On extending actions of groups

S. M. Ageev   ; D. Repovš

Sbornik. Mathematics, Tome 201 (2010) no. 2, pp. 159-182

Résumé

Problems of dense and closed extension of actions of compact transformation groups are solved. The method developed in the paper is applied to problems of extension of equivariant maps and of construction of equivariant compactifications. Bibliography: 27 titles.

Détail

Keywords: extension of action, equivariant absolute extensor
Mots-clés : equivariant compactification.

Bibliographie

[1] S. M. Ageev, D. Repovš, E. V. Ščepin, “The extension problem for complete $UV^n$-preimages”, Tsukuba J. Math., 23:1 (1999), 97–111 | MR | Zbl

[2] V. V. Fedorchuk, “Extension of fiberings”, Moscow Univ. Math. Bull., 1985, no. 40, 92–95 | MR | Zbl

[3] S. M. Ageev, “Extensor properties of orbit spaces and the problem of continuation of an action”, Moscow Univ. Math. Bull., 49:1, 9–12 | MR | Zbl

[4] S. M. Ageev, “Zadacha o prodolzhenii deistviya i ekstenzornye svoistva prostranstv orbit”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1994, no. 6, 80

[5] S. A. Antonyan, “Retraction properties of an orbit space”, Math. USSR-Sb., 65:2 (1990), 305–321 | DOI | MR | Zbl

[6] J. W. Jaworowski, “Symmetric products of ANR's associated with a permutation group”, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 20 (1972), 649–651 | MR | Zbl

[7] S. M. Ageev, D. Repovš, “The Jaworowski method in the problem of the preservation of extensor properties by the orbit functor”, Math. Notes, 71:3–4 (2002), 428–431 | DOI | MR | Zbl

[8] S. M. Ageev, “On extending the action”, Mosc. Univ. Math. Bull., 47:5 (1992), 17–19 | MR | Zbl

[9] S. M. Ageev, “On a problem of Zambakhidze–Smirnov”, Math. Notes, 58:1 (1995), 679–684 | DOI | MR | Zbl

[10] R. Cauty, “Une généralisation du théorème de Borsuk–Whitehead–Hanner aux espaces stratifiables”, C. R. Acad. Sci. Paris Sér. A-B, 275:4 (1972), 271–274 | MR | Zbl

[11] M. Murayama, “On $G$-ANR's and their $G$-homotopy types”, Osaka J. Math., 20:3 (1983), 479–512 | MR | Zbl

[12] H. Torunczyk, J. E. West, “The fine structure of $S^1/S^1$; a $Q$-manifold hyperspace localization of the integers”, Proceedings of the International Conference on Geometric Topology (Warsawa, 1978), PWN, Warsaw, 1980, 439–449 | MR | Zbl

[13] G. E. Bredon, Introduction to compact transformation groups, 46, Academic Press, New York–London, 1972 | MR | MR | Zbl | Zbl

[14] T. tom Dieck, Transformation groups and representation theory, Lecture Notes in Math., 766, Springer-Verlag, Berlin, 1979 | DOI | MR | MR | Zbl | Zbl

[15] T. Matumoto, “Equivariant $K$-theory and Fredholm operators”, J. Fac. Sci. Univ. Tokyo Sect. I A Math., 18 (1971), 109–125 | MR | Zbl

[16] Ch. N. Maxwell, “Homomorphisms of topological transformation groups into function spaces”, Duke Math. J., 33:3 (1966), 567–574 | DOI | MR | Zbl

[17] C. H. Dowker, “On a theorem of Hanner”, Ark. Mat., 2:4 (1952), 307–313 | DOI | MR | Zbl

[18] R. S. Palais, “The classification of $G$-spaces”, Mem. Amer. Math. Soc., 1960, no. 36 | MR | Zbl

[19] R. Engelking, General topology, PWN, Warsawa, 1977 | MR | MR | Zbl

[20] L. S. Pontryagin, Topological groups, Gordon and Breach, New York–London–Paris, 1966 | MR | MR | Zbl | Zbl

[21] E. H. Spanier, Algebraic topology, McGraw-Hill, New York–Toronto, ON–London, 1966 | MR | MR | Zbl | Zbl

[22] J. Szenthe, “On the topological characterization of transitive Lie group actions”, Acta Sci. Math. (Szeged), 36:3–4 (1974), 323–344 | MR | Zbl

[23] P. S. Aleksandrov, B. A. Pasynkov, Vvedenie v teoriyu razmernosti, Nauka, M., 1973 | MR | Zbl

[24] G. Villalobos, “Inversion of the theorem on the existence of slices”, Moscow Univ. Math. Bull., 55:1 (2000), 26–29

[25] “The equivariant theorem of Dugundji”, Russian Math. Surveys, 45:5 (1990), 219–220 | DOI | MR | Zbl

[26] H. Abels, “A universal proper $G$-space”, Math. Z., 159:2 (1978), 143–158 | DOI | MR | Zbl

[27] C. J. R. Borges, “On stratifiable spaces”, Pacif. J. Math., 17:1 (1966), 1–16 | MR | Zbl