Email this article On equivariant fibrations of $G$-CW-complexes
- Authors: Gevorgyan P.S.1, Jimenez R.B.2
-
Affiliations:
- Moscow State Pedagogical University
- National Autonomous University of Mexico, Institute of Mathematics
- Issue: Vol 210, No 10 (2019)
- Pages: 91-98
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/133287
- DOI: https://doi.org/10.4213/sm9133
- ID: 133287
Cite item
Open Access
Access granted
Subscription Access
Abstract
It is proved that if $G$ is a compact Lie group, then an equivariant Serre fibration of $G$-CW-complexes is an equivariant Hurewicz fibration in the class of compactly generated $G$-spaces. In the nonequivariant setting, this result is due to Steinberger, West and Cauty. The main theorem is proved using the following key result: a $G$-CW-complex can be embedded as an equivariant retract in a simplicial $G$-complex. It is also proved that an equivariant map $p\colon E\to B$ of $G$-CW-complexes is a Hurewicz $G$-fibration if and only if the $H$-fixed point map $p^H\colon E^H \to B^H$ is a Hurewicz fibration for any closed subgroup $H$ of $G$. This gives a solution to the problem of James and Segal in the case of $G$-CW-complexes. Bibliography: 9 titles.
About the authors
Pavel Samvelovich Gevorgyan
Moscow State Pedagogical University
Email: pgev@yandex.ru
Doctor of physico-mathematical sciences, Professor
Rolando Benitez Jimenez
National Autonomous University of Mexico, Institute of MathematicsCandidate of physico-mathematical sciences
References
- M. Steinberger, J. West, “Covering homotopy properties of maps between C.W. complexes or ANR's”, Proc. Amer. Math. Soc., 92:4 (1984), 573–577
- R. Cauty, “Sur les ouverts des CW-complexes et les fibres de Serre”, Colloq. Math., 63:1 (1992), 1–7
- J. P. May, J. Sigurdsson, Parametrized homotopy theory, Math. Surveys Monogr., 132, Amer. Math. Soc., Providence, RI, 2006, x+441 pp.
- I. M. James, G. B. Segal, “On equivariant homotopy type”, Topology, 17:3 (1978), 267–272
- G. E. Bredon, Equivariant cohomology theories, Lecture Notes in Math., 34, Springer-Verlag, Berlin–New York, 1967, vi+64 pp. (not consecutively paged)
- S. Illman, “The equivariant triangulation theorem for actions of compact Lie groups”, Math. Ann., 262:4 (1983), 487–501
- R. Cauty, “Sur les sous-espaces des complexes simpliciaux”, Bull. Soc. Math. France, 100 (1972), 129–155
- T. tom Dieck, Transformation groups, De Gruyter Stud. Math., 8, Walter de Gruyter & Co., Berlin, 1987, x+312 pp.
- W. Lück, Transformation groups and algebraic $K$-theory, Lecture Notes in Math., 1408, Springer-Verlag, Berlin, 1989, xii+443 pp.
Supplementary files
