Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-26T06:26:44.330Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivariant shape theory

Published online by Cambridge University Press:  24 October 2008

Zvonko Čerin
Zvonko Čerin
Affiliation:
Department of Mathematics, University of Zagreb, Zagreb, Croatia. E-mail: zcerin@x400.srce.hr
Get access Rights & Permissions [Opens in a new window]

Abstract

Equivariant shape theory is an improvement of equivariant homotopy theory which could be regarded as an equivariant version of Borsuk's shape theory. The main result in this paper is a description of the equivariant shape category ShG whose objects are equivariant spaces or G-spaces, i.e. topological spaces endowed with an action of a given topological group G, and whose morphisms are equivariant homotopy or G-homotopy classes of families of multi-valued functions which we call equivariant multi-nets or G-multi-nets. Previously equivariant shape theories have been described only under the assumptions that the group G is either finite or compact. We also study classes of G-spaces on which equivariant shape and equivariant homotopy coincide, look for conditions under which a G-map f: X → Y is an equivariant shape equivalence, and give some characterizations of G-spaces with trivial equivariant shape.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 2 , March 1995 , pp. 303 - 320
Copyright
Copyright © Cambridge Philosophical Society 1995

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]Aló, R. A. and Shapiro, M. L.. Normal Topological Spaces (Cambridge University Press, 1972).Google Scholar
[2]Antonian, S. A. and Mardešić, S.. Equivariant shape. Fund. Math. 127 (1987), 213–224.CrossRefGoogle Scholar
[3]Borsuk, K.. Concerning homotopy properties of compacta. Fund. Math. 62 (1968), 223–254.CrossRefGoogle Scholar
[4]Bredon, G. E.. Introduction to compact transformation groups (Academic Press, 1972).Google Scholar
[5]Čerin, Z.. Shape theory intrinsically. Publicacions Matemàtiques 37 (1993), 317–334.CrossRefGoogle Scholar
[6]De Vries, J.. Topological transformation groups 1 (Mathematisch Centrum, Amsterdam, 1975).Google Scholar
[7]Dold, A.. Lectures on Algebraic Topology (Springer-Verlag, 1972).CrossRefGoogle Scholar
[8]Fox, R. H.. On shape. Fund. Math. 74 (1972), 47–71.CrossRefGoogle Scholar
[9]Kozlowski, G.. Images of ANR's. Trans. Amer. Math. Soc., to appear.Google Scholar
[10]Mardešić, S. and Segal, J.. Shape Theory (North Holland, 1982).Google Scholar
[11]Palais, R. S.. The classification of G-spaces. Mem. Amer. Math. Soc. 36 (1960).Google Scholar
[12]Pop, I.. An equivariant shape theory. Anal. Sti. Univ. laŞi, Sec. la 30 (1984), 53–67.Google Scholar
[13]Sanjurjo, J.. An intrinsic description of shape. Trans. AMS 329 (1992), 625–636.CrossRefGoogle Scholar
[14]Smirnov, Yu. M.. Shape theory and continuous transformation groups, in Russian. Uspehi Mat. Nauk 34 (1979), 119–123.Google Scholar
[15]Smirnov, Yu. M.. Equivariant shape. Serdica 10 (1984), 223–228.Google Scholar
[16]Smirnov, Yu. M.. Shape theory for G-spaces, in Russian. Uspehi Mat. Nauk 40 (1985), 151–165.Google Scholar