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.Comment: 38 pages, AMSTe

Asymptotic dimension of finitely presented groups

“…It is a large scale analog of topological dimension and it is invariant by quasiisometries. This notion has proved relevant in the context of Novikov's higher signature conjecture and it was investigated further by other people (see [11], [1], [9]). …”

“…We consider now the van Kampen diagram D that corresponds to the path w and the function f from D (1) to the Cayley graph G. So f (∂D) = w. For notational convenience we label vertices and edges of ∂D in the same way as w. So for example we denote the vertex on ∂D which is mapped to x 0 ∈ w by f also by x 0 .…”

Untitled

“…There is a natural map f from the 1-skeleton D (1) of D to the Cayley graph of G; f sends the base point to a vertex v of the Cayley graph and edges of D (1) to edges of the Cayley graph with the same label. Obviously f is determined by the image of the base point, v; f is not necessarily injective.…”

Resolvability of ball structures

“…The results from [10], [11], [12] show that the ball structures (with the isomorphisms defined above) are the natural asymptotic counterparts of topological spaces. A good motivation to study ball structures related to metric spaces is in the survey [5].…”