Alexander N. Dranishnikov

Alexander N. Dranishnikov
University of Florida | UF · Department of Mathematics

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155
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Publications (155)
Article
We introduce the notion of spaces with weak relative cohomology and show the acyclicity of the complement $Q\setminus X$ in the Hilbert cube $Q$ of a compactum $X$ with weak relative cohomology. As a corollary we obtain the acyclicity of the complement results when (a) $X$ is weakly infinite dimensional; (b) $X$ has finite cohomological dimension.
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We show that the Lusternik-Schnirelmann category of the homotopy cofiber of the diagonal map for non-orientable surfaces equals three.
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We present a reduction of the Hilbert-Smith conjecture in the case of the finite dimensional orbit space to some algebraic topology problems.
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We reduced Rudyak's conjecture that a degree one map between closed manifolds cannot raise the Lusternik-Schnirelmann category to the computation of the category of the product of two lens spaces $L^n_p\times L_q^n$ with relatively prime $p$ and $q$. We have computed $cat(L^n_p\times L^n_q)$ for values of $p,q>n/2$. It turns out that our computatio...
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We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity TC(X) and monoidal topological complexity TCM (X). Using these results we provide lower and upper bounds for the topological complexity of the wedge X∨Y. We use these bounds to give a counterexample to the conjecture asserting that TC(X′) ≤...
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The notion of the decomposition complexity was introduced in [14] using a game theoretic approach. We introduce a notion of straight decomposition complexity and compare it with the original as well with the asymptotic property C. Then we define a game theoretic analog of Haver's property C in the classical dimension theory and compare it with the...
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Gromov's Conjecture states that for a closed $n$-manifold $M$ with positive scalar curvature the macroscopic dimension of its universal covering $\tilde M$ satisfies the inequality $\dim_{mc}\tilde M\le n-2$\cite{G2}. We prove this inequality for totally non-spin $n$-manifolds whose fundamental group is a virtual duality group with $vcd\ne n$. In t...
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We present a proof of the following theorem of Levin: For every connected CW complex $K$ there is a simply connected CW complex $K^+$ obtained from $K$ by attaching cells of dimension 2 and 3 such that the inclusion $K\to K^+$ induces isomorphisms of homology groups in dimension $>1$.
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We prove the inequality $$ \dim_{mc}\Wi M\le n-2$$ for the macroscopic dimension of the universal covers $\Wi M$ of almost spin $n$-manifolds $M$ with positive scalar curvature whose fundamental group $\pi_1(M)$ is a virtual duality group that satisfies the coarse Baum-Connes conjecture.
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We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate $$TC(X)\le TC(\pi_1(X))+\dim X$$ of the topological complexity of a space in terms of its dimension and the complexity of its fundamental group.
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We give a homological characterization of n-manifolds whose universal covering (formula presented) has Gromov’s macroscopic dimension dimmc (formula presented) <n. As a result, we distinguish dimmc from the macroscopic dimension dimMC defined by the author in an earlier paper. We prove the inequality dimmc (image found) < dimMC (formula presented)...
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The notion of the decomposition complexity was introduced in [GTY] using a game theoretical approach. We introduce a notion of straight decomposition complexity and compare it with the original as well with the asymptotic property C. Then we define a game theoretical analog of Haver's property C in the classical dimension theory and compare it with...
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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 $\dim(X\times X)=2\dim X-1$. I...
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We show that for a rationally inessential orientable closed $n$-manifold $M$ whose fundamental group $\pi$ is a duality group the macroscopic dimension of its universal cover is strictly less than $n$:$$ \dim_{MC}\Wi M<n.$$ As a corollary we obtain the following 0.1 Theorem. The inequality $ \dim_{MC}\Wi M<n$ holds for the universal cover of a clos...
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We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity $TC(X)$ and monoidal topological complexity $TC^M(X)$. Using these results we provide lower and upper bounds for the topological complexity of the wedge $X\vee Y$. We use these bounds to give a counterexample to the conjecture asserting tha...
