Papers by Sergey Antonyan
Topology and its Applications, Apr 1, 2022
Topology and its Applications, Dec 1, 2020
Topology and its Applications, Apr 1, 2023
arXiv (Cornell University), Feb 26, 2017
In his seminal work [14], R. Palais extended a substantial part of the theory of compact transfor... more In his seminal work [14], R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results well known for compact group actions. In particular, we prove that if H is a compact subgroup of a locally compact group G and S is a small (in the sense of Palais) H-slice in a proper G-space, then the action map G × S → G(S) is open. This is applied to prove that the slicing map f S : G(S) → G/H is continuos and open, which provides an external characterization of a slice. Also an equivariant extension theorem is proved for proper actions. As an application, we give a short proof of the compactness of the Banach-Mazur compacta.
Russian Mathematical Surveys, Dec 31, 1980
Topology and its Applications, Nov 1, 2017
Let G be a matrix Lie group. We prove that a proper Gspace X of finite structure, which is metriz... more Let G be a matrix Lie group. We prove that a proper Gspace X of finite structure, which is metrizable by a G-invariant metric, is a G-ANR (resp., a G-AR) iff for any compact subgroup H ⊂ G the Hfixed point set X H is an ANR (resp., an AR). An equivariant embedding result for proper G-spaces of finite structure is also obtained. Problem 1.1 (Jaworowski's Problem). Let G be a compact Lie group and X a metrizable G-space that has a finitely many G-orbit types (e.g., when G is a finite group). Assume that for each compact subgroup H ⊂ G, the H-fixed point set X H = {x ∈ X | hx = x, ∀h ∈ H} is an ANR (resp., an AR). Is then X a G-ANR (resp., a G-AR)? This problem still remains open. Moreover, it is open even in a very special case when G = Z 2 , X = Q = [−1, 1] ∞ , the Hilbert cube, and X Z 2 = { * }, a sigleton. In this case the problem is equivalent to the following old problem of R.D. Anderson posed in the mid-sixties (see [39, p. 292], see also [40, p. 656] and [41, Problem 930]): Problem 1.2 (Anderson's Problem). Let α : Q → Q be a based-free involution, i.e., α 2 = 1 Q and α has a unique fixed point. Is then α conjugate
Advances in Mathematics, Mar 1, 2020
We investigate the hyperspace GH(R n) of the isometry classes of all non-empty compact subsets of... more We investigate the hyperspace GH(R n) of the isometry classes of all non-empty compact subsets of a Euclidean space in the Gromov-Hausdorff metric. It is proved that for any n ≥ 1, GH(R n) is homeomorphic to the orbit space 2 R n /E(n) of the hyperspace 2 R n of all non-empty compact subsets of a Euclidean space R n equipped with the Hausdorff metric and the natural action of the Euclidean group E(n). This is further applied to prove that 2 R n /E(n) is homeomorphic to the open cone OCone Ch(B n)/O(n) , where Ch(B n) stands for the set of all A ∈ 2 R n for which the closed Euclidean unit ball B n is the least circumscribed ball (the Chebyshev ball). These results lead to determine the complete topological structure of GH(R n) for n ≤ 2, namely, we prove that GH(R n) is homeomorphic to the Hilbert cube with a removed point. We also prove that for n ≤ 2, GH(B n) is homeomorphic to the Hilbert cube.
Advances in Mathematics, Dec 1, 2021
Topology and its Applications, 2017
This issue of Topology and its Applications is dedicated to the memory of Alex Chigogidze, who pa... more This issue of Topology and its Applications is dedicated to the memory of Alex Chigogidze, who passed away on December 15, 2014. Alex was a world-renowned topologist with broad interests not only in Topology, but also in Functional Analysis, Measure Theory and Operator Algebras. According to MatSciNet, Alex authored 105 publications, including two monographs, and has more than 300 citations. Alex served as a member of editorial board of the Tbilisi Mathematical Journal and the JP Journal of Geometry and Topology. He was also a managing editor of Topology and its Applications He was born in Tbilisi, Georgia, on January 1, 1955 and earned his MSc and PhD degrees from Tbilisi State University. Like his research, the list of Alex's positions and university's appointments is very impressive.
Математические заметки, 1999
Успехи математических наук, 1996
Pseudocompact Topological Spaces, 2018
In what follows, G is a Hausdorff topological group with identity element e. A topological transf... more In what follows, G is a Hausdorff topological group with identity element e. A topological transformation group or a G-space with phase group G is a triple \(\langle X, G, \alpha \rangle \), where X is a topological space and \(\alpha \).
Let n be a natural number equal or greater than 2. In this paper we study the topological structu... more Let n be a natural number equal or greater than 2. In this paper we study the topological structure of certain hyperspaces of convex subsets of constant width, equipped with the Hausdorff metric topology. We focus our attention on the hyperspace cw_D(R^n) of all compact convex subsets with constant width d∈ D, where D is a convex subset of [0,∞). Our main result states that cw_D(R^n) is homeomorphic to D× R^n× Q, where Q denotes the Hilbert cube. We also prove that the hyperspace crw_D(R^n), consisting of all pairs of compact convex sets of constant relative width d∈ D is homeomorphic to cw_D(R^n). In particular, we prove that the hyperspace cw(R^n) of all compact convex bodies of constant width, as well as the hyperspace crw(R^n) of all pairs of compact convex sets of constant relative positive width are homeomorphic to R^n+1× Q.
