21 research outputs found
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: CannonāÅ tan'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 ā M between closed manifolds which are induced by cell-like maps p: N ā X and q: M ā 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> 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 Ļ2(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 Ļ and a precompact subset C of Gromov-Hausdorff space such that for every Ē«> 0 there are nonhomeomorphic Riemannian manifolds with contractibility function Ļ which lie within Ē« of each other in C. 1