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We show that for every pair (X,Y)(X,Y) of ANR compacta, Y⊂XY⊂X, the free abelian topological group applied to a collapsing map q:X→X/Yq:X→X/Y produces a locally trivial bundle A(q):A(X)→A(X/Y)A(q):A(X)→A(X/Y) with the fiber A(Y)A(Y). As a result we obtain a short proof of the classical Dold–Thom theorem which states that H˜n(X)=πn(A(X)) for all com...
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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 $\sup \{\dim (Y \times f^{-1}(y)): y \in Y \}$}. We disprove this conjectu...
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We show that in dimensions $>1$ the cohomology groups of the Higson compactification of the hyperbolic space $\H^n$ with respect to the $C_0$ coarse structure are trivial. Also we prove that the cohomology groups of the Higson compactification of $\H^n$ for the bounded coarse structure are trivial in all even dimensions.
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We show that, for a rationally inessential orientable closed n-manifold M whose fundamental group is a duality group, the macroscopic dimension of its universal cover [(M)\tilde]\tilde M is strictly less than n: dim MC [(M)\tilde] < n\tilde M < n. As a corollary, we obtain the following partial result towards Gromov’s conjecture The inequality di...
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M. Gromov asked whether the macroscopic dimension of rationally essential n-dimensional manifolds equals n. We show that the answer depends only on the corresponding group homology class and give an affirmative answer for certain classes. In particular, the answer is positive for manifolds with amenable fundamental groups.
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Dimension growth functions of groups have been introduced by Gromov in 1999. We prove that every solvable finitely generated subgroups of the R. Thompson group $F$ has polynomial dimension growth while the group $F$ itself, and some solvable groups of class 3 have exponential dimension growth with exponential control. We describe connections betwee...
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We construct a counterexamples in dimensions $n>3$ to Gromov's conjecture \cite{Gr1} that the macroscopic dimension of rationally essential $n$-dimensional manifolds equals $n$.
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We prove the Gromov conjecture on the macroscopic dimension of the universal covering of a closed spin manifold with a positive scalar curvature under the following assumptions on the fundamental group. 0.1 0.1 . Theorem. Suppose that a discrete group π \pi has the following properties: 1 1 . The Strong Novikov Conjecture holds for π \pi . 2 2 . Th...
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We prove that $$ \cat X\le cd(\pi_1(X))+\bigg\lceil\frac{\dim X-1}{2}\bigg\rceil$$ for every CW complex $X$ where $cd(\pi_1(X))$ denotes the cohomological dimension of the fundamental group of $X$.
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We give a short answer to the question in the title: dendrits. Precisely we show that the C*-algebra C(X) of all complex-valued continuous functions on a compactum X is projective in the category C1 of all (not necessarily commutative) unital C*-algebras if and only if X is an absolute retract of dimension dimX⩽1 or, equivalently, that X is a dendr...
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It follows from a theorem of Gromov that the stable systolic category of a closed manifold is bounded from below by the rational cup-length of the manifold. In the paper we study the inequality in the opposite direction. In particular, combining our results with Gromov's theorem, we prove the equality of stable systolic category and rational cup-le...
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Given a closed manifold M , we prove the upper bound cat sys (M) ≤ dim M+cd(π1M) 2 for the systolic category of M , where "cd" is the cohomological dimension. We apply this upper bound to deduce the inequality cat sys (M) ≤ cat LS (M) between the systolic category and the Lusternik–Schnirelmann cat-egory of 4-manifolds. Furthermore we obtain a lowe...
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Given a closed manifold M, we prove the upper bound of $${1 \over 2}(\dim M + {\rm{cd}}({{\rm{\pi }}_1}M))$$ for the number of systolic factors in a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov’s systolic inequalities. Here “cd” is the cohomological dimension. We apply this upper bound to show that, in the case o...
