Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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A PROSET STRUCTURE INDUCED FROM HOMOTOPY CLASSES OF MAPS AND A CLASSIFICATION OF FIBRATIONS

  • Yamaguchi, Toshihiro (Faculty of Education Kochi University) ;
  • Yokura, Shoji (Department of Mathematics and Computer Science Graduate School of Science and Engineering Kagoshima University)
  • Received : 2018.07.30
  • Accepted : 2018.10.12
  • Published : 2019.07.31
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Abstract

Firstly we consider preorders (not necessarily partial orders) on a canonical quotient of the set of the homotopy classes of continuous maps between two spaces induced by a certain equivalence relation ${\sim}_{{\varepsilon}R}$. Secondly we apply it to a classification of orientable fibrations over Y with fibre X. In the classification theorem of J. Stasheff [22] and G. Allaud [3], they use the set $[Y,\;Baut_1X]$ of homotopy classes of continuous maps from Y to $Baut_1X$, which is the classifying space for fibrations with fibre X due to A. Dold and R. Lashof [11]. In this paper we give a classification of fibrations using a preordered set (abbr., proset) structure induced by $[Y,\;Baut_1X]_{{\varepsilon}R}:=[Y,\;Baut_1X]/{\sim}_{{\varepsilon}R}$.

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References

  1. P. Alexandroff, Diskrete Raume, Mat. Sb. (N.S.) (1937), 501-518.
  2. P. Alexandroff, Combinatorial topology. Vol. 1, 2 and 3, translated from the Russian, reprint of the 1956, 1957 and 1960 translations, Dover Publications, Inc., Mineola, NY, 1998.
  3. G. Allaud, On the classification of fiber spaces, Math. Z. 92 (1966), 110-125. https://doi.org/10.1007/BF01312430
  4. F. G. Arenas, Alexandroff spaces, Acta Math. Univ. Comenian. (N.S.) 68 (1999), no. 1, 17-25.
  5. J. A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Lecture Notes in Mathematics, 2032, Springer, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-22003-6
  6. K. Borsuk, Sur la notion de diviseur et de multiple des transformations, Bull. Acad. Polon. Sci. Cl. III. 3 (1955), 81-85.
  7. J. W. Bruce, T. Gaffney, and A. A. Du Plessis, On left equivalence of map germs, Bull. London Math. Soc. 16 (1984), no. 3, 303-306. https://doi.org/10.1112/blms/16.3.303
  8. J. M. Curry, Sheaves, cosheaves and applications, ProQuest LLC, Ann Arbor, MI, 2014.
  9. A. du Plessis and L. Wilson, On right-equivalence, Math. Z. 190 (1985), no. 2, 163-205. https://doi.org/10.1007/BF01160458
  10. P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan, Real homotopy theory of Kahler manifolds, Invent. Math. 29 (1975), no. 3, 245-274. https://doi.org/10.1007/BF01389853
  11. A. Dold and R. Lashof, Principal quasi-fibrations and bre homotopy equivalence of bundles, Illinois J. Math. 3 (1959), 285-305. http://projecteuclid.org/euclid.ijm/1255455128 https://doi.org/10.1215/ijm/1255455128
  12. Y. Felix, S. Halperin, and J.-C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4613-0105-9
  13. M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Springer-Verlag, New York, 1973.
  14. P. A. Griffiths and J. W. Morgan, Rational Homotopy Theory and Differential Forms, Progress in Mathematics, 16, Birkhauser, Boston, MA, 1981.
  15. P. Hilton, G. Mislin, and J. Roitberg, Localization of Nilpotent Groups and Spaces, North-Holland Publishing Co., Amsterdam, 1975.
  16. J. Lurie, Higher Algebra, http://www.math.harvard.edu/lurie/papers/HA.pdf, September18, 2017.
  17. J. P. May, Finite topological spaces, http://math.uchicago.edu/-may/FINITE/REUNotes2010/FiniteSpaces.pdf
  18. M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. 33 (1966), 465-474. http://projecteuclid.org/euclid.dmj/ 1077376525 https://doi.org/10.1215/S0012-7094-66-03352-7
  19. H. Nishinobu and T. Yamaguchi, Sullivan minimal models of classifying spaces for non-formal spaces of small rank, Topology Appl. 196 (2015), part A, 290-307. https://doi.org/10.1016/j.topol.2015.10.003
  20. M. Schlessinger and J. Stasheff, Deformation theory and rational homotopy type, 2012, arXiv:1211.1647v1.
  21. T. Speer, A short study of Alexandroff spaces, arXiv:0708.2136.
  22. J. Stasheff, A classification theorem for bre spaces, Topology 2 (1963), 239-246. https://doi.org/10.1016/0040-9383(63)90006-5
  23. A. K. Steiner, The lattice of topologies: Structure and complementation, Trans. Amer. Math. Soc. 122 (1966), 379-398. https://doi.org/10.2307/1994555
  24. D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Etudes Sci. Publ. Math. No. 47 (1977), 269-331 (1978). https://doi.org/10.1007/BF02684341
  25. D. Tamaki, Cellular stratified spaces, in Combinatorial and toric homotopy, 305-435, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 35, World Sci. Publ., Hackensack, NJ, 2018.
  26. D. Tanre, Homotopie rationnelle: modeles de Chen, Quillen, Sullivan, Lecture Notes in Mathematics, 1025, Springer-Verlag, Berlin, 1983. https://doi.org/10.1007/BFb0071482
  27. J. Woolf, The fundamental category of a stratied space, J. Homotopy Relat. Struct. 4 (2009), no. 1, 359-387.