Self-closeness number and weak homotopy decomposition

Choi, Ho Won; Lee, Kee Young

doi:10.1090/proc/15897

Self-closeness number and weak homotopy decomposition
HTML articles powered by AMS MathViewer

by Ho Won Choi and Kee Young Lee PDF

Proc. Amer. Math. Soc. 150 (2022), 3189-3198 Request permission

Abstract:

For any CW-complex $X$, there exists a weak homotopy decomposition $X^{(m)}$ and a self-closeness number of $X$. In this paper, we study the self-closeness number of a weak homotopy decomposition of $X$. We prove that the self-closeness number of $X^{(m)}$ is dominated by the self-closeness number of $X$. Moreover, we determine the set of self-homotopy classes and the group of self-homotopy equivalence classes of a weak homotopy decomposition of $X$, and herein we provide some examples of the self-closeness number for homotopy m-sections.

References

Shôrô Araki and Hirosi Toda, Multiplicative structures in $\textrm {mod}\,q$ cohomology theories. I, Osaka Math. J. 2 (1965), 71–115. MR 182967
Martin Arkowitz, The group of self-homotopy equivalences—a survey, Groups of self-equivalences and related topics (Montreal, PQ, 1988) Lecture Notes in Math., vol. 1425, Springer, Berlin, 1990, pp. 170–203. MR 1070585, DOI 10.1007/BFb0083840
Martin Arkowitz, Introduction to homotopy theory, Universitext, Springer, New York, 2011. MR 2814476, DOI 10.1007/978-1-4419-7329-0
M. Arkowitz and C. R. Curjel, Groups of homotopy classes. Rank formulas and homotopy-commutativity, Second revised edition, Lecture Notes in Mathematics, Vol. 4, Springer-Verlag, Berlin-New York, 1967. MR 0214061, DOI 10.1007/BFb0099647
W. D. Barcus and M. G. Barratt, On the homotopy classification of the extensions of a fixed map, Trans. Amer. Math. Soc. 88 (1958), 57–74. MR 97060, DOI 10.1090/S0002-9947-1958-0097060-7
Ho Won Choi and Kee Young Lee, Certain numbers on the groups of self-homotopy equivalences, Topology Appl. 181 (2015), 104–111. MR 3303934, DOI 10.1016/j.topol.2014.12.004
Ho Won Choi and Kee Young Lee, Length of a chain composed by certain monoids of self maps, Topology Appl. 301 (2021), Paper No. 107498, 10. MR 4312949, DOI 10.1016/j.topol.2020.107498
Donald W. Kahn, Induced maps for Postnikov systems, Trans. Amer. Math. Soc. 107 (1963), 432–450. MR 150777, DOI 10.1090/S0002-9947-1963-0150777-X
Donald W. Kahn, The group of stable self-equivalences, Topology 11 (1972), 133–140. MR 286103, DOI 10.1016/0040-9383(72)90027-4
Kee Young Lee, Postnikov sections and groups of self pair homotopy equivalences, Bull. Korean Math. Soc. 41 (2004), no. 3, 393–401. MR 2081531, DOI 10.4134/BKMS.2004.41.3.393
Nobuyuki Oda and Toshihiro Yamaguchi, Self-maps of spaces in fibrations, Homology Homotopy Appl. 20 (2018), no. 2, 289–313. MR 3825015, DOI 10.4310/hha.2018.v20.n2.a15
John W. Rutter, The group of homotopy self-equivalence classes of CW complexes, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 2, 275–293. MR 691998, DOI 10.1017/S0305004100060576