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We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We also obtain some general results on the relations between the fundamental group of a closed manifold M...
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The asymptotic dimension theory was founded by Gromov [M. Gromov, Asymptotic invariants of infinite groups,295] in the early 90s. In this paper we give a survey of its recent history where we emphasize two of its features: an analogy with the dimension theory of compact metric spaces and applications to the theory of discrete groups. Counting dimen...
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We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactification of an EΓ. We then show that for groups of finite asymptotic dimension, the Higson compactification is mod p acyclic for all p and deduce the integral Novikov conjecture for these groups. © 2007 Wiley...
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We prove that for any group π with cohomological dimension at least n the n th power of the Berstein class of π is nontrivial. This allows us to prove the following Berstein–Svarc theorem for all n : Theorem . For a connected complex X with dim X = cat X = n , we have $\ber_X^n$ ≠ 0 where $\ber_X$ is the Berstein class of X . Previously it was know...
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The following inequality \[ c a t L S X ≤ c a t L S Y + ⌈ h d ( X ) − r r + 1 ⌉ \mathrm {cat}_{\mathrm {LS}} X\le \mathrm {cat}_{\mathrm {LS}} Y+\bigg \lceil \frac {hd(X)-r}{r+1}\bigg \rceil \] holds for every locally trivial fibration f : X → Y f:X\to Y between A N E ANE spaces which admits a section and has the r r -connected fiber, where h d ( X...
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The classical Eilenberg-Borsuk theorem on extension of partial mappings into a sphere is generalized to the case of an arbitrary complex . It is formulated in terms of extraordinary dimension theory, which is developed in the present paper. When is an Eilenberg-MacLane complex, the result can be expressed in terms of cohomological dimension theory....
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For any positive integer the author constructs a continuous mapping of the -dimensional Menger compactum onto itself that is universal in the class of mappings between -dimensional compacta, i.e., for any continuous mapping between -dimensional compacta there exist imbeddings of and in such that the restriction of to is homeomorphic to . The mappin...
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The article provides a construction of an infinite-dimensional compact space of dimension 2 modulo p for any p. A characterization of compact spaces n-dimensional modulo p in terms of inverse spectrum of polyhedra is given. It is proved that compact spaces n-dimensional modulo p, and only these spaces, are images of n-dimensional compact spaces und...
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It is proved that there exists a free action of an arbitrary zero-dimensional compact group on every Menger manifold. It is shown that in the case of a finite group G such a G-action on the n-dimensional Menger compactum is unique and universal in the class of free G-actions on n-dimensional compacta. Bibliography: 22 titles.
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It is determined under what conditions the standard problem of extension of a mapping to the whole space is solvable for any closed subset . For finite-dimensional metric compacta and -complexes this is equivalent to the system of inequalities -. The result is applied to finding conditions for general position of a compactum in a Euclidean space.
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We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient condition for ensuring a lowe...
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We show that every Gromov hyperbolic group Γ admits a quasi-isometric embedding into the product of n+1 binary trees, where n=dim∂∞Γ is the topological dimension of the boundary at infinity of Γ.
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We prove the inequality $$ \as A\ast_CB\le\max\{\as A,\as B,\as C+1\} $$ and we apply this inequality to show that the asymptotic dimension of any right-angled Coxeter group does not exceed the dimension of its Davis' complex.
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We construct a limit aperiodic coloring of hyperbolic groups. Also we construct limit strongly aperiodic strictly balanced tilings of the Davis complex for all Coxeter groups. In [BDS] we constructed a quasi-isometric embedding of hyperbolic groups into a finite product of binary trees. First we implemented such construction for hyperbolic Coxeter...
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We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $\bL$-classes. In dimension $>5$ we identify all such homotopy equivalences to $M$ w...