We prove that for a compact subgroup H of an almost connected locally compact Hausdorff group G, ... more We prove that for a compact subgroup H of an almost connected locally compact Hausdorff group G, the following properties are mutually equivalent: (1) H is a maximal compact subgroup of G, (2) G/H is contractible, (3) G/H is homeomorphic to a Euclidean space, (4) G/H is an AE for paracompact spaces, (5) G/H is a G-AE for paracompact proper G-spaces having a paracompact orbit space.
Fundamenta Mathematicae
Let J(n) be the hyperspace of all centrally symmetric compact convex bodies A ⊆ R n , n ≥ 2, for ... more Let J(n) be the hyperspace of all centrally symmetric compact convex bodies A ⊆ R n , n ≥ 2, for which the ordinary Euclidean unit ball is the ellipsoid of maximal volume contained in A (the John ellipsoid). Let J 0 (n) be the complement of the unique O(n)-fixed point in J(n). We prove that: (1) the Banach-Mazur compactum BM (n) is homeomorphic to the orbit space J(n)/O(n) of the natural action of the orthogonal group O(n) on J(n); (2) J(n) is an O(n)-AR; (3) J 0 (2)/SO(2) is an Eilenberg-MacLane space K(Q, 2); (4) BM 0 (2) = J 0 (2)/O(2) is noncontractible; (5) BM (2) is a nonhomogeneous absolute retract. Other models for BM (n) are established. 0. Introduction. In [30, Chapter 30, Problem 899, ANR 11] the following problems of A. Pe lczyński were posed: (a) Are the Banach-Mazur compacta BM (n) AR's? (b) Are they Hilbert cubes?
We characterize coset spaces of topological groups which are coset spaces of (separable) metrizab... more We characterize coset spaces of topological groups which are coset spaces of (separable) metrizable groups and complete metrizable (Polish) groups. Besides, it is shown that for a G-space X with a d-open action there is a topological group H of weight and cardinality less than or equal to the weight of X such that H admits a d-open action on X. This is further applied to show that if X is a separable metrizable coset space then X has a Polish extension which is a coset space of a Polish group.
Let G be a compact Lie group. We prove that a metrizable G-space X is a G-ANE (resp., a G-AE) iff... more Let G be a compact Lie group. We prove that a metrizable G-space X is a G-ANE (resp., a G-AE) iff X is an ANE (resp., an AE) and, for any closed subgroup H ⊂ G, the H-fixed point set X[H] is a strong neighborhood H-deformation retract (resp., a strong H-deformation retract) of X. If a G-space X has no G-fixed point, then X is a G-ANE provided that X is an H-ANE for any subgroup H ⊂ G that occurs as a stabilizer in X. As an application, we give a new proof of the equivariant Dugundji extension theorem in the metrizable case.
Dedicated to Professor Sibe Mardeˇsić on the occasion of his 80th birthday Abstract. Let G be a c... more Dedicated to Professor Sibe Mardeˇsić on the occasion of his 80th birthday Abstract. Let G be a compact Lie group. We prove that if each point x ∈ X of a G-space X admits a Gx-invariant neighborhood U which is a Gx-ANE then X is a G-ANE, where Gx stands for the stabilizer of x. This result is further applied to give two equivariant homotopy characterizations of G-ANR’s. One of them sounds as follows: a metrizable G-space Y is a G-ANR iff Y is locally G-contractible and every metrizable closed G-pair (X, A) has the G-equivariant homotopy extension property with respect to Y. In the same terms we also characterize G-ANR subsets of a given G-ANR space. 1.
Let X be a Hausdorff topological group and G a locally compact subgroup of X. We show that the na... more Let X be a Hausdorff topological group and G a locally compact subgroup of X. We show that the natural action of G on X is proper in the sense of R. Palais. This is applied to prove that there exists a closed set F ⊂ X such that FG = X and the restriction of the quotient projection X → X/G to F is a perfect map F → X/G. This is a key result to prove that many topological properties (among them, paracompactness and normality) are transferred from X to X/G, and some others are transferred from X/G to X. Yet another application leads to the inequality dimX ≤ dimX/G+dimG for every paracompact topological group X and a locally compact subgroup G of X having a compact group of connected components.
We prove that for a compact subgroup H of a locally compact Hausdorff group G, the following prop... more We prove that for a compact subgroup H of a locally compact Hausdorff group G, the following properties are mutually equivalent: (1) G/H is a manifold, (2) G/H is finite-dimensional and locally connected, (3) G/H is locally contractible, (4) G/H is an ANE for paracompact spaces, (5) G/H is a metrizable G-ANE for paracompact proper G-spaces having a paracompact orbit space. A new version of the Approximate slice theorem is also proven in the light of these results.
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Papers by Sergey Antonyan