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Search Results: 1 - 10 of 461868 matches for " A. Chigogidze "
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Nonmetrizable ANR's admitting a group structure are manifolds
A. Chigogidze
Mathematics , 2005,
Abstract: It is shown that a nonmetrizable ANR-space of weight $\tau > \omega$, admitting a group structure, is (topologically) an $\text{\f R}^{\tau}$-manifold.
Extraordinary dimension theories generated by complexes
A. Chigogidze
Mathematics , 2003,
Abstract: We study the extraordinary dimension function dim_{L} introduced by \v{S}\v{c}epin. An axiomatic characterization of this dimension function is obtained. We also introduce inductive dimensions ind_{L} and Ind_{L} and prove that for separable metrizable spaces all three coincide. Several results such as characterization of dim_{L} in terms of partitions and in terms of mappings into $n$-dimensional cubes are presented. We also prove the converse of the Dranishnikov-Uspenskij theorem on dimension-raising maps.
Retracts of sigma-products of Hilbert cubes
A. Chigogidze
Mathematics , 2006,
Abstract: We consider the sigma-product of the $\omega_{1}$-power of the Hilbert cube. This space is characterized among its retracts as the only one without $G_{\delta}$-points.
Valdivia compact groups are products
A. Chigogidze
Mathematics , 2007,
Abstract: It is shown that every Valdivia compact group is homeomorphic to a product of metrizable compacta.
Extension properties of Stone-?ech coronas and proper absolute extensors
A. Chigogidze
Mathematics , 2012,
Abstract: We characterize, in terms of $X$, extensional dimension of the Stone-\v{C}ech corona $\beta X \setminus X$ of locally compact and Lindel\"{o}f space $X$. The non-Lindel\"{o}f case case is also settled in terms of extending proper maps with values in $I^{\tau}\setminus L$, where $L$ is a finite complex. Further, for a finite complex $L$, an uncountable cardinal $\tau$ and a $Z_{\tau}$-set $X$ in the Tychonov cube $I^{\tau}$ we find necessary and sufficient condition, in terms of $I^{\tau}\setminus X$, for $X$ to be in the class $\operatorname{AE}([L])$. We also introduce a concept of a proper absolute extensor and characterize the product $[0,1)\times I^{\tau}$ as the only locally compact and Lindel\"{o}f proper absolute extensor of weight $\tau > \omega$ which has the same pseudocharacter at each point.
Notes on two conjectures in Extension Theory
A. Chigogidze
Mathematics , 2002,
Abstract: It is noted that conjectures about the non-existence of universal compacta and compactifications of the given extension dimension for non finitely dominated complexes are not valid for all CW complexes of the form $L \vee S^{2}$, where $L$ is of finite type and has a finite fundamental group, but is not finitely dominated.
$Z$-set unknotting in large cubes
A. Chigogidze
Mathematics , 2006,
Abstract: This paper has been withdrawn by the author. Much simpler proof of the main result was obtained which led to major changes in the presentation.
Near-homeomorphisms of Nobeling manifolds
A. Chigogidze,A. Nagorko
Mathematics , 2007,
Abstract: We characterize maps between $n$-dimensional N\"obeling manifolds that can be approximated by homeomorphisms.
Topological model categories generated by finite complexes
A. Chigogidze,A. Karasev
Mathematics , 2002,
Abstract: Our main result states that for each finite complex L the category ${\bf TOP}$ of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov's notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them -- describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups -- is obtained by letting $L = \{{\rm point}\}$. The other -- describing n-homomotopy equivalences between at most $(n+1)$-dimensional CW-complexes as maps inducing isomorophisms of k-dimensional homotopy groups with $k \leq n$ -- by letting $L = S^{n+1}$, $n \geq 0$.
Hurewicz theorem for extension dimension
N. Brodsky,A. Chigogidze
Mathematics , 2002,
Abstract: We prove a new selection theorem for multivalued mappings of C-space. Using this theorem we prove extension dimensional version of Hurewicz theorem for a closed mapping $f\colon X\to Y$ of $k$-space $X$ onto paracompact $C$-space $Y$: if for finite $CW$-complex $M$ we have $\ed Y\le [M]$ and for every point $y\in Y$ and every compactum $Z$ with $\ed Z\le [M]$ we have $\ed(f^{-1}(y)\times Z)\le [L]$ for some $CW$-complex $L$, then $\ed X\le [L]$.
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