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The Silences of the Archives, the Reknown of the Story. The Martin Guerre affair has been told many times since Jean de Coras and Guillaume Lesueur published their stories in 1561. It is in many ways a perfect intrigue with uncanny resemblance, persuasive deception and a surprizing end when the two Martin stood face to face, memory to memory, befor...
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We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdimZ X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic...
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For a large class of metric spaces X including discrete groups we prove that the asymptotic Assouad–Nagata dimension AN-asdimX of X coincides with the covering dimension dim(νLX) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality AN-asdim(X×R)=AN-asdimX+1. We note that the si...
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We show that the universal cover of an aspherical manifold whose fundamental groups has finite asymptotic dimension in sense of Gromov is hypereuclidean after crossing with some Euclidean space
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We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite-dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound...
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We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [B-D2] to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their Hirsch length.
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There are two basic types of approximation problems in topology: to approximate a given topological space by better spaces; and to approximate a given map by better maps. This chapter discusses the examples of both the types. The two approximation problems that appear in different areas of topology are in place as keystone problems. The classical t...
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Let M be a nilpotent CW-complex. We give necessary and sufficient cohomological dimension theory conditions for a finite-dimensional metric compactum X so that every map A→M, where A is a closed subset of X, can be extended to a map X→M.This is a generalization of a result by Dranishnikov [Mat. Sb. (1991)] where such conditions were found for simpl...
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This survey was compiled from lectures and problem sessions at the International Conference on Geometric Topology at the Mathematical Research and Conference Center in Bedlewo, Poland in July 2005.
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We prove that a finitely generated, right-angled, hyperbolic Coxeter group can be quasiisometrically embedded into the product of n binary trees, where n is the chromatic number of the group. As application we obtain certain strongly aperiodic tilings of the Davis complex of these groups.
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Bestvina and Mess [BM] proved a remarkable formula for torsion free hyperbolic groups $$ \dim_L\partial\Gamma=cd_L\Gamma-1 $$ connecting the cohomological dimension of a group $\Gamma$ with the cohomological dimension of its boundary $\partial\Gamma$. In [Be] Bestvina introduced a notion of $\sZ$-structure on a discrete group and noticed that his f...
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This is a detailed introductory survey of the cohomological dimension theory of compact metric spaces.
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We construct closed (k - 1)-connected manifolds of dimensions $\geq 4k - 1$ that possess non-trivial rational Massey triple products. We also construct examples of manifolds M such that all the cup-products of elements of $H^{k}(M)$ vanish, while the group $H^{3k-1}(M; \mathbb{Q})$ is generated by Massey products: such examples are useful for the t...
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We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper boun...
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We construct a universal space for the class of proper metric spaces of bounded geometry and of given asymptotic dimension. As a consequence of this result, we establish coincidence of asymptotic dimension with asymptotic inductive dimension.
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We show that finitely generated groups with a polynomial dimension growth have Yu’s property A and give an example of such groups.
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We prove that a right angled Coxeter group with chromatic number n can be embedded in a bilipschitz way into the product of n locally finite trees. We give applications of this result to various embedding problems and determine the hyperbolic rank of products of exponentially branching trees.
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For every prime p and each n=2,3,…,∞, we constructed in [A.N. Dranishnikov, J.B. West, Topology Appl. 80 (1997) 101–114] an action of G=∏∞i=1(Z/pZ) on a two-dimensional compact metric space X with n-dimensional orbit space. The argument of [A.N. Dranishnikov, J.B. West, Topology Appl. 80 (1997) 101–114] had a gap in Lemma 15 which affected the main...
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We prove an exact formula for the asymptotic dimension of a free product. Our main theorem states that if A and B are finitely gen- erated groups and with asdimA = n, and asdimB n then asdim(A B) = max{n,1}.
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We construct closed $(k-1)$-connected manifolds of dimensions $\ge 4k-1$ that possess non-trivial rational Massey triple products. We also construct examples of manifolds $M$ such that all the cup-products of elements of $H^k(M)$ vanish, while the group $H^{3k-1}(M;\Q)$ is generated by Massey products: such examples are useful for theory of systols...
