arXiv subjects

M. Cencelj

Publications and source records attributed to M. Cencelj.

18 recordsLinked to original sources

arXiv2021

On the Browder-Levine-Novikov embedding theorems

In this survey we present applications of the ideas of complement and neighborhood in the theory embeddings of manifolds into Euclidean space (in codimension at least three). We describe how the combination of these ideas gives a reduction of embeddability and isotopy problems to algebraic problems. We present a more clarified exposition of the Browder-Levine theorem on realization of normal systems. Most of the survey is accessible to non-specialists in the theory of embeddings.

math.GT↗

arXiv2015

Asymptotic dimension of coarse spaces via maps to simplicial complexes

It is well-known that a paracompact space $X$ is of covering dimension at most $n$ if and only if any map $f\colon X\to K$ from $X$ to a simplicial complex $K$ can be pushed into its $n$-skeleton $K^{(n)}$. We use the same idea to characterize asymptotic dimension in the coarse category of arbitrary coarse spaces. Continuity of the map $f$ is replaced by variation of $f$ on elements of a uniformly bounded cover. The same way one can generalize Property A of G.Yu to arbitrary coarse spaces.

math.GT↗

arXiv2013

Coarse amenability versus paracompactness

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using partitions of unity. In this paper we go deeper into divulging analogies between coarse amenability and paracompactness. In particular, we define a new coarse analog of paracompactness modelled on the defining characteristics of expanders. That analog gives an easy proof of three categories of spaces being coarsely non-amenable: expander sequences, graph spaces with girth approaching infinity, and unions of powers of a finite non-trivial group.

math.MG↗

arXiv2012

Classification of knotted tori in the 2-metastable dimension

This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings $N\to S^m$. We study the specific case of knotted tori, i. e. the embeddings $S^p \times S^q \to S^m$. The classification of knotted tori up to isotopy in the metastable dimension range $m>p+\frac{3}{2}q+2$, $p\le q$, was given by A. Haefliger, E. Zeeman and A. Skopenkov. We consider the dimensions below the metastable range, and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension: Theorem. Assume that $p+\frac{4}{3}q+2 2p+q+2$. Then the set of smooth embeddings $S^p \times S^q \to S^m$ up to isotopy is infinite if and only if either $q+1$ or $p+q+1$ is divisible by 4. Our approach to the classification is based on an analogue of the Koschorke exact sequence from the theory of link maps. This sequence involves a new $β$-invariant of knotted tori. The exactness is proved using embedded surgery and the Habegger-Kaiser techniques of studying the complement.

math.GT↗

arXiv2009

Asymptotic dimension, Property A, and Lipschitz maps

It is well-known that a paracompact space X is of covering dimension n if and only if any map f from X to a simplicial complex K can be pushed into its n-skeleton. We use the same idea to define dimension in the coarse category. It turns out the analog of maps f from X to K is related to asymptotically Lipschitz maps, the analog of paracompact spaces are spaces related to Yu's Property A, and the dimension coincides with Gromov's asymptotic dimension.

math.MG↗

arXiv2009

A combinatorial approach to coarse geometry

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. That embedding can be realized by either Rips complexes or analogs of Roe's anti-\v\{C}ech approximations of spaces. In that model the asymptotic dimension being at most n means that for each k there is m > k such that the bonding map from K_k to K_m factors (up to contiguity) through an n-dimensional complex. One can give a similar characterization of Property A of G.Yu. Using our approach we give a simple proof of a characterization of geodesic spaces that are coarsely equivalent to simplicial trees (a result of Fujiwara and Whyte).

math.MG↗

arXiv2008

Property A and asymptotic dimension

The purpose of this note is to characterize the asymptotic dimension $asdim(X)$ of metric spaces $X$ in terms similar to Property A of Yu: If $(X,d)$ is a metric space and $n\ge 0$, then the following conditions are equivalent: [a.] $asdim(X,d)\leq n$, [b.] For each $R,ε> 0$ there is $S > 0$ and finite non-empty subsets $A_x\subset B(x,S)\times N$, $x\in X$, such that $\frac{| A_xΔA_y|}{| A_x\cap A_y|} < ε$ if $d(x,y) < R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\in X$, [c.] For each $R > 0$ there is $S > 0$ and finite non-empty subsets $A_x\subset B(x,S)\times N$, $x\in X$, such that $\frac{| A_xΔA_y|}{| A_x\cap A_y|} < \frac{1}{n+1}$ if $d(x,y) < R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\in X$.

math.MG↗

arXiv2008

Bockstein theorem for nilpotent groups

We extend the definition of Bockstein basis $σ(G)$ to nilpotent groups $G$. A metrizable space $X$ is called a {\it Bockstein space} if $\dim_G(X) = \sup\{\dim_H(X) | H\in σ(G)\}$ for all Abelian groups $G$. Bockstein First Theorem says that all compact spaces are Bockstein spaces. Here are the main results of the paper: Let $X$ be a Bockstein space. If $G$ is nilpotent, then $\dim_G(X) \leq 1$ if and only if $\sup\{\dim_H(X) | H\inσ(G)\}\leq 1$. $X$ is a Bockstein space if and only if $\dim_{\Z_{(l)}} (X) = \dim_{\hat{Z}_{(l)}}(X)$ for all subsets $l$ of prime numbers.

