arXiv subjects

Sergey A. Antonyan

Publications and source records attributed to Sergey A. Antonyan.

8 recordsLinked to original sources

arXiv2026

Equivariant homotopy dense subsets in the realm of uniform G-ANR spaces

Let $G$ be a compact group. The existence of certain $G$-homotopy dense subsets in a metrizable $G$-space $X$ plays a fundamental role, as it is equivalent to $X$ being a $G$-ANR. From this perspective, the present paper develops several applications of this class of $G$-subsets. In particular, we prove that for a compact $G$-space $X$ and a metric space $Y$, the mapping space $C(X,Y)$ is a $G$-UA(N)R if and only if $Y$ is a UA(N)R in the sense of Michael. This result is significant because it enables the construction of examples of Lawson metric $G$-semilattices for which the property of being a $G$-UANR is equivalent to uniform local path-connectedness. Moreover, we show that this equivalence holds for every Lawson metric $G$-semilattice whenever $G$ is finite. Finally, we analyze the behavior of $G$-homotopy dense subsets when the ambient space is a $G$-A(N)R, thereby introducing the notion of a $G$-A(N)R-pair.

math.GN↗

arXiv2026

Characterizations of $G$-ANR spaces and inverse limits

In this paper we prove that, for a compact group $G$, a metrizable $G$-space is a $G$-ANR under the following asumptions: (1) if it dominates a $G$-ANR space through a fine $G$-homotopy equivalence; (2) if it is $G$-homotopy dense in a $G$-ANR; (3) if it contains a $G$-ANR as a $G$-homotopy dense subset; (4) if it is the inverse limit of an inverse sequence of $G$-ANR spaces with bonding maps that are fine $G$-homotopy equivalences.

math.GT↗

arXiv2023

Extension properties of orbit spaces for proper actions revisited

Let $G$ be a locally compact Hausdorff group. We study orbit spaces of equivariant absolute neighborhood extensors ($G$-${\rm ANE}$'s) for the class of all proper $G$-spaces that are metrizable by a $G$-invariant metric. We prove that if a $G$-space $X$ is a $G$-${\rm ANE}$ and all $G $-orbits in $X$ are metrizable, then the $G$-orbit space $X/G$ is an {\rm ANE}. If $G$ is either a Lie group or an almost connected group, then for any closed normal subgroup $H$ of $G$, the $H$-orbit space $X/H$ is a $G/H$-{\rm ANE} provided that all $H$-orbits in $X$ are metrizable.

math.GN↗

arXiv2017

Characterizing slices for proper actions of locally compact groups

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 compact group $G$ and $S$ is a small (in the sense of Palais) $H$-slice in a proper $G$-space, then the action map $G\times S\to G(S)$ is open. This is applied to prove that the slicing map $f_S:G(S)\to G/H$ 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.

math.GN↗

arXiv2012

Proper actions on topological groups: Applications to quotient spaces

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 prove that many topological properties (among them, paracompactness and normality) are transferred from X to ferred from X/G to X. Yet another application leads to the inequality dim X<= dim X/G + dim G for every paracompact group X and its locally compact subgroup G.

math.GN↗

arXiv2011

Characterizing maximal compact subgroups

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.

math.GN↗

arXiv2011

Locally compact subgroup actions on topological groups

Let $X$ be a Hausdorff topological group and $G$ a locally compact subgroup of $X$. We show that $X$ admits a locally finite $σ$-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 addition, the space of connected components of $G$ is compact and $X$ is normal, then $X$ itself is $G$-homeomorphic to a twisted product $G\times_KS$, where $K$ is a maximal compact subgroup of $G$. This implies that $X$ is $K$-homeomorphic to the product $G/K\times S$, and in particular, $X$ is homeomorphic to the product $\Bbb R^n\times S$, where $n={\rm dim\,} G/K$. Using these results we prove the inequality $ {\rm dim}\, X\le {\rm dim}\, X/G + {\rm dim}\, G$ for every Hausdorff topological group $X$ and a locally compact subgroup $G$ of $X$.

math.GN↗

arXiv2011

Equivariant extension properties of coset spaces of locally compact groups and approximate slices

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.

math.GN↗