Asymptotic topology

Abstract

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.

38 pages, AMSTeX

Cited by in corpus (34)

• Hurewicz Theorem for Assouad-Nagata dimension
• On asymptotic dimension of groups
• Relative commutant pictures of Roe algebras
• Growth of the asymptotic dimension function for groups
• Coarse geometry and asymptotic dimension
• The asymptotic dimension of quotients by finite groups
• The General Notion of Descent in Coarse Geometry
• Weak coherence of groups and finite decomposition complexity
• On constructions preserving the asymptotic topology of metric spaces
• A totally bounded Uniformity on coarse metric Spaces
• Nagata-Assouad dimension via Lipschitz extensions
• Coarse amenability and discreteness
• Normality conditions of structures in coarse geometry and an alternative description of coarse proximities
• Coarse Homotopy Groups
• Decomposition of Cellular Balleans
• The normality and bounded growth of balleans
• Higson Compactification and Dimension Raising
• Examples of metric spaces with asymptotic property $C$
• Boundaries of coarse proximity spaces and boundaries of compactifications
• Functional boundedness of balleans
• On Large Scale Properties of Manifolds
• Coarse dimensions and partitions of unity
• Sublinear Higson corona and Lipschitz extensions
• Sublinear Higson corornae of Euclidean cones
• Asymptotic Dimension of Finitely Presented Groups
• Lipschitz Cohomology, Novikov conjecture, and Expanders
• Classification of metric spaces with infinite asymptotic dimension
• Simply connected homogeneous continua are not separated by arcs
• Convex hyperspaces of probability measures and extensors in the asymptotic category
• Decomposition theorems for asymptotic property C and property A
• Asymptotic property C of the wreath product ZwrZ
• Property {A} and duality in linear programming
• Asymptotic property C of the countable direct sum of uniformly discrete $0$-hyperbolic spaces
• An asymptotic dimension for metric spaces, and the 0-th Novikov-Shubin invariant