math.GT↗

arXiv2008

Homotopy type of the complement of an immersion and classification of embeddings of tori

This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings $S^p\times S^q\to S^m$, which we call knotted tori. The set of knotted tori in the the space of sufficiently high dimension, namely in the metastable range $m\ge p+3q/2+2$, $p\le q$, which is a natural limit for the classical methods of embedding theory, has been explicitely described earlier. The aim of this note is to present an approach which allows for results in lower dimension.

math.GT↗

arXiv2008

Preserving $Z$-sets by Dranishnikov's resolution

We prove that Dranishnikov's $k$-dimensional resolution $d_k\colon μ^k\to Q$ is a UV$^{n-1}$-divider of Chigogidze's $k$-dimensional resolution $c_k$. This fact implies that $d_k^{-1}$ preserves $Z$-sets. A further development of the concept of UV$^{n-1}$-dividers permits us to find sufficient conditions for $d_k^{-1}(A)$ to be homeomorphic to the Nöbeling space $ν^k$ or the universal pseudoboundary $σ^k$. We also obtain some other applications.

math.GN↗

arXiv2008

Compact maps and quasi-finite complexes

The simplest condition characterizing quasi-finite CW complexes $K$ is the implication $Xτ_h K\implies β(X)τK$ for all paracompact spaces $X$. Here are the main results of the paper: Theorem: If $\{K_s\}_{s\in S}$ is a family of pointed quasi-finite complexes, then their wedge $\bigvee\limits_{s\in S}K_s$ is quasi-finite. Theorem: If $K_1$ and $K_2$ are quasi-finite countable complexes, then their join $K_1\ast K_2$ is quasi-finite. Theorem: For every quasi-finite CW complex $K$ there is a family $\{K_s\}_{s\in S}$ of countable CW complexes such that $\bigvee\limits_{s\in S} K_s$ is quasi-finite and is equivalent, over the class of paracompact spaces, to $K$. Theorem: Two quasi-finite CW complexes $K$ and $L$ are equivalent over the class of paracompact spaces if and only if they are equivalent over the class of compact metric spaces. Quasi-finite CW complexes lead naturally to the concept of $Xτ{\mathcal F}$, where ${\mathcal F}$ is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept.

math.GT↗

arXiv2007

Classification of framed links in 3-manifolds

We present a short proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in details: {\bf Theorem.} Let $M$ be a connected oriented closed smooth 3-manifold. Let $L_1(M)$ be the set of framed links in $M$ up to a framed cobordism. Let $°:L_1(M)\to H_1(M;\Z)$ be the map taking a framed link to its homology class. Then for each $α\in H_1(M;\Z)$ there is a 1-1 correspondence between the set $°\nolimits^{-1}α$ and the group $\Bbb Z_{2d(α)}$, where $d(α)$ is the divisibility of the projection of $α$ to the free part of $H_1(M;\Bbb Z)$.

math.GT↗

arXiv2007

On $π- π$ theorem for manifold pairs with boundaries

Surgery obstruction of a normal map to a simple Poincare pair $(X,Y)$ lies in the relative surgery obstruction group $L_*(π_1(Y)\toπ_1(X))$. A well known result of Wall, the so called $π$-$π$ theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincare pair with $π_1(X)\congπ_1(Y)$ is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced surgery obstruction group for manifold pairs $LP_*$ and splitting obstruction groups $LS_*$. In the present paper we formulate and prove for manifold pairs with boundaries the results which are similar to the $π$-$π$ theorem. We give direct geometric proofs, which are based on the original statements of Wall's results and apply obtained results to investigate surgery on filtered manifolds.

math.GT↗

arXiv2006

On the splitting problem for manifold pairs with boundaries

The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category of manifolds with boundaries. In the $ rel \partial$ case the surgery on a manifold pair is considered with the given fixed manifold structure on the boundary. In the relative case the surgery on the manifold with boundary is considered without fixing maps on the boundary. Consider a normal map to a manifold pair $(Y, \partial Y)\subset (X, \partial X)$ with boundary which is a simple homotopy equivalence on the boundary $\partial X$. This map defines a mixed structure on the manifold with the boundary in the sense of Wall. We introduce and study groups of obstructions to splitting of such mixed structures along submanifold with boundary $(Y, \partial Y)$. We describe relations of these groups to classical surgery and splitting obstruction groups. We also consider several geometric examples.

