Approximations of Stone-Čech compactifications by Higson compactifications (Q5890215)

Let \(X\) be a locally compact metric space with \(d\) a proper metric on \(X\) inducing its topology. The Higson compactification of \(X\) depends on the metric \(d\). This compactification depends on the metric \(d\), but is non-metrizable and has many properties in common with the Stone-Čech compactification. In this paper the authors investigate some comparisons of the Higson compactification and the Stone-Čech compactification. NEWLINENEWLINENEWLINEIn section 2 the authors determine precisely when the Stone-Čech and Higson compactifications are equal for a proper metric space \(X\). In section 3, the authors show that the Stone-Čech compactification is the supremum of all Higson compactifications for all proper metrics inducing the topology of \(X\). In this section it is also shown that the Higson compactification of the product \(X \times Y\) with any proper metric is not equivalent to the product of the Higson compactifications. This is a Glicksberg type theorem for the Higson compactification. NEWLINENEWLINENEWLINEIn section 3 there is also some discussion of the homotopy classes of mappings of the Higson compactification to \(S^1\). There are also some theorems involving the asymptotic dimension of proper metric spaces and their products. It is also shown that the Stone-Čech compactification is the supremum of all Higson compactifications based on proper metrics with the Higson corona having dimension \(\leq 3\). NEWLINENEWLINENEWLINESection 4 is an appendix which discusses metric compactification approximations of the Stone-Čech compactification.