# Sergey AntonyanUniversidad Nacional Autónoma de México | UNAM · Department of Mathematics

Sergey Antonyan

Doctor of Philosophy

## About

93

Publications

3,509

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746

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Introduction

Group actions on infinite-dimensional manifolds

**Skills and Expertise**

## Publications

Publications (93)

For a locally compact group G, proper G-spaces in the sense of R. Palais are studied. One of our results states that each strongly metrizable proper G-space admits a G-invariant metric (compatible with its topology) provided G is an almost connected group. This extends several results about the existence of invariant metrics. We also prove that if...

We investigate the hyperspace GH(Rn) 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(Rn) is homeomorphic to the orbit space 2Rn/E(n) of the hyperspace 2Rn of all non-empty compact subsets of a Euclidean space Rn equipped with the Hausdorff metric and the...

For a Lie group G, we prove the existence of a universal G-space in the class of all paracompact (respectively, metrizable, and separable metrizable) free proper G-spaces which have a paracompact orbit space. Our approach is based on Milnor's construction EG. We show that EG and the universal free G-spaces in question are G-AE's. Besides, we show h...

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 \(\alpha \).

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 ac...

Let G be a matrix Lie group. We prove that a proper G-space 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 H-fixed point set XH is an ANR (resp., an AR). An equivariant embedding result for proper G-spaces of finite structure is also obtained.

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...

In his seminal work \cite{pal:61}, 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 co...

Let G be a compact group acting on a Polish group X by means of automorphisms. It is proved that the orbit space is an -manifold (resp., homeomorphic to ) provided X is a G-ANR (resp., G-AR) and the fixed point set is not locally compact. It is also proved that if a compact group G acts affinely on a separable closed convex subset K of a Fréchet sp...

It is proved that for G a locally compact group of weight less than or equal to a given infinite cardinal number τ, there exists a metric proper G-space of weight τ which is a G-AR and every metrizable proper G-space X of weight ≤τ can equivariantly be embedded in . As a by-product we prove that every compact subgroup of a locally compact group G i...

A compact convex subset K of a topological linear space is called a Keller compactum if it is affinely homeomorphic to an infinite-dimensional compact convex subset of the Hilbert space ℓ2ℓ2. Let G be a compact topological group acting affinely on a Keller compactum K and let 2K2K denote the hyperspace of all non-empty compact subsets of K endowed...

For a locally compact group G we consider the class G-MG-M of all proper (in the sense of R. Palais) G-spaces that are metrizable by a G-invariant metric. We show that each X∈G-MX∈G-M admits a compatible G-invariant metric whose closed unit balls are small subsets of X. This is a key result to prove that X admits a closed equivariant embedding into...

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\in
D, where D is a convex subset of [0,\in...

For a locally compact Hausdorff group G we introduce the notion of a uniformly proper G-space. We prove that a uniformly proper G-space X admits a closed fundamental set F⊂XF⊂X; in particular, the restriction of the orbit projection X→X/GX→X/G to F is a perfect surjective map F→X/GF→X/G. This is a key result to prove the existence of a compatible G...

We prove that for any closed subgroup H of a locally compact
Hausdorff group G the following properties are mutually
equivalent:
(1) the coset space G/H is locally contractible,
(2) G/H is finite-dimensional and locally connected, (3) G/H is
a manifold. Assume that G is a locally compact group with
compact space of connected components. If the natu...

For every n ≥ 2, let cc(ℝn) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space ℝn endowed with the Hausdorff metric topology. Let cb(ℝn) be the subset of cc(ℝn) consisting of all compact convex bodies. In this paper we discover several fundamental properties of the natural action of the affine group Aff(n) on cb(ℝn)...

We prove that if G is a locally compact group acting properly (in the sense of R. Palais) on a Tychonoff space X, then X can be embedded equivariantly into a linear G-space L endowed with a linear G-action which is proper on the complement L∖{0}L∖{0}. If, in addition, G is a Lie group and τ an infinite cardinal number, then the linearizing G-space...

We establish the existence of universal G-spaces for proper actions of locally compact groups on Tychonoff spaces. A typical result sounds as follows: for each infinite cardinal number τ every locally compact, non-compact, σ-compact group G of weight w(G)⩽τ, can act properly on Rτ∖{0} such that Rτ∖{0} contains a G-homeomorphic copy of every Tychono...

Let G be a locally compact Hausdorff group, L a linear G-space, and Y ⊂ L a metrizable convex subset of L endowed with a proper action of G. Let X be a paracompact proper Gspace with a paracompact orbit space. We give conditions for Y in order that every equivariant lower semicontinuous multivalued map ø : X ⇒ Y with complete convex values admits a...

