Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio


IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2018: 0.27

Online
ISSN
1572-9176
See all formats and pricing
More options …
Volume 26, Issue 2

Issues

Finite spaces and an axiomatization of the Lefschetz number

Paweł Bilski
  • Corresponding author
  • Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-05-11 | DOI: https://doi.org/10.1515/gmj-2017-0012

Abstract

In [1] Arkowitz and Brown presented an axiomatization of the reduced Lefschetz number of self-maps of finite CW-complexes. By the results of McCord [8], finite simplicial complexes are closely related to finite T0-spaces. This connection and the axioms given by Arkowitz and Brown suggest an axiomatization of the reduced Lefschetz number of maps of finite T0-spaces. However, using the notion of the subdivision of a finite T0-space, we consider the degree and the Lefschetz number of not only self-maps. We also present some properties of the degree of maps between finite models of the circle 𝕊1.

Keywords: Euler characteristic; finite topological spaces; Lefschetz number

MSC 2010: 54H25; 55M20

References

  • [1]

    M. Arkowitz and R. F. Brown, The Lefschetz–Hopf theorem and axioms for the Lefschetz number, Fixed Point Theory Appl. 2004 (2004), no. 1, 1–11. Google Scholar

  • [2]

    J. A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Lecture Notes in Math. 2032, Springer, Heidelberg, 2011. Google Scholar

  • [3]

    D. L. Gonçalves and J. Weber, Axioms for the equivariant Lefschetz number and for the Reidemeister trace, J. Fixed Point Theory Appl. 2 (2007), no. 1, 55–72. CrossrefWeb of ScienceGoogle Scholar

  • [4]

    A. Granas and J. Dugundji, Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003. Google Scholar

  • [5]

    A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. Google Scholar

  • [6]

    T. Leinster, The Euler characteristic of a category, Doc. Math. 13 (2008), 21–49. Google Scholar

  • [7]

    J. P. May, Finite spaces and larger contexts, preprint (2016), http://math.uchicago.edu/~may/FINITE/FINITEBOOK/FINITEBOOKCollatedDraft.pdf.

  • [8]

    M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. 33 (1966), 465–474. Google Scholar

  • [9]

    E. H. Spanier, Algebraic Topology, Springer, New York, 1966. Google Scholar

  • [10]

    R. E. Stong, Finite topological spaces, Trans. Amer. Math. Soc. 123 (1966), 325–340. CrossrefGoogle Scholar

  • [11]

    C. E. Watts, On the Euler characteristic of polyhedra, Proc. Amer. Math. Soc. 13 (1962), 304–306. CrossrefGoogle Scholar

  • [12]

    P. Wruck, Axiomatic description of Lefschetz type equivariant homotopy invariants, preprint (2013), http://arxiv.org/abs/1301.7308v1.

About the article

Received: 2015-02-12

Revised: 2015-08-02

Accepted: 2015-10-23

Published Online: 2017-05-11

Published in Print: 2019-06-01


Citation Information: Georgian Mathematical Journal, Volume 26, Issue 2, Pages 165–175, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2017-0012.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in