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For each k ∈ , we construct a uniformly contractible metric on Euclidean space which is not mod k hypereuclidean. We also construct a pair of uniformly contractible Riemannian metrics on n, n ≥ 11, so that the resulting mani- folds Z and Zare bounded homotopy equivalent by a homotopy equivalence which is not boundedly close to a homeomorphism. We s...
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THEOREM. For every prime $p$ and each $n=2, 3, ... \infty$, there is an action of $G=\prod_{i=1}^{\infty}(Z/ pZ)$ on a two-dimensional compact metric space $X$ with $n$-dimensional orbit space. This theorem was proved in [DW: A.N. Dranishnikov and J.E. West, Compact group actions that raise dimension to infinity, Topology and its Applications 80 (1...
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We prove the following embedding theorems in the coarse geometry: Theorem A. Every metric space X with bounded geometry whose asymptotic dimension does not exceed n admits a large scale uniform embedding into the product of n + l locally finite trees.Corollary. Every metric space X with bounded geometry whose asymptotic dimension does not exceed n...
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We present sufficient conditions for the cohomology of a closed aspherical manifold to be proper Lipschitz in sense of Connes-Gromov-Moscovici [CGM]. The conditions are stated in terms of the Stone-\v{C}ech compactification of the universal cover of a manifold. We show that these conditions are formally weaker than the sufficient conditions for the...
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Let A, B and X be finite-dimensional ANR compacta and let α :A→X and f:A→B be maps such that α restricted to the set Sf={x∈A∣f−1f(x)≠x} is one-to-one. Then the pushout Y of the diagram Xα←Af→B is ANR. We apply this result to a construction of ANRs Mp, p is prime, for which dim(Mp×Mq)≠dimMp+dimMq.
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Gromov introduced the concept of uniform embedding into Hilbert space and asked if ev- ery separable metric space admits a uniform embedding into Hilbert space. In this paper, we study uniform embedding into Hilbert space and answer Gromov's question negatively.
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We prove the following theorem: Let $\pi$ be the fundamental group of a finite graph of groups with finitely generated vertex groups $G_v$ having asdim $G_v\le n$ for all vertices $v$. Then asdim$\pi\le n+1$. This gives the best possible estimate for the asymptotic dimension of an HNN extension and the amalgamated product.
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Let M be a nilpotent CW-complex with finitely generated fundamental group. We give necessary and sufficient cohomological dimension theory conditions for a finite-dimensional metric compactum X so that every map A→M, where A is a closed subset of X can be extended to a map X→M.This is a generalization of a result by Dranishnikov [Mat. Sb. 182 (1991...
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We present a technique for construction of infinite-dimensional compacta with given extensional dimension. We then apply this technique to construct some examples of compact metric spaces for which the equivalence XτM(G,n)⇔XτK(G,n) fails to be true for some torsion Abelian groups G and n⩾1.
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We develop some basic Lipschitz homotopy technique and apply it to manifolds with finite asymptotic dimension. In particular we show that the Higson compactification of a uniformly contractible manifold is mod $p$ acyclic in the finite dimensional case. Then we give an alternative proof of the Higher Signature Novikov Conjecture for the groups with...
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Assume that the nerve K of a hyperbolic Coxeter group Γ is n-connected and the complement K⧹Δ to every simplex is n-connected. Then the boundary ∂Γ is n-connected and locally n-connected.
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We prove a version of the countable union theorem for asymptotic dimension and we apply it to groups acting on asymptotically finite dimensional metric spaces. As a consequence we obtain the following finite dimensionality theorems. A) An amalgamated product of asymptotically finite dimensional groups has finite asymptotic dimension: asdim A *_C B...
