Coarse Structures and Higson Compactification

FOLLOWING John Roe in his Lectures on Coarse Geometry, we begin by describing the large-scale structure of metric spaces by means of coarse maps between them, those being maps which preserve distances at large scales. Using these techniques, we demonstrate that the real numbers and the integers have the same large scale structure--or are coarsely equivalent--but that the real line is coarsely equivalent to neither the Euclidean plane nor the set of positive real numbers. Following a generalization of these concepts for general topological spaces with the introduction of an abstract coarse structure on the space, we show, among other things, that the real line is not coarsely equivalent to the long line of the countable ordinals with the order topology.

We depart from Roe to describe a connection between locally compact Hausdorff spaces and a sub-class of Banach algebras known as C*-algebras, where we find that every such algebra can be described as a set of continuous functions on a particular locally compact Hausdorff space. In particular, we see that there is a very strong relationship between the two: the categories of C*-algebras and *-homomorphisms, and the opposite of locally compact Hausdorff spaces and continuous functions are dual.

Returning to coarse structures, we examine compactifications of locally compact Hausdorff spaces with a view to construction of a topological coarse structure, one in which those maps which are coarse are precisely those which may be continuously extended to the boundary of the space. We complete our investigation by describing the inverse process: given a coarse structure, can we find a compactification possessing the same properties? We provide an partial answer to this question, called a Higson compactification, and end by calculating Higson compactifications of some familiar spaces.