In this paper, for G a locally compact group (or a Lie group), we study the relationship between the covering dimensions of a proper G-space X and its orbit space X/G. We prove also that dimX=IndX for every proper G-space X with a metrizable orbit space provided that G is either pro-Lie or σ-compact or has a metrizable quotient group of connected c...

We prove that for a compact subgroup H of a locally compact almost connected group G, the following properties are mutually equivalent: (1) H is a maximal compact subgroup of G, (2) the coset space G/H is \({\mathbb{Q}}\) -acyclic and \({\mathbb{Z}/2\mathbb{Z}}\) -acyclic in Čech cohomology, (3) G/H is contractible, (4) G/H is homeomorphic to a Euc...

Let $X$ be a Hausdorff topological group and $G$ a locally compact subgroup
of $X$. We show that $X$ admits a locally finite $\sigma$-discrete
$G$-functionally open cover each member of which is $G$-homeomorphic to a
twisted product $G\times_H S_i$, where $H$ is a compact large subgroup of $G$
(i.e., the quotient $G/H$ is a manifold). If, in additi...

We prove that for a compact subgroup H of a locally compact Hausdorff group G. the following properties are mutually equivalent: (I) G/H is finite-dimensional and locally connected. (2) G/H is a smooth manifold. (3) G/H satisfies the following equivariant extension property: for every paracompact proper G-space X having a paracompact orbit space, e...

We apply equivariant joins to give a new and more transparent proof of the following result: if G is a compact Hausdorff group and X a G-ANR (respectively, a G-AR), then for every closed normal subgroup H of G, the H-orbit space X/H is a G/H-ANR (respectively, a G/H-AR). In particular, X/G is an ANR (respectively, an AR).

We prove that if G is a locally compact group acting properly (in the sense of R. Palais) on a space X that is metrizable by a G-invariant metric, then X can be embedded equivariantly into a normed linear G-space E endowed with a linear isometric G-action which is proper on the complement E∖{0}. If, in addition, G is a Lie group then E∖{0} is a G-e...

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 of 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 pro...

Let G be a locally compact Hausdorff group. We study absolute (neighborhood) extensors (G-AE's and G-ANE's) in the category G-M of all proper G-spaces that are metrizable by a G-invariant metric. We first solve the linearization problem for proper group actions by proving that each X ∈ G-M admits an equivariant embedding in a Banach G-space L such...

In this paper there is an investigation, for the case of a compact group G, of the orbit space X/G of a given G-space X, from the point of view of the theory of retracts. A particular case of the main result asserts that if one of the spaces X and G has countable weight and X is a G-A(N)R for metrizable spaces, then X/G is an A(N)R for metrizable s...

Sufficient conditions are given for an action of the orthogonal group O(n) on the Hilbert cube Q in order that the corresponding orbit space Q/O(n) be homeomorphic to the Banach-Mazur compactum BM(n). This result is applied to obtain simple topological models for BM(2).

The authors analyzed the results of binocular refractive lens exchange with a pseudoaccomodative AcrySof ReSTOR multifocal diffractive intraocular lens (IOL) in 24 patients (48 eyes) aged 40 to 61 years. Preoperative hyperopic refraction ranged from 1.0 to 6.0 diopters. One week postoperatively, a high percentage of the operated eyes had fine resul...

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-A...

We prove a general theorem about the orbit spaces of compact Lie group actions which are Hilbert cube manifolds. This result is further applied to prove that the Banach–Mazur compactum BM(2)BM(2) is homeomorphic to the orbit space (expS1)/O(2)(expS1)/O(2), where expS1expS1 is the hyperspace of all nonempty closed subsets of the unit circle S1S1 end...

We prove that if G is a compact Hausdorff group then every G-ANR has the G-homotopy type of a G-CW complex. This is applied to extend the James–Segal G-homotopy equivalence theorem to the case of arbitrary compact group actions.

Let G be a compact group and X a G-ANR. Then X is a G-AR iff the H-fixed point set XH is homotopy trivial for each closed subgroup H ⊂ G.

Let G be a compact Lie group, X a metric G-space, and exp X the hyperspace of all nonempty compact subsets of X endowed with the Hausdorff metric topology and with the induced action of G. We prove that the following three assertions are equivalent: (a) X is locally continuum-connected (resp., connected and locally continuum-connected); (b) exp X i...

A variant of the Bourgin-Yang theorem for a-compact perturbations of a closed linear operator (in general,
unbounded and having an infinite-dimensional kernel) is proved.
An application to integrodifferential equations is discussed.

