arXiv research

New space of tensorial bodies defined, properties and representatives studied.

We call a value $y=f(x)$ of a map $f:X\to Y$ dimensionally regular if $\dim X\le \dim(Y\times f^{-1}(y))$. It was shown in \cite{first-exotic} that if a map $f:X\to Y$ between compact metric spaces does not have dimensionally regular values, then $X$ is a Boltyanskii compactum, i.e. a compactum satisfying the equality …

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

Study on the topology of tensorial bodies, showing they are homeomorphic to a product space.

The paper proves a Whitehead theorem for fine shape spaces.

Let X be a compactum such that dim_Q X < n+1, n>1. We prove that there is a Q-acyclic resolution r: Z-->X from a compactum Z of dim < n+1. This allows us to give a complete description of all the cases when for a compactum X and an abelian group G such that dim_G X < n+1, n>1 there is a G-acyclic resolution r: Z-->X fr…

Author simplifies and generalizes p-adic integer action construction.

We investigate some topological properties of a normal functor $H$ introduced earlier by Radul which is some functorial compactification of the Hartman--Mycielski construction HM. We prove that the pair ($HX$, HM$Y$) is homeomorphic to the pair $(Q,σ)$ for each nondegenerated metrizable compactum $X$ and each dense $σ$…

We prove that for every $n>2$, the Banach-Mazur compactum Q(n) is the compactification of a Hilbert cube manifold by the Euclidean point. For $n=2$ this result was proved earlier.

We show that an n-dimensional compactum X embeds in R^m, where m>3(n+1)/2, if and only if X x X - Δadmits an equivariant map to S^{m-1}. In particular, X embeds in R^{2n}, n>3, iff the top power of the (twisted) Euler class of the factor-exchanging involution on X x X - Δis trivial. Assuming that X quasi-embeds in R^{2…

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^…

We prove that for every compactum $X$ and every integer $n \geq 2$ there are a compactum $Z$ of $\dim \leq n+1$ and a surjective $UV^{n-1}$-map $r: Z \lo X$ such that for every abelian group $G$ and every integer $k \geq 2$ such that $\dim_G X \leq k \leq n$ we have $\dim_G Z \leq k$ and $r$ is $G$-acyclic.

In this paper we further investigate the geometry of monads of order-preserving functionals and of positively homogeneous functionals. We prove that for any compactum X with $w(X) = τ$ the map $μ_F X$, where $F\in\{O,OH\}$, is homeomorphic to trivial $I^τ$-fibration if and only if $X$ is openly generated $χ$-homogeneou…

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…

The main results of this paper are: (1) If a space $X$ can be embedded as a cellular subspace of $\mathbb{R}^n$ then $X$ admits arbitrary fine open coverings whose nerves are homeomorphic to the $n$-dimensional cube $\mathbb{D}^n$; (2) Every $n$-dimensional cell-like compactum can be embedded into $(2n+1)$-dimensional …

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…

In extension theory, in particular in dimension theory, it is frequently useful to represent a given compact metrizable space X as the limit of an inverse sequence of compact polyhedra. We are going to show that, for the purposes of extension theory, it is possible to replace such an X by a better metrizable compactum …

In this note we introduce the concept of a quasi-finite complex. Next, we show that for a given countable and locally finite CW complex L the following conditions are equivalent: (i) L is quasi-finite. (ii) There exists a [L]-invertible mapping of a metrizable compactum X with e-dim X = [L] onto the Hilbert cube. Final…

New representation of PSL2(R) on infinite hyperbolic space via convex bodies.

In this work we define a new pseudometric in $\mathcal K^n_*$, the hyperspace of all non-degenerated compact convex sets of $\mathbb R^n$, which is invariant under similarities. We will prove that the quotient space generated by this pseudometric (which is the orbit space generated by the natural action of the group of…

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

Fine shape of local compacta represented by ordinary maps.

The paper shows how to use fine shape to understand infinite-dimensional spaces.

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.

Raymond and Wiliams in "Examples of p-adic transformation groups", Ann. of Math. (2) 78 (1963) 92-106, constructed an action of the p-adic integers on an n-dimensional compactum, n>1, with the orbit space of dimension n+2. We present a simpler construction of such an example.

We introduce and investigate the notion of (strong) $K^n_G$-manifolds, where $G$ is an abelian group. One of the result related to that notion (Theorem 3.4) implies the following partial answer to the Bing-Borsuk problem \cite{bb}, whether any partition of a homogeneous metric $ANR$-space $X$ of dimension $n$ is cyclic…

We study a model situation in which direct limit ($\text{colim}$) and inverse limit ($\lim$) do not commute, and offer some computations of their "commutator". The homology of a separable metrizable space $X$ has two well-known approximants: $qH_n(X)$ ("Čech homology") and $pH_n(X)$ ("Čech homology with compact support…

We show that every automorphism $α$ of a free group $F_k$ of finite rank $k$ has {\it asymptotically periodic} dynamics on $F_k$ and its boundary $\partial F_k$: there exists a positive power $α^q$ such that every element of the compactum $F_k \cup \partial F_k$ converges to a fixed point under iteration of $α^q$.

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 …

In this paper we investigate the functors of OH of positively homogenous functionals and OS of semiadditive functionals. We show that OH(X) is AR if and only if X is openly generated, and OS(X) is AR if and only if X is an openly generated compactum of weight less than $ω_1$. Also, we investigate the multiplication map…

Proves existence of equivariant maps avoiding diagonal in n-space.

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…

Spaces containing compact subsets with polyhedral complements are studied.

A compactum X is an `absolute cone' if, for each of its points x, the space X is homeomorphic to a cone with x corresponding to the cone point. In 1971, J. de Groot conjectured that each n-dimensional absolute cone is an n-cell. In this paper, we give a complete solution to that conjecture. In particular, we show that …

Study bounds topological entropy of toroidal attractors.

Let $M$ be a complete metric $ANR$-space such that for any metric compactum $K$ the function space $C(K,M)$ contains a dense set of Bing (resp., Krasinkiewicz) maps. It is shown that $M$ has the following property: If $f\colon X\to Y$ is a perfect surjection between metric spaces, then $C(X,M)$ with the source limitati…

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…

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…

A classical theorem of Alexandroff states that every $n$-dimensional compactum $X$ contains an $n$-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz manifolds,…

Paper calculates homotopy types of non-compact surfaces using groupoids.

The paper studies time-optimal problems on specific Lie groups, describing orbits and integrals.

In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore,…

Study characterizes compact Einstein-type manifolds with boundary.

Develops Poisson and Dirac manifolds of compact types with applications.

We prove the following result announced in Todorov and Valov: Any homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum with $\check{H}^n(X;G)\neq…

We compute the index of a Callias-type operator with APS boundary condition on a manifold with compact boundary in terms of combination of indexes of induced operators on a compact hypersurface. Our result generalizes the classical Callias-type index theorem to manifolds with compact boundary.