Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T13:50:30.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Localization of seif-homotopy equivalences inducing the identity on homology

Published online by Cambridge University Press:  24 October 2008

Ken-Ichi Maruyama
Ken-Ichi Maruyama
Affiliation:
Department of Mathematics, Faculty of Education, Chiba University, Yayoicho, Chiba, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

Let us denote the group of based homotopy classes of seif-homotopy equivalences of a space X by E(X). We consider E0(X), the subgroup of E(X) consisting of elements which induce the identity map on homology. Dror and Zabrodsky have shown that E0(X) and the subgroup E#(X) consisting of elements inducing the identity on homotopy are both nilpotent groups for finite-dimensional nilpotent spaces, or finite-dimensional spaces respectively ([4], theorem D, theorem A). The theory of localization for nilpotent groups has been developed by several authors (see [8]). The aim of this paper is to prove the following theorem. The corresponding result for E#(X) is obtained in [9], theorem 0·1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 2 , September 1990 , pp. 291 - 297
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arkowitz, M. and Curjel, C. R.. Groups of Hornotopy Classes. Lecture Notes in Math. vol. 4 (Springer-Verlag, 1967).CrossRefGoogle Scholar
[2]Barcus, W. D. and Barratt, . On the homotopy classification of the extensions of a fixed map. Trans. Amer. Math. Soc. 88 (1958), 57–74.CrossRefGoogle Scholar
[3]Bousfield, A. K.. Localization of spaces with respect to homology. Topology 14 (1975), 133–150.CrossRefGoogle Scholar
[4]Dror, E. and Zabrodsky, A.. Unipotency and nilpotency in homotopy equivalences. Topology 18 (1979), 187–197.CrossRefGoogle Scholar
[5]Ganea, T.. A generalization of the homology and homotopy suspension. Comment. Math. Helv. 39 (1965), 295–321.CrossRefGoogle Scholar
[6]Hilton, P.. On orbit sets for group actions and localization. In Algebraic Topology, Lecture Notes in Math. vol. 673 (Springer-Verlag, 1978), pp. 185–201.CrossRefGoogle Scholar
[7]Hilton, P.. Nilpotent actions on nilpotent groups. In Algebra and Logic, Lecture Notes in Math. vol. 450 (Springer-Verlag, 1975), pp. 174–196.CrossRefGoogle Scholar
[8]Hilton, P., Mislin, G. and Roitberg, J.. Localization of Nilpotent Groups and Spaces. Mathematics Studies no. 15 (North Holland, 1975).Google Scholar
[9]Maruyama, K.. Localization of a certain subgroup of seif-homotopy equivalences. Pacific J. Math. 136 (1989), 293–301.CrossRefGoogle Scholar
[10]Nishida, C.. Homotopy Theory (in Japanese) (Kyôritu, 1985).Google Scholar
[11]Nomura, Y.. Homotopy equivalences in a principal fibre space. Math. Z. 92 (1966), 380–388.CrossRefGoogle Scholar
[12]Oka, S.. On the group of self homotopy equivalences of H-spaces of low rank II. Mem. Fac. Sci. Kyushu Univ. Ser. A 35 (1981), 307–323.Google Scholar
[13]Oka, S., Sawashita, N. and Sugawara, M.. On the group of self-equivalences of a mapping cone. Hiroshima Math. J. 4 (1974), 9–28.CrossRefGoogle Scholar
[14]Rutter, J. W.. The group of homotopy self-equivalence classes of CW complexes. Math. Proc. Cambridge Philos. Soc. 93 (1983), 275–293.CrossRefGoogle Scholar
[15]Rutter, J. W.. Self-equivalences and principal morphisms. Proc. London Math. Soc. 20 (1970), 644–658.CrossRefGoogle Scholar
[16]Sieradski, A. J.. Twisted self-homotopy equivalences. Pacific J. Math. 34 (1970), 789–802.CrossRefGoogle Scholar