×

Homogeneous spaces and transitive actions by Polish groups. (English) Zbl 1153.54021

Author’s abstract: We prove that for every homogeneous and strongly locally homogeneous Polish space \(X\) there is a Polish group admitting a transitive action on \(X\). We also construct an example of a homogeneous Polish space which is not a coset space and on which no separable metrizable topological group acts transitively.

MSC:

54H20 Topological dynamics (MSC2010)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. M. Aarts and L. G. Oversteegen, The product structure of homogeneous spaces, Indagationes Mathematicae (New Series) 1 (1990), 1–5. · Zbl 0696.54008 · doi:10.1016/0019-3577(90)90027-K
[2] F. D. Ancel, An alternative proof and applications of a theorem of E. G. Effros, The Michigan Mathematical Journal 34 (1987), 39–55. · Zbl 0626.54036 · doi:10.1307/mmj/1029003481
[3] R. D. Anderson, On topological infinite deficiency, The Michigan Mathematical Journal 14 (1967), 365–383. · Zbl 0148.37202 · doi:10.1307/mmj/1028999787
[4] H. Becker and A. S. Kechris, The Descriptive Set Theory of Polish Group Actions, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996. · Zbl 0949.54052
[5] M. Bestvina, Characterizing k-dimensional universal Menger compacta, Memoirs of the American Mathematical Society 71 (1988), no. 380, vi+110. · Zbl 0645.54029
[6] R. H. Bing and F. B. Jones, Another homogeneous plane continuum, Transactions of the American Mathematical Society 90 (1959), 171–192. · Zbl 0084.18903 · doi:10.1090/S0002-9947-1959-0100823-3
[7] J. J. Dijkstra and J. van Mill, Homeomorphism groups of manifolds and Erdos space, Electronic Research Announcments of the American Mathematical Society 10 (2004), 29–38. · Zbl 1078.57033 · doi:10.1090/S1079-6762-04-00127-1
[8] E. G. Effros, Transformation groups and C*-algebras, Annals of Mathematics 81 (1965), 38–55. · Zbl 0152.33203 · doi:10.2307/1970381
[9] R. Engelking, Theory of Dimensions Finite and Infinite, Heldermann Verlag, Lemgo, 1995. · Zbl 0872.54002
[10] P. Erdos, The dimension of the rational points in Hilbert space, Annals of Mathematics 41 (1940), 734–736. · Zbl 0025.18701 · doi:10.2307/1968851
[11] L. R. Ford, Jr., Homeomorphism groups and coset spaces, Transactions of the American Mathematical Society 77 (1954), 490–497. · Zbl 0058.17302 · doi:10.1090/S0002-9947-1954-0066636-1
[12] M. Fort, Homogeneity of infinite products of manifolds with boundary, Pacific Journal of Mathematics 12 (1962), 879–884. · Zbl 0112.38004
[13] E. Glasner and M. Megrelishvili, Some new algebras of functions on toopological groups arising from G-spaces, 2006, preprint.
[14] A. Hohti, Another alternative proof of Effros’ theorem, Topology Proceedings 12 (1987), 295–298. · Zbl 0674.54009
[15] K. Kawamura, L. G. Oversteegen and E. D. Tymchatyn, On homogeneous totally disconnected 1-dimensional spaces, Fundamenta Mathematicae 150 (1996), 97–112. · Zbl 0861.54028
[16] A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. · Zbl 0819.04002
[17] O. H. Keller, Die Homoiomorphie der kompakten konvexen Mengen in Hilbertschen Raum, Mathematische Annalen 105 (1931), 748–758. · Zbl 0003.22401 · doi:10.1007/BF01455844
[18] K. Kunen, Set Theory. An Introduction to Independence Proofs, Studies in Logic and the foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1980. · Zbl 0443.03021
[19] W. Lewis, Continuous curves of pseudo-arcs, Houston Journal of Mathematics 11 (1985), 91–99. · Zbl 0577.54039 · doi:10.1016/0315-0860(85)90079-5
[20] M. G. Megrelishvili, A Tikhonov G-space admitting no compact Hausdorff G-extension or G-linearization, Russian Mathematical Surveys 43 (1988), 177–178. · Zbl 0664.54024 · doi:10.1070/RM1988v043n02ABEH001733
