Asymptotically exact spaces and coarse assembly

TL;DR

This paper introduces the class of asymptotically exact metric spaces, showing that the coarse Baum-Connes assembly map is split surjective for them, revealing new insights into their geometric and analytical properties.

Contribution

It defines asymptotically exact spaces and proves the coarse Baum-Connes assembly map is split surjective for this class, expanding understanding beyond known categories.

Findings

The coarse Baum-Connes assembly map is split surjective for asymptotically exact spaces.

Asymptotically exact spaces form an intermediate category between known classes.

The kernel of the assembly map is generally non-zero for these spaces.

Abstract

Between the category of exact metric spaces with bounded geometry (about which much is known) and the larger category of arbitrary exact metric spaces (about which little is known) lies the intermediate category of asymptotically exact metric spaces. We show that the coarse Baum-Connes assembly map is naturally split surjective for this class, with generally non-zero kernel.