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 \Gamma is Davis' group constructed from a group \pi with asdim\pi < infinity. Then asdim\Gamma < infinity.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-4.abs.htm

Cohomological dimension with respect to perfect groups

AbstractWe introduce new classes of compact metric spaces: Cannon—Štan'ko, Cainian, and nonabelian compacta. In particular, we investigate compacta of cohomological dimension one with respect to certain classes of nonabelian groups, e.g., perfect groups. We also present a new method of constructing compacta with certain extension properties

CELL-LIKE MAPS AND TOPOLOGICAL STRUCTURE GROUPS ON MANIFOLDS

Abstract. We show that there are homotopy equivalences h: N → M between closed manifolds which are induced by cell-like maps p: N → X and q: M → X but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain L-classes. In dimension&gt; 5 we identify all such homotopy equivalences to M with a torsion subgroup S CE (M) of the topological structure group S(M). In the case of simply connected M with finite π2(M), the subgroup S CE (M) coincides with the odd torsion in S(M). For general M, the group S CE (M) admits a description in terms of the second stage of the Postnikov tower of M. As an application, we show that there exist a contractibility function ρ and a precompact subset C of Gromov-Hausdorff space such that for every ǫ&gt; 0 there are nonhomeomorphic Riemannian manifolds with contractibility function ρ which lie within ǫ of each other in C. 1