Geodesic
On equivariant fibrations of $G$-CW-complexes
Sbornik. Mathematics, Tome 210 (2019) no. 10, pp. 1428-1433 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: $G$-CW-complex, simplicial $G$-complex, equivariant fibration, $H$-fixed points. 
@article{SM_2019_210_10_a3,
     author = {P. S. Gevorgyan and R. Jimenez},
     title = {On equivariant fibrations of $G${-CW-complexes}},
     journal = {Sbornik. Mathematics},
     pages = {1428--1433},
     year = {2019},
     volume = {210},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2019_210_10_a3/}
}
TY  - JOUR
AU  - P. S. Gevorgyan
AU  - R. Jimenez
TI  - On equivariant fibrations of $G$-CW-complexes
JO  - Sbornik. Mathematics
PY  - 2019
SP  - 1428
EP  - 1433
VL  - 210
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_2019_210_10_a3/
LA  - en
ID  - SM_2019_210_10_a3
ER  - 
%0 Journal Article
%A P. S. Gevorgyan
%A R. Jimenez
%T On equivariant fibrations of $G$-CW-complexes
%J Sbornik. Mathematics
%D 2019
%P 1428-1433
%V 210
%N 10
%U http://geodesic.mathdoc.fr/item/SM_2019_210_10_a3/
%G en
%F SM_2019_210_10_a3
P. S. Gevorgyan; R. Jimenez. On equivariant fibrations of $G$-CW-complexes. Sbornik. Mathematics, Tome 210 (2019) no. 10, pp. 1428-1433. http://geodesic.mathdoc.fr/item/SM_2019_210_10_a3/

[1] 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 | DOI | MR | Zbl

[2] R. Cauty, “Sur les ouverts des CW-complexes et les fibrés de Serre”, Colloq. Math., 63:1 (1992), 1–7 | DOI | MR | Zbl

[3] J. P. May, J. Sigurdsson, Parametrized homotopy theory, Math. Surveys Monogr., 132, Amer. Math. Soc., Providence, RI, 2006, x+441 pp. | DOI | MR | Zbl

[4] I. M. James, G. B. Segal, “On equivariant homotopy type”, Topology, 17:3 (1978), 267–272 | DOI | MR | Zbl

[5] G. E. Bredon, Equivariant cohomology theories, Lecture Notes in Math., 34, Springer-Verlag, Berlin–New York, 1967, vi+64 pp. (not consecutively paged) | DOI | MR | Zbl

[6] S. Illman, “The equivariant triangulation theorem for actions of compact Lie groups”, Math. Ann., 262:4 (1983), 487–501 | DOI | MR | Zbl

[7] R. Cauty, “Sur les sous-espaces des complexes simpliciaux”, Bull. Soc. Math. France, 100 (1972), 129–155 | DOI | MR | Zbl

[8] T. tom Dieck, Transformation groups, De Gruyter Stud. Math., 8, Walter de Gruyter Co., Berlin, 1987, x+312 pp. | DOI | MR | Zbl

[9] W. Lück, Transformation groups and algebraic $K$-theory, Lecture Notes in Math., 1408, Springer-Verlag, Berlin, 1989, xii+443 pp. | DOI | MR | Zbl