Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SELF-HOMOTOPY EQUIVALENCES RELATED TO COHOMOTOPY GROUPS

  • Received : 2015.12.22
  • Published : 2017.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

Given a topological space X and a non-negative integer k, we study the self-homotopy equivalences of X that do not change maps from X to n-sphere $S^n$ homotopically by the composition for all $n{\geq}k$. We denote by ${\varepsilon}^{\sharp}_k(X)$ the set of all homotopy classes of such self-homotopy equivalences. This set is a dual concept of ${\varepsilon}^{\sharp}_k(X)$, which has been studied by several authors. We prove that if X is a finite CW complex, there are at most a finite number of distinguishing homotopy classes ${\varepsilon}^{\sharp}_k(X)$, whereas ${\varepsilon}^{\sharp}_k(X)$ may not be finite. Moreover, we obtain concrete computations of ${\varepsilon}^{\sharp}_k(X)$ to show that the cardinal of ${\varepsilon}^{\sharp}_k(X)$ is finite when X is either a Moore space or co-Moore space by using the self-closeness numbers.

Keywords

Acknowledgement

Supported by : Korea University

References

  1. S. Araki and H. Toda, Multiplicative structures in mod q cohomology theories. I, Osaka J. Math. 2 (1965), 71-115.
  2. M. Arkowitz, The group of self-homotopy equivalences - a survey, Groups of self-equivalences and related topics (Montreal, PQ, 1988), 170-203, Lecture Notes in Math., 1425, Springer, Berlin, 1990.
  3. M. Arkowitz and K. Maruyama, Self-homotopy equivalences which induce the identity on homology, cohomology or homotopy groups, Topology Appl. 87 (1998), no. 2, 133-154. https://doi.org/10.1016/S0166-8641(97)00162-4
  4. H. Choi and K. Lee, Certain self homotopy equivalences on wedge products on Moore spaces, Pacific J. Math. 272 (2014), no. 1, 35-57. https://doi.org/10.2140/pjm.2014.272.35
  5. H. Choi and K. Lee, Certain numbers on the groups of self-homotopy equivalences, Topology Appl. 181 (2015), 104-111. https://doi.org/10.1016/j.topol.2014.12.004
  6. B. Gray, Homotopy Theory, Academic Press, Inc., 1975.
  7. K. Lee, The groups of self pair homotopy equivalences, J. Korean Math. Soc. 43 (2006), no. 3, 491-506. https://doi.org/10.4134/JKMS.2006.43.3.491
  8. J. Rutter, The group of homotopy self-equivalence classes of CW-complexes, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 2, 275-293. https://doi.org/10.1017/S0305004100060576
  9. N. Sawashita, On the group of self-equivalences of the product spheres, Hiroshima Math. J. 5 (1975), 69-86.
  10. A. Sieradski, Twisted self-homotopy equivalences, Pacific J. Math. 34 (1970), 789-802. https://doi.org/10.2140/pjm.1970.34.789
  11. A. Sieradski, Stabilization of self-equivalences of the pesudoprojective spaces, Michigan. Math. J. 19 (1972), 109-119. https://doi.org/10.1307/mmj/1029000840
  12. H. Toda, Composition methods in homotopy groups of spheres, Princeton Univ., 1962.