020-00964-7
dc.identifier.urihttps://hdl.handle.net/20.500.14352/7571
dc.issue.number1
dc.journal.titleRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
dc.language.isoeng
dc.publisherSpringer
dc.relation.projectID(MTM2015-63612-P; PGC2018-098321-B-100; BES-2016-076669)
dc.rights.accessRightsopen access
dc.subject.cdu512.56
dc.subject.cdu512.541.5
dc.subject.cdu515.143
dc.subject.keywordAutomorphisms
dc.subject.keywordHomotopy equivalence
dc.subject.keywordAlexandroff spaces
dc.subject.keywordPosets
dc.subject.ucmÁlgebra
dc.subject.ucmGrupos (Matemáticas)
dc.subject.ucmTopología
dc.subject.unesco1201 Álgebra
dc.subject.unesco1210 Topología
dc.titleTopological realizations of groups in Alexandroff spaces
dc.typejournal article
dc.volume.number115
dspace.entity.typePublication
relation.isAuthorOfPublicatione4503677-6189-4b18-be22-23d31611f2a4
relation.isAuthorOfPublication.latestForDiscoverye4503677-6189-4b18-be22-23d31611f2a4
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