arXiv research

We rephrase Gromov's definition of Markov compacta, introduce a subclass of Markov compacta defined by one building block and study cohomological dimensions of these compacta. We show that for a Markov compactum $X$, $\dim_{\Z_{(p)}}X=\dim_{\Q}X$ for all but finitely many primes $p$ where $\Z_{(p)}$ is the localization…

The study describes how topological properties of Markov compacta can be inferred from their diagrammatic structures.

The study examines compact spaces resolvable by p-adic actions.

Fine shape of local compacta represented by ordinary maps.

Defines finite type Multivalued Shape using hyperspaces.

An important "stability" theorem in shape theory, due to D.A. Edwards and R. Geoghegan, characterizes those compacta having the same shape as a finite CW complex. In this note we present straightforward and self-contained proof of that theorem.

New embeddings show answer to Baker-Laidacker question can be yes or no.

Finite approximations help reconstruct countable metric and ultrametric spaces.

New compacta with unique embedding properties found.

We present some results on n-dimensional compacta lying in n-dimensional products of compacta, in particular, in products of n 1-dimensional compacta. Most of our basic results are proven under the assumption that the compacta X admit essential maps into the n-sphere. The results of the present paper may be viewed as a…

The homological dimension $d_G$ of metric compacta was introduced by Alexandroff. In this paper we provide some general properties of $d_G$, mainly with an eye towards describing the dimensional full-valuedness of compact metric spaces. As a corollary of the established properties of $d_G$, we prove that any two-dimens…

Spaces are classified as almost homology n-manifolds if their homology groups are trivial for all but the last dimension.

We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.

We prove that the Gromov boundary of every hyperbolic group is homeomorphic to some Markov compactum. Our reasoning is based on constructing a sequence of covers of $\partial G$, which is quasi-$G$-invariant wrt. the ball $N$-type (defined by Cannon) for $N$ sufficiently large. We also ensure certain additional propert…

Compacta X and Y are said to admit a stable intersection in R^n if there are maps f : X -> R^n and g : Y -> R^n such that for every sufficiently close continuous approximations f' : X -> R^n and g' : Y -> R^n of f and g we have f'(X)\cap g'(Y)\neq\emptyset. The well-known conjecture asserting that X and Y do not admit …

Classifies compact spaces by shape, finite spaces by weak homotopy.

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.

Cencelj and Dranishnikov showed that for certain nilpotent groups $G$, $K(G_{ab},1) \in \text{AE}(X)$ is equivalent to $K(G,1) \in \text{AE}(X)$ for any compacta $X$ (here $G_{ab}$ is the abelianization of $G$). We examine the same problem for solvable groups. We also give an elementary proof of this fact for any nilpo…

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 a…

For arbitrary integer n, we describe a large class of right-angled Coxeter systems for which the visual baundary (of the corresponding Coxeter-Davis complex) is homeomorphic to the n-dimensional Sierpiński compactum. We also provide a necessary and sufficient condition for a planar simplicial complex L under which the …

Paper reconstructs compact metric spaces using finite approximations and inverse persistence.

Study local properties of homogeneous ANR-spaces, proving dimension full-valuedness.

Fine shape theory extends strong shape to noncompact metrizable spaces.

Smooth knots can be embedded into a specific Menger continuum.

It is an open question (Pawlikowski) whether every finitely generated group can be realized as a fundamental group of a compact metric space. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of four dimensional Euclidean space. According to theorems of Shela…

The paper uses polyhedral expansions to capture the shape of compact metric spaces.

We prove the existence of a center, or continuous selection of a point, in the relative interior of $C^1$ embedded $k$-disks in Riemannian $n$-manifolds. If $k\le 3$ the center can be made equivariant with respect to the isometries of the manifold, and under mild assumptions the same holds for $k=4=n$. By contrast, for…

Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homot…

New groups with Menger curve boundaries found.

Spaces containing compact subsets with polyhedral complements are studied.

In accordance with the Bing-Borsuk conjecture, we show that if X is an n-dimensional homogeneous metric ANR compactum and x\in X, then there is a local basis at x consisting of connected open sets U such that the cohomological properties of \overline U and bdU are similar to the properties of the closed ball \mathbb B^…

Generalizes Milnor's axiomatic homology theorems to Polish spaces.

Introduces fine shape theory to simplify shape and antishape invariants.

A dense amalgam connects boundaries of groups split by finite subgroups.

We present a construction, called the limit of a tree system of spaces (or, less formally, a tree of spaces). The construction is designed to produce compact metric spaces that resemble fractals, out of more regular spaces, such as closed manifolds, compact polyhedra, compact Menger manifolds, etc. Such spaces are pote…

Let $f : X \lo Y$ be a map of compact metric spaces. A classical theorem of Hurewicz asserts that $\dim X \leq \dim Y +\dim f$ where $\dim f =\sup \{\dim f^{-1}(y): y \in Y \}$. The first author conjectured that {\em $\dim Y + \dim f$ in Hurewicz's theorem can be replaced by $\sup \{\dim (Y \times f^{-1}(y)): y \in Y \…

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 co…

In accordance with the Bing-Borsuk conjecture \cite{bb}, we show that if $X$ is an $n$-dimensional homogeneous metric $ANR$ compactum and $x\in X$, then there is a local basis at x consisting of connected open sets U such that the homological properties of \bar U and bdU are similar to the properties of the closed ball…

There are different definitions of homological dimension of metric compacta involving either Čech homology or exact (Steenrod) homology. In this paper we investigate the relation between these homological dimensions with respect to different groups. It is shown that all homological dimensions of a metric compactum X wi…

We introduce and study the operation, called dense amalgam, which to any tuple X_1,...,X_k of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted $\widetilde\sqcup(X_1,...,X_k)$, in which there are many appropriately distributed copies of the spaces X_1,...,X_k. We then sh…

The paper characterizes toroidal sets as attractors for flows and homeomorphisms of R^3.

New Cantor sets with high-dimensional projections discovered.

In this paper we discuss four problems regarding Markov equivalences for subclasses of loopless mixed graphs. We classify these four problems as finding conditions for internal Markov equivalence, which is Markov equivalence within a subclass, for external Markov equivalence, which is Markov equivalence between subclas…

Study Markov cubature rules for polynomial processes.

The paper estimates key metrics for linear models with Markov or hidden Markov sources.

New method improves convergence of gradient descent for non-convex, non-reversible Markov chains.

Proves Markov theorem for trivalent braids using L-move approach.

Expands Hidden Markov Model to include Markov chain observations.