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.
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J. M. Aarts and L. G. Oversteegen, The product structure of homogeneous spaces, Indagationes Mathematicae (New Series) 1 (1990), 1–5.
F. D. Ancel, An alternative proof and applications of a theorem of E. G. Effros, The Michigan Mathematical Journal 34 (1987), 39–55.
R. D. Anderson, On topological infinite deficiency, The Michigan Mathematical Journal 14 (1967), 365–383.
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.
M. Bestvina, Characterizing k-dimensional universal Menger compacta, Memoirs of the American Mathematical Society 71 (1988), no. 380, vi+110.
R. H. Bing and F. B. Jones, Another homogeneous plane continuum, Transactions of the American Mathematical Society 90 (1959), 171–192.
J. J. Dijkstra and J. van Mill, Homeomorphism groups of manifolds and Erdős space, Electronic Research Announcments of the American Mathematical Society 10 (2004), 29–38.
E. G. Effros, Transformation groups and C*-algebras, Annals of Mathematics 81 (1965), 38–55.
R. Engelking, Theory of Dimensions Finite and Infinite, Heldermann Verlag, Lemgo, 1995.
P. Erdős, The dimension of the rational points in Hilbert space, Annals of Mathematics 41 (1940), 734–736.
L. R. Ford, Jr., Homeomorphism groups and coset spaces, Transactions of the American Mathematical Society 77 (1954), 490–497.
M. Fort, Homogeneity of infinite products of manifolds with boundary, Pacific Journal of Mathematics 12 (1962), 879–884.
E. Glasner and M. Megrelishvili, Some new algebras of functions on toopological groups arising from G-spaces, 2006, preprint.
A. Hohti, Another alternative proof of Effros’ theorem, Topology Proceedings 12 (1987), 295–298.
K. Kawamura, L. G. Oversteegen and E. D. Tymchatyn, On homogeneous totally disconnected 1-dimensional spaces, Fundamenta Mathematicae 150 (1996), 97–112.
A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995.
O. H. Keller, Die Homoiomorphie der kompakten konvexen Mengen in Hilbertschen Raum, Mathematische Annalen 105 (1931), 748–758.
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.
W. Lewis, Continuous curves of pseudo-arcs, Houston Journal of Mathematics 11 (1985), 91–99.
M. G. Megrelishvili, A Tikhonov G-space admitting no compact Hausdorff G-extension or G-linearization, Russian Mathematical Surveys 43 (1988), 177–178.
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.
M. G. Megrelishvili, Every semitopological semigroup compactification of the group H + [0, 1] is trivial, Semigroup Forum 63 (2001), 357–370.
J. van Mill, The Infinite-Dimensional Topology of Function Spaces, North-Holland Publishing Co., Amsterdam, 2001.
J. van Mill, A note on Ford’s Example, Topology Proceedings 28 (2004), 689–694.
J. van Mill, A note on the Effros Theorem, The American Mathematical Monthly 111 (2004), 801–806.
J. van Mill, Strong local homogeneity and coset spaces, Proceedings of the American Mathematical Society 133 (2005), 2243–2249.
J. van Mill, Not all homogeneous Polish spaces are products, Houston Journal of Mathematics 32 (2006), 489–492.
J. van Mill, Homogeneous spaces and transitive actions by analytic groups, The Bulletin of the London Mathematical Society 39 (2007), 329–336.
M. W. Mislove and J. T. Rogers, Jr., Local product structures on homogeneous continua, Topology and its Applications 31 (1989), 259–267.
M. W. Mislove and J. T. Rogers, Jr., Addendum: “Local product structures on homogeneous continua”, Topology and its Applications 34 (1990), 209.
P. S. Mostert, Reasonable topologies for homeomorphism groups, Proceedings of the American Mathematical Society 12 (1961), 598–602.
V. Pestov, Dynamics of Infinite-dimensional Groups and Ramsey-type Phenomena, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2005.
M. Rubin, On the reconstruction of topological spaces from their groups of homeomorphisms, Transactions of the Amrecian Mathematical Society 312 (1989), 487–538.
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.
S. Teleman, Sur la représentation linéaire des groupes topologiques, Annales Scientifiques de l’École Normale Supérieure 74 (1957), 319–339.
G. S. Ungar, On all kinds of homogeneous spaces, Transactions of the Amrecian Mathematical Society 212 (1975), 393–400.
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.
V. V. Uspenskii, Topological groups and Dugundji compact spaces, Matematicheskii Sbornik 180 (1989), 1092–1118.
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.
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.
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van Mill, J. Homogeneous spaces and transitive actions by Polish groups. Isr. J. Math. 165, 133–159 (2008). https://doi.org/10.1007/s11856-008-1007-0
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DOI: https://doi.org/10.1007/s11856-008-1007-0