math.GT↗

arXiv2006

Sublinear Higson corona and Lipschitz extensions

The purpose of the paper is to characterize the dimension of sublinear Higson corona $ν_L(X)$ of $X$ in terms of Lipschitz extensions of functions: Theorem: Suppose $(X,d)$ is a proper metric space. The dimension of the sublinear Higson corona $ν_L(X)$ of $X$ is the smallest integer $m\ge 0$ with the following property: Any norm-preserving asymptotically Lipschitz function $f'\colon A\to \R^{m+1}$, $A\subset X$, extends to a norm-preserving asymptotically Lipschitz function $g'\colon X\to \R^{m+1}$. One should compare it to the result of Dranishnikov \cite{Dr1} who characterized the dimension of the Higson corona $ν(X)$ of $X$ is the smallest integer $n\ge 0$ such that $\R^{n+1}$ is an absolute extensor of $X$ in the asymptotic category $\AAA$ (that means any proper asymptotically Lipschitz function $f\colon A\to \R^{n+1}$, $A$ closed in $X$, extends to a proper asymptotically Lipschitz function $f'\colon X\to \R^{n+1}$). \par In \cite{Dr1} Dranishnikov introduced the category $\tilde \AAA$ whose objects are pointed proper metric spaces $X$ and morphisms are asymptotically Lipschitz functions $f\colon X\to Y$ such that there are constants $b,c > 0$ satisfying $|f(x)|\ge c\cdot |x|-b$ for all $x\in X$. We show $\dim(ν_L(X))\leq n$ if and only if $\R^{n+1}$ is an absolute extensor of $X$ in the category $\tilde\AAA$. \par As an application we reprove the following result of Dranishnikov and Smith \cite{DRS}: Theorem: Suppose $(X,d)$ is a proper metric space of finite asymptotic Assouad-Nagata dimension $\asdim_{AN}(X)$. If $X$ is cocompact and connected, then $\asdim_{AN}(X)$ equals the dimension of the sublinear Higson corona $ν_L(X)$ of $X$.

math.MG↗

arXiv2006

Codimension two PL embeddings of spheres with nonstandard regular neighborhoods

For a given polyhedron $K\subset M$ the notation $R_M(K)$ denotes a regular neighborhood of $K$ in $M$. We study the following problem: find all pairs $(m,k)$ such that if $K$ is a compact $k$-polyhedron and $M$ a PL $m$-manifold, then $R_M(fK)\cong R_M(gK)$, for each two homotopic PL embeddings $f,g:K\to M$. We prove that $R_{S^{k+2}}(S^k)\not\cong S^k\times D^2$ for each $k\ge2$ and {\it some} PL sphere $S^k\subset S^{k+2}$ (even for {\it any} PL sphere $S^k\subset S^{k+2}$ having an isolated non-locally flat point with the singularity $S^{k-1}\subset S^{k+1}$ such that $π_1(S^{k+1}-S^{k-1})\not\cong\Z$).

math.GT↗

arXiv2006

Hurewicz-Serre Theorem in extension theory

The paper is devoted to generalizations of Cencelj-Dranishnikov theorems relating extension properties of nilpotent CW complexes to its homology groups. Here are the main results of the paper: \par {\bf Theorem}. Suppose $L$ is a nilpotent CW complex and $F$ is the homotopy fiber of the inclusion $i$ of $L$ into its infinite symmetric product $SP(L)$. If $X$ is a metrizable space such that $XτK(H_k(L),k)$ for all $k\ge 1$, then $XτK(π_k(F),k)$ and $XτK(π_k(L),k)$ for all $k\ge 2$. \par {\bf Theorem}. Let $X$ be a metrizable space such that $\dim(X) < \infty$ or $X\in ANR$. Suppose $L$ is a nilpotent CW complex and $SP(L)$ is its infinite symmetric product. If $XτSP(L)$, then $XτL$ in the following cases: \begin{itemize} \item[a.] $H_1(L)$ is finitely generated. \item[b.] $H_1(L)$ is a torsion group. \end{itemize}

math.AT↗

arXiv2005

Algebraic properties of quasi-finite complexes

A countable CW complex $K$ is quasi-finite (as defined by A.Karasev) if for every finite subcomplex $M$ of $K$ there is a finite subcomplex $e(M)$ such that any map $f:A\to M$, where $A$ is closed in a separable metric space $X$ satisfying $XτK$, has an extension $g:X\to e(M)$. Levin's results imply that none of the Eilenberg-MacLane spaces $K(G,2)$ is quasi-finite if $G\ne 0$. In this paper we discuss quasi-finiteness of all Eilenberg-MacLane spaces. More generally, we deal with CW complexes with finitely many nonzero Postnikov invariants. Here are the main results of the paper: Suppose $K$ is a countable CW complex with finitely many nonzero Postnikov invariants. If $π_1(K)$ is a locally finite group and $K$ is quasi-finite, then $K$ is acyclic. Suppose $K$ is a countable non-contractible CW complex with finitely many nonzero Postnikov invariants. If $π_1(K)$ is nilpotent and $K$ is quasi-finite, then $K$ is extensionally equivalent to $S^1$.

math.GT↗