[21] M. G. Megrelishvili, Compactification and factorization in the category of G-spaces, in Categorical Topology and its Relation to Analysis, Algebra and Combinatorics (Prague, 1988), World Sci. Publishing, Teaneck, NJ, 1989, pp. 220–237.
[22] M. G. Megrelishvili, Every semitopological semigroup compactification of the group H + [0, 1] is trivial, Semigroup Forum 63 (2001), 357–370. · Zbl 1009.22004
[23] J. van Mill, The Infinite-Dimensional Topology of Function Spaces, North-Holland Publishing Co., Amsterdam, 2001. · Zbl 0969.54003
[24] J. van Mill, A note on Ford’s Example, Topology Proceedings 28 (2004), 689–694. · Zbl 1088.54015
[25] J. van Mill, A note on the Effros Theorem, The American Mathematical Monthly 111 (2004), 801–806. · Zbl 1187.54030 · doi:10.2307/4145191
[26] J. van Mill, Strong local homogeneity and coset spaces, Proceedings of the American Mathematical Society 133 (2005), 2243–2249. · Zbl 1063.22006 · doi:10.1090/S0002-9939-05-07808-1
[27] J. van Mill, Not all homogeneous Polish spaces are products, Houston Journal of Mathematics 32 (2006), 489–492. · Zbl 1094.54018
[28] J. van Mill, Homogeneous spaces and transitive actions by analytic groups, The Bulletin of the London Mathematical Society 39 (2007), 329–336. · Zbl 1181.54043 · doi:10.1112/blms/bdm018
[29] M. W. Mislove and J. T. Rogers, Jr., Local product structures on homogeneous continua, Topology and its Applications 31 (1989), 259–267. · Zbl 0674.54024 · doi:10.1016/0166-8641(89)90022-9
[30] M. W. Mislove and J. T. Rogers, Jr., Addendum: ”Local product structures on homogeneous continua”, Topology and its Applications 34 (1990), 209. · Zbl 0689.54022 · doi:10.1016/0166-8641(90)90083-E
[31] P. S. Mostert, Reasonable topologies for homeomorphism groups, Proceedings of the American Mathematical Society 12 (1961), 598–602. · Zbl 0102.38001 · doi:10.1090/S0002-9939-1961-0130681-7
[32] V. Pestov, Dynamics of Infinite-dimensional Groups and Ramsey-type Phenomena, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2005. · Zbl 1076.37005
[33] M. Rubin, On the reconstruction of topological spaces from their groups of homeomorphisms, Transactions of the Amrecian Mathematical Society 312 (1989), 487–538. · Zbl 0677.54029 · doi:10.1090/S0002-9947-1989-0988881-4
[34] S. Solecki, Polish group topologies, in Sets and proofs, London Math. Soc. Lecture Note Series, vol. 258 (S. B. Cooper and J. K. Truss, eds.), Cambridge University Press, Cambridge, 1999, pp. 339–364. · Zbl 0941.54034
[35] S. Teleman, Sur la représentation linéaire des groupes topologiques, Annales Scientifiques de l’École Normale Supérieure 74 (1957), 319–339.
[36] G. S. Ungar, On all kinds of homogeneous spaces, Transactions of the Amrecian Mathematical Society 212 (1975), 393–400. · Zbl 0318.54037 · doi:10.1090/S0002-9947-1975-0385825-3
[37] V. V. Uspenskii, Why compact groups are dyadic, in General Topology and its Relations to Modern Analysis and Algebra, VI (Prague, 1986), Research and Exposition in Mathematics, vol. 16, Heldermann, Berlin, 1988, pp. 601–610.
[38] V. V. Uspenskii, Topological groups and Dugundji compact spaces, Matematicheskii Sbornik 180 (1989), 1092–1118. · Zbl 0684.22001
[39] J. de Vries, On the existence of G-compactifications, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 26 (1978), 275–280.
[40] J. de Vries, Linearization, compactification and the existence of non-trivial compact extensors for topological transformation groups, in Topology and Measure III, Ernst-Moritz-Arndt-Universität zu Greifswald, 1982, pp. 339–346. · Zbl 0506.54032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.