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We introduce the notion of asymptotic inductive dimension aaInd X for metric spaces X and establish connections between aaInd X, Gromov’s asymptotic dimension aadim X and the inductive dimension of the Higson corona Ind νX.
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The paper deals with generalizing several theorems of the covering dimension theory to the extension theory of separable metrizable spaces. Here are some of the main results: Generalized Eilenberg-Borsuk Theorem. Let L be a countable CW complex. If X is a separable metrizable space and K * L is an absolute extensor of X for some CW complex K, then...
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We show that a space with a finite asymptotic dimension is embeddable in a non-positively curved manifold. Then we prove that if a uniformly contractible manifold X is uniformly embeddable in R n or non-positively curved n-dimensional simply connected manifold then X × R n is integrally hyperspherical. If a uniformly contractible manifold X of boun...
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The negative answer to the following problem of V. I. Arnold is given: Is the number of topologically dierent k-manifolds of bounded total curvature nite?
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For every two compact metric spaces X and Y , both with dimen- sion at most n 3, there are dense G-subsets of mappings f : X! Rn and g : Y !Rn with dimf(X)\g(Y ) dim(XY ) n.
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Coxeter groups admit amenable actions on compact spaces. Moreover, they have finite asymptotic dimension.
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In this paper we study the similarity between local topology and its global analogue, so-called asymptotic topology. In the asymptotic case, the notions of dimension, cohomological dimension, and absolute extensor are introduced and some basic facts about them are proved. The Higson corona functor establishing a connection between macro-and micro-t...
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There is a word metric $d$ on countably generated free group $\Gamma$ such that $(\Gamma,d)$ does not admit a coarse uniform embedding into a Hilbert space.
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We prove the Countable Extension Basis (CEB) Theorem for noncompact metric spaces. As an application of the CEB Theorem we give alternative proofs of some classical results in the area. In particular, we construct a universal complete metric space XK in the class of spaces with e dimX K together with a K-soft map onto the Hilbert cube. As a corolla...
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We prove that every compactum X, having rational dimension n, is an image under a rationally acyclic map of an (n+1)-dimensional compactum Y. If n>1, then additionally dimQY=n. As a consequence we obtain existence of rational homology k-manifolds with covering dimension greater than k.
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1.(1) We construct hyperbolic Coxeter groups with boundaries homeomorphic to Pontryagin surfaces.2.(2) For any given finite simplicial complex L we construct a finite simplicial complex K such that the link of every vertex of K is homeomorphic to L.3.(3) We construct an effective action of p-adic integers on boundaries of certain Coxeter groups .
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Contents § 1. Introduction § 2. The Rubin-Shapiro theorem and its dual § 3. The wedge theorem Bibliography
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Let X be a proper metric space and let νX be its Higson corona. We prove that the covering dimension of νX does not exceed the asymptotic dimension asdimX of X introduced by M. Gromov. In particular, it implies that dim νRn = n for euclidean and hyperbolic metrics on Rn. We prove that for finitely generated groups Γ′ ⊃ Γ with word metrics the inequ...
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The main purpose of this paper is to present a unified treatment of the formula for dimension of the transversal intersection of compacta in Euclidean spaces. A new contribution is the proof of inequality dim(X∩Y)≥dim(X×Y)-n for transversally intersecting compacta X,Y⊂ℝ n , based on a correct interpretation of the classical Chogoshvili theorem [G....
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Contents §1. The Higson-Roe compactification §2. Principle of descent §3. Large Riemannian manifolds §4. Cohomology of the Higson-Roe compactification of Euclidean space §5. Cohomology of the Higson-Roe compactification of hyperbolic space Bibliography
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We show that any light map f: X → Y between compact spaces admits a decomposition f = gh, where g : Z → Y is a finite-to-one map of a special type and h : X → Z has arbitrarily small fibers. It follows that light maps between compact spaces do not lower extensional dimension. Our theorem yields a positive answer to Problem 423 from “Open Problems i...

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