Let G be a locally compact Hausdorff group. We study orbit spaces of equivariant absolute neighborhood extensors (G-ANE's) in the category of all proper G-spaces that are metrizable by a G-invariant metric. We prove that if a proper G-space X is a G-ANE (respectively, a G-ANE(n),n⩾0), and H a closed normal subgroup of G such that all the H-orbits i...

We prove that if G is a compact Lie group, Y a G-space equipped with a topological local convex structure compatible with the action of G, then Y is a G-ANE for metrizable G-spaces. If, in addition, Y has a G-fixed point and admits a global convex structure compatible with the action of G, then Y is a G-AE. This is applied to show that certain hype...

For a compact Lie group G we prove that every free (resp., semifree) G-space admits a based-free (resp., semifree) G-compactification. Examples of finite- and infinite-dimensional G-spaces are presented that do not admit a free or based-free G-compactification. We give several characterizations of the maximal G-compactification βG
X that are furthe...

Let G be a locally compact Hausdorff group. We study orbit spaces and unions of equivariant absolute neighborhood extensors (G-ANEs) in the category of all proper G-spaces that are metrizable by a G-invariant metric. We prove that if a proper G-space X is a G-ANE such that all the G-orbits in X are metrizable, then the G-orbit space X/G is an ANE....

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...

Let G be a compact Lie group, X a metric G-space, and exp X the hyperspace of all nonempty compact subsets of X endowed with the Hausdorff metric topology and with the induced action of G. We prove that the following three assertions are equivalent: (a) X is locally continuum-connected (resp., connected and locally continuum-connected); (b) exp X i...

Let G be a locally compact Hausdorff group. It is proved that: 1. on each Palais proper G-space X there exists a compatible family of G-invariant pseudometrics; 2.the existence of a compatible G-invariant metric on a metrizable proper G-space X is equivalent to the paracompactness of the orbit space X/G; 3. if in addition G is either almost connect...

It is proved that: (1) every Lie group G can act properly (in sense of Palais) on each infinite-dimensional Hilbert space l2(τ) of a given weight τ such that (G,l2(τ)) becomes a universal G-space for all metrizable proper G-spaces admitting an invariant metric and having weight ⩽τ; (2) every Lie group G can act properly on such that ) becomes a uni...

It is shown that: (1) any action of a Moscow group G on a first countable, Dieudonné complete (in particular, on a metrizable) space X can uniquely be extended to an action of the Dieudonné completion γG on X, (2) any action of a locally pseudocompact topological group G on a b
f
-space (in particular, on a first countable space) X can uniquely be...

We give a short proof of Wojdyslawski's famous theorem.

Let J(n) be the hyperspace of all centrally symmetric compact convex bodies A ⊆ ℝn, n ≥ 2, for which the ordinary Euclidean unit ball is the ellipsoid of maximal volume contained in A (the John ellipsoid). Let J0(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 o...

From the point of view of retracts and shape theory, the category G-TOPB of G-spaces over a G-space B, where G is a compact group, is investigated. In particular, we prove that if B has only one orbit type and E is a metric G-ANR over B, then the orbit space E/G is an ANR over B/G. As an application we construct a fiberwise G-orbit functor μ : G-SH...

Extensorial properties of orbit spaces of locally compact proper group actions are investigated.

It is proved that a based-free action alpha of a given compact Lie group G on the Hilbert cube Q is equivalent to the standard based-free action sigma if and only if the orbit space Q(0)/alpha of the free part Q(0) = is a Q-manifold having the proper homotopy type of the orbit space Q(0)/sigma. The existence of an equivariant retraction (Q(0), sigm...

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The theorem proved earlier by A.N. Dranishnikov is generalized on the case of arbitrary metrized spaces and gained in an equivariant sense. It is proved that a functor of G-symmetrical n-degree preserves a property of metrized space to be LCk-space (k≥0). The functor is appeared to be N(G)-equivariant absolute extensor in dimensionality k+1, where...

It is proved that the functor SP G n of G-symmetric nth power preserves the properties of a metrizable space X to be an LC k ∩C k space and to be an LC k space for k≥0. Moreover, it turns out that SP G n X is an N(G)-equivariant absolute (neighborhood) extensor in dimension k+1, where N(G) is the normalizer of G in the symmetric group S n .

In this paper there is an investigation, for the case of a compact group G, of the orbit space X/G of a given G-space X, from the point of view of the theory ofretracts. A particular case of the main result asserts that if one of the spaces X and G has countable weight and X is a G-A(N)R for metrizable spaces, then X/G is an A(N)R for metrizable sp...

Not Available Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints

Not Available Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints