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A. Chigogidze

Publications and source records attributed to A. Chigogidze.

At least 19 recordsLinked to original sources
arXiv2013

Extension properties of Stone-Čech coronas and proper absolute extensors

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 product $[0,1)\times I^τ$ as the only locally compact and Lindelöf proper absolute extensor of weight $τ> ω$ which has the same pseudocharacter at each point.

math.GN↗
arXiv2012

Periodic and fixed points of multivalued maps on Euclidean spaces

We show, in particular, that a multivalued map $f$ from a closed subspace $X$ of $\mathbb R^n$ to ${\rm exp}_k(\mathbb R^n)$ has a point of period exactly $M$ if and only if its continuous extension $\tilde f: βX\to {\rm exp}_k(β\mathbb R^n)$ has such a point. The result also holds if one repace $\mathbb R^n$ by a locally compact Lindelöf space of finite dimension. We also show that if $f$ is a colorable map froma normal space $X$ to the space ${\mathcal K}(X)$ of all compact subsets of $X$ then its extension $\tilde f:βX\to {\mathcal K}(βX)$ is fixed-point free.

math.GN↗
arXiv2010

Fixed-point free maps of Euclidean spaces

Our main result states that every fixed-point free continuous self-map of ${\mathbb R}^{n}$ is colorable. This result can be re-formulated as follows: A continuous map $f: {\mathbb R}^{n}\to {\mathbb R}^{n}$ is fixed-point free iff $\widetilde f: β{\mathbb R}^{n}\to β{\mathbb R}^{n}$ is fixed-point free. We also obtain a generalization of this fact and present some examples.

math.GN↗
arXiv2009

$Z$-set unknotting in large cubes

This paper has been withdrawn by the author.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.

math.GN↗
arXiv2009

Local sections of Serre fibrations with 2-manifold fibers

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. Results of this paper extend Whitney theorem to the case when all fibers are homeomorphic to a given compact two-dimensional manifold.

math.GT↗
arXiv2009

Local section of Serre fibrations with 3-manifold fibers

It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. An extension of the Whitney's theorem to the case when all fibers are homeomorphic to some fixed compact two-dimensional manifold was proved by the authors \cite{BCS}. The main result of this paper proves the existence of local sections in a Serre fibration with all fibers homeomorphic to some fixed compact three-dimensional manifold.

math.GT↗
arXiv2009

Which compacta are noncommutative ARs?

We give a short answer to the question in the title: {\em dendrits}. Precisely we show that the $C^{\ast}$-algebra $C(X)$ of all complex-valued continuous functions on a compactum $X$ is projective in the category ${\mathcal C}^{1}$ of all (not necessarily commutative) unital $C^{\ast}$-algebras if and only if $X$ is an absolute retract of dimension $\dim X \leq 1$ or, equivalently, that $X$ is a dendrit.

math.OA↗
arXiv2005

On commutative and non-commutative C*-algebras with the approximate n-th root property

We say that a C*-algebra X has the approximate n-th root property (n\geq 2) if for every a\in X with ||a||\leq 1 and every ε>0 there exits b\in X such that ||b||\leq 1 and ||a-b^n||<ε. Some properties of commutative and non-commutative C*-algebras having the approximate n-th root property are investigated. In particular, it is shown that there exists a non-commutative (resp., commutative) separable unital C*-algebra X such that any other (commutative) separable unital C*-algebra is a quotient of X. Also we illustrate a commutative C*-algebra, each element of which has a square root such that its maximal ideal space has infinitely generated first Cech cohomology.

math.OA↗
arXiv2003

Extraordinary dimension theories generated by complexes

We study the extraordinary dimension function dim_{L} introduced by Ščepin. 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.

math.GN↗
arXiv2002

Notes on two conjectures in Extension Theory

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.

math.AT↗
arXiv2002

Sections of Serre fibrations with low-dimensional fibers

It was proved by H. Whitney in 1933 that it is possible to mark a point in all curves in a continuous way. The main result of this paper extends the Whitney theorem to dimensions 2 and 3. Namely, we prove that it is possible to choose a point continuously in all two-dimensional surfaces sufficiently close to a given surface, and in all 3-manifolds sufficiently close to a given 3-manifold.

math.GT↗
arXiv2002

Topological model categories generated by finite complexes

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 $L = S^{n+1}$, $n \geq 0$.

math.AT↗
arXiv2002

C*-algebras of infinite real rank

We introduce the notion of weakly (strongly) infinite real rank for unital $C^{\ast}$-algebras. It is shown that a compact space $X$ is weakly (strongly) infine-dimensional if and only if $C(X)$ has weakly (strongly) infinite real rank. Some other properties of this concept are also investigated. In particular, we show that the group $C^{\ast}$-algebra $C^{\ast}({\mathbb F}_{\infty})$ of the free group on countable number of generators has strongly infinite real rank.

math.GN↗