arXiv research

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

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…

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

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…

In this paper we study compacta Y that are resolvable by a free p-adic action on a compactum of a lower dimension and focus on compacta Y whose cohomological dimension with respect to the group Z[1/p] is 1.

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^…

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…

Fine shape of local compacta represented by ordinary maps.

We develop a formalism that allows us to describe Markov compacta with finite sets of diagrams that are building blocks of the entire sequence. This encodes complex, continuous spaces with discrete collections of combinatorial objects. We show that topological properties of the limit (such as $k$-connectedness, local $…

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…

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

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.

The aim of this paper is to show how the homotopy type of compact metric spaces can be reconstructed by the inverse limit of an inverse sequence of finite approximations of the corresponding space. This recovering allows us to define inverse persistence as a new kind of persistence process.

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 …

Fine shape theory extends strong shape to noncompact metrizable spaces.

Smooth knots can be embedded into a specific Menger continuum.

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve …

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…

Milnor proved two uniqueness theorems for axiomatic (co)homology: one for pairs of compacta (1960) and another, in particular, for pairs of countable simplicial complexes (1961). We obtain their common generalization: the Eilenberg-Steenrod axioms along with Milnor's map excision axiom and a (non-obvious) common genera…

We introduce and develop fine shape, which has a very simple definition and aims to supersede all previously known shape theories for metrizable spaces. The problem with known shape theories of metrizable spaces is illustrated by the following bizarre situation. Čech cohomology is an invariant of shape, and a fortiori …

Spaces containing compact subsets with polyhedral complements are studied.

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…

In this paper we focus on compacta $K \subseteq \mathbb{R}^3$ which possess a neighbourhood basis that consists of nested solid tori $T_i$. We call these sets toroidal. In \cite{hecyo1} we defined the genus of a toroidal set as a generalization of the classical notion of genus from knot theory. Here we introduce the se…

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…

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…

Introduces new types of homogeneous spaces and their properties.

Introduces homogeneity supermanifolds for studying graded structures.

Recently, it is shown that each regular homogeneous Finsler space $M$ admits at least one homogeneous geodesic through any point $o\in M$. The purpose of this article is to study the existence of homogeneous geodesics on singular homogeneous $(α,β)$-spaces, specially, homogeneous Kropina spaces. We show that any homoge…

In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogen…

We prove that the Penrose limit of a spacetime along a homogeneous geodesic is a homogeneous plane wave spacetime and that the Penrose limit of a reductive homogeneous spacetime along a homogeneous geodesic is a Cahen--Wallach space. We then consider several homogenous examples to show that these results are indeed sha…

We prove that under some purely algebraic conditions every locally homogeneous structure modelled on some homogeneous space is induced by a locally homogeneous structure modelled on a different homogeneous space.

Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.

We show that a Lorentzian homogeneous space admitting a homogeneous structure of type T1 + T3 is either a (locally) symmetric space or a singular homogeneous plane wave.