A finitely generated group that does not satisfy the generalized Burghelea Conjecture
We construct a finitely generated group that does not satisfy the generalized Burghelea conjecture.
math.KT↗
arXiv subjects
A. Dranishnikov
Publications and source records attributed to A. Dranishnikov.
An infinite-dimensional phenomenon in finite-dimensional metric topology
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 $\mathbb L$-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension $>6$ we classify all such homotopy equivalences. As an application, we show that such homotopy equivalences are realized by deformations of Riemannian manifolds in Gromov-Hausdorff space preserving a contractibility function.
math.GT↗
On the Lusternik-Schnirelmann category of spaces with 2-dimensional fundamental group
The following inequality \cat X\le \cat Y+\lceil\frac{hd(X)-r}{r+1}\rceil holds for every locally trivial fibration between $ANE$ spaces $f:X\to Y$ which admits a section and has the $r$-connected fiber where $hd(X)$ is the homotopical dimension of $X$. We apply this inequality to prove that \cat X\le \lceil\frac{\dim X-1}{2}\rceil+cd(π_1(X)) for every complex $X$ with $cd(π_1(X))\le 2$.
math.GT↗
Asymptotic Dimension
The asymptotic dimension theory was founded by Gromov 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.
math.GT↗
Asymptotic Dimension of Discrete Groups
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.
math.GT↗
An etale approach to the Novikov conjecture
We show that the rational Novikov conjecture for a group $Γ$ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an E$Γ$. 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.
math.GT↗
Asymptotic dimension in Bedlewo
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.
math.GR↗
Lipschitz Cohomology, Novikov conjecture, and Expanders
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-Čech compactification of the universal cover of a manifold. We show that these conditions are formally weaker than the sufficient conditions for the Novikov conjecture given in [CP]. Also we show that the Cayley graph of the fundamental group of a closed aspherical manifold with proper Lipschitz cohomology cannot contain an expander in the coarse sense. In particular, this rules out a Lipschitz cohomology approach to the Novikov Conjecture for recent Gromov's examples of exotic groups.
math.GT↗
Universal spaces for asymptotic dimension
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 the asymptotic dimension with the asymptotic inductive dimensions.
math.GT↗
On asymptotic dimension of groups acting on trees
We prove the following theorem: Let $π$ 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$π\le n+1$. This gives the best possible estimate for the asymptotic dimension of an HNN extension and the amalgamated product.
math.GR↗
On asymptotic dimension of groups
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 < infinity. B) Suppose that G' is an HNN extension of a group G with asdim G < infinity. Then asdim G'< infinity. C) Suppose that Γis Davis' group constructed from a group πwith asdimπ< infinity. Then asdimΓ< infinity.
math.GR↗
HyperEuclidean manifolds and the Novikov Conjecture
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 finite asymptotic dimension. Finally we define an asymptotically piecewise Euclidean metric space as a space which admits an approximation by Euclidean asymptotic polyhedra. We show that the Gromov-Lawson conjecture holds for the asymptotically piecewise Euclidean groups. Also we prove that expanders are not asymptotically piecewise Euclidean
math.GT↗