Alex Chigogidze
We discuss compact Hausdorff groups from the point of view of the general theory of absolute extensors. In particular, we characterize the class of simple, connected and simply connected compact Lie groups as AE(2)-groups the third homotopy group of which is $\text{\f Z}$. This is the converse of the corresponding result of R. Bott.
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We describe the proper absolute (neighborhood) extensors for the class of at most $n$-dimensional spaces, notation $\rm{A(N)E}_p(n)$. For example, the unique locally compact $n$-dimensional separable metric space $X\in\rm{AE}_p(n)$ satisfiyng the $\rm{DD^nP}$-property is the $n$-dimensional Menger compactum without a point. Non-metrizable $\rm{A(N)E}_p(n)$-spaces are also described.
We characterize, in terms of $X$, extensional dimension of the Stone-Čech corona $βX \setminus X$ of locally compact and Lindelöf space $X$. The non-Lindelöf case case is also settled in terms of extending proper maps with values in $I^τ\setminus L$, where $L$ is a finite complex. Further, for a finite complex $L$, an uncountable cardinal $τ$ and a $Z_τ$-set $X$ in the Tychonov cube $I^τ$ we find necessary and sufficient condition, in terms of $I^τ\setminus X$, for $X$ to be in the class $\operatorname{AE}([L])$. We also introduce a concept of a proper absolute extensor and characterize the pr…
Let L be a countable and locally finite CW complex. Suppose that the class of all metrizable compacta of extension dimension not greater than L contains a universal element which is an absolute extensor in dimension L. Our main result shows that L is quasi-finite.
Twisted products: Enveloping actions and equivariant absolute neighborhood extensors
本文在部分作用语境下研究了扭曲积，证明了部分作用整体化即扭曲积，并探讨了其可度量化条件、同伦与范畴性质，以及包络空间成为等变绝对邻域扩张子的充分条件。
Let G be a compact group acting on a Banach space L by means of linear isometries. The action of G on L induces a natural continuous action on cc(L), the hyperspace of all compact convex subsets of L endowed with the Hausdorff metric topology. The main result of this paper states that the G-space cc(L) is a G-AE. Under some extra assumptions, this result can be extended to CB(L), the hyperspace of all closed and bounded convex subsets of L.
We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper: \proclaim{Theorem} Suppose X is a paracompact space. There is a CW complex K such that {a.} K is an absolute extensor of X up to homotopy, {b.} If a CW complex L is an absolute extensor of X up to homotopy, then L is an absolute extensor of Y up to homotopy of any paracompact space Y such that K is an absolute extensor of Y up to homotopy. proclaim The proof is based on the following simple result (see 1.6). \proclaim{Theorem} Suppose X be a paracompact space and $f:A\to Y$ is a map from a c…
We develop a method of extending actions of compact transformation groups which is then applied to the problem of preservation of equivariant extensor property by passing to a subspace of given orbit types.
It is proved that for a 3-dimensional compact metrizable space X the infinite real projective space is an absolute extensor of X if and only if the real projective plane is an absolute extensor of X.
This paper is devoted to dualization of dimension-theoretical results from the small scale to the large scale. So far there are two approaches for such dualization: one consisting of creating analogs of small scale concepts and the other amounting to the covering dimension of the Higson corona $ν(X)$ of $X$. The first approach was used by M.Gromov when defining the asymptotic dimension $\asdim(X)$ of metric spaces $X$. The second approach was implicitly contained in the paper \cite{Dran AsyTop} by Dranishnikov on asymptotic topology. It is not known if the two approaches yield the same concept…
We show that a Moore space M(Z_m,1) is an absolute extensor for finite dimensional metrizable spaces of cohomological dimension dim_{Z_m} \leq 1.
We prove the projective plane $\rp^2$ is an absolute extensor of a finite-dimensional metric space $X$ if and only if the cohomological dimension mod 2 of $X$ does not exceed 1. This solves one of the remaining difficult problems (posed by A.N.Dranishnikov) in extension theory. One of the main tools is the computation of the fundamental group of the function space $\Map(\rp^n,\rp^{n+1})$ (based at inclusion) as being isomorphic to either $\Z_4$ or $\Z_2\oplus\Z_2$ for $n\ge 1$. Double surgery and the above fact yield the proof.
Our main result states that the hyperspace of convex compact subsets of a compact convex subset $X$ in a locally convex space is an absolute retract if and only if $X$ is an absolute retract of weight $\leω_1$. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube $I^{ω_1}$ is homeomorphic to $I^{ω_1}$. An analogous result is also proved for the cone over $I^{ω_1}$. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.
Extension properties of orbit spaces for proper actions revisited
本文研究真G-空间中等变绝对邻域扩张子轨道空间的扩张性质，证明在轨道可度量条件下，G-轨道空间为ANE，并推广到Lie群或几乎连通群的商群情形。
Group topologies on groups of bi-absolutely continuous homeomorphisms
本文研究双绝对连续同胚群上的群拓扑，证明 τ_ac 与 τ_co 间不存在中间可分群拓扑，并给出 Roelcke 预紧性结果。
An example is given of a compact absolute retract that is not a Hilbert cube manifold but whose second symmetric porduct is the Hilbert cube. A factor theorem is given for nth symmetric product of the cartesian product of any absolute neighborhood retract with the Hilbert cube. A short proof is included of the known fact that symmetric products preserve the property of being a compact Hilbert cube manifold (the theorem is proved here for all Hilbert cube manifolds).
We show that a compact n-polyhedron PL embeds in a product of n trees if and only if it collapses onto an (n-1)-polyhedron. If the n-polyhedron is contractible and n\ne 3 (or n=3 and the Andrews-Curtis Conjecture holds), the product of trees may be assumed to collapse onto the image of the embedding. In contrast, there exists a 2-dimensional compact absolute retract X such that X\times I^k does not embed in any product of 2+k dendrites for each k.
We study locally compact group topologies on semisimple Lie groups. We show that the Lie group topology on such a group $S$ is very rigid: every 'abstract' isomorphism between $S$ and a locally compact and $σ$-compact group $Γ$ is automatically a homeomorphism, provided that $S$ is absolutely simple. If $S$ is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all.
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