Equivariant embedding of metrizable $G$-spaces in linear $G$-spaces
HTML articles powered by AMS MathViewer
- by Aasa Feragen PDF
- Proc. Amer. Math. Soc. 136 (2008), 2985-2995 Request permission
Abstract:
Given a Lie group $G$ we study the class $\mathcal {M}_G$ of proper metrizable $G$-spaces with metrizable orbit spaces, and show that any $G$-space $X \in \mathcal {M}_G$ admits a closed $G$-embedding into a convex $G$-subset $C$ of some locally convex linear $G$-space, such that $X$ has some $G$-neighborhood in $C$ which belongs to the class $\mathcal {M}_G$. As a corollary we see that any $G$-ANR for $\mathcal {M}_G$ is a $G$-ANE for $\mathcal {M}_G$.References
- S. Antonian, Equivariant embeddings into $G$-ARs, Glas. Mat. Ser. III 22(42) (1987), no. 2, 503–533 (English, with Serbo-Croatian summary). MR 957632
- S. A. Antonyan, Retraction properties of an orbit space, Mat. Sb. (N.S.) 137(179) (1988), no. 3, 300–318, 432 (Russian); English transl., Math. USSR-Sb. 65 (1990), no. 2, 305–321. MR 976513, DOI 10.1070/SM1990v065n02ABEH001311
- Sergey A. Antonyan, Extensorial properties of orbit spaces of proper group actions, Topology Appl. 98 (1999), no. 1-3, 35–46. II Iberoamerican Conference on Topology and its Applications (Morelia, 1997). MR 1719992, DOI 10.1016/S0166-8641(99)00039-5
- Sergey Antonyan, Universal proper $G$-spaces, Topology Appl. 117 (2002), no. 1, 23–43. MR 1874002, DOI 10.1016/S0166-8641(00)00115-2
- S. Antonyan and S. de Neymet, Invariant pseudometrics on Palais proper $G$-spaces, Acta Math. Hungar. 98 (2003), no. 1-2, 59–69. MR 1958466, DOI 10.1023/A:1022853304454
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
- Erik Elfving, The $G$-homotopy type of proper locally linear $G$-manifolds, Ann. Acad. Sci. Fenn. Math. Diss. 108 (1996), 50. MR 1413841
- Erik Elfving, The $G$-homotopy type of proper locally linear $G$-manifolds. II, Manuscripta Math. 105 (2001), no. 2, 235–251. MR 1846619, DOI 10.1007/s002290170004
- Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
- Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
- J. Jaworowski, $G$-spaces with a finite structure and their embedding in $G$-vector spaces, Acta Math. Acad. Sci. Hungar. 39 (1982), no. 1-3, 175–177. MR 653689, DOI 10.1007/BF01895230
- Katsuo Kawakubo, The theory of transformation groups, Translated from the 1987 Japanese edition, The Clarendon Press, Oxford University Press, New York, 1991. MR 1150492
- Richard S. Palais, The classification of $G$-spaces, Mem. Amer. Math. Soc. 36 (1960), iv+72. MR 0177401
- Richard S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. MR 126506, DOI 10.2307/1970335
Additional Information
- Aasa Feragen
- Affiliation: Department of Mathematics, University of Helsinki, F-I-00014 Helsinki, Finland
- Address at time of publication: Department of Mathematical Sciences, University of Aarhus, NY Munkegade, Building 1530, DK-8000 Aarhus, Denmark
- Email: aasa.feragen@helsinki.fi
- Received by editor(s): August 7, 2006
- Received by editor(s) in revised form: July 3, 2007
- Published electronically: April 15, 2008
- Additional Notes: The research leading to this article was financed by the Magnus Ehrnrooth Foundation.
- Communicated by: Paul Goerss
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2985-2995
- MSC (2000): Primary 57S20
- DOI: https://doi.org/10.1090/S0002-9939-08-09307-6
- MathSciNet review: 2399067