Coarse geometry is the study of the large-scale structure of geometric spaces, in contrast to the better known field of topology, which is the study of small-scale structures. Coarse geometry has been around for about twenty-five years and has significant applications in geometric topology and in the study of curvature.
The Novikov conjecture concerns homotopy invariance of higher signatures and has generated a huge quantity of research in fields ranging from algebraic K-theory to geometric functional analysis. One way of obtaining a positive solution to this conjecture is to prove the coarse Baum--Connes conjecture using coarse geometric methods and then to apply the notion of descent.
The