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We establish some basic theorems in dimension theory and absolute extensor theory in the coarse category of metric spaces. Some of the statements in this category can be translated in general topology language by applying the Higson corona functor. The relation of problems and results of this `Asymptotic Topology' to Novikov and similar conjectures is discussed.Comment: 38 pages, AMSTe

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

Introduction 1 1. General properties of the cohomological dimension 2 2. Bockstein theory 6 3. Cohomological dimension of Cartesian product 10 4. Dimension type algebra 15 5. Realization theorem 19 6. Test spaces 24 7. Infinite-dimensional compacta of finite cohomological dimension 28 8. Resolution theorems 33 9. Resolutions preserving cohomological dimensions 41 10. Imbedding and approximation 47 11. Classifying spaces for cohomological dimension 50 12. Cohomological dimension of ANR compacta 54 References 59Beverley L. Brechner asked Alexander Dranishnikov to write this survey for Topology Atlas. It was written in 1998. This survey has not been refereed.The opposite inequality follows from Theorem 1.6 applied to G = Lim → i s=1 G s and the fact that sup i {dim i s=1 Gs X} = sup s {dim Gs X} imply the proof. Definition.A compactum X has an r-dimensional obstruction at its point x with respect to a coefficient group G if there is a neighborhood U of x such that for every smaller neighborhood V of x the image of the inclusion homomorphism i V,U : H r c (V ; G) → H r c (U ; G) is nonzero. Theorem 1.8. Let X be a compact with dim G X = r then X contains a compact subset Y of dim G Y = r such that at every point x ∈ Y the compact X has an r-dimensional obstruction with respect to G.Proof. Let W be an open subset of X with H r c (W ; G) = 0. Because of the continuity of cohomology there is a closed in U set Z minimal with respect the property: the inclusion homomorphism H r c (W ; G) → H r c (Z; G) is nonzero. Then dim G Z = r and by the Countable Union Theorem there exists a compact subset Y ⊂ Z with dim G Y = r. For every x ∈ Y we take U = W . Let V ⊂ U be a neighborhood of x.

Topology and its Applications 140 (2004) 203-225. doi:10.1016/j.topol.2003.07.0092016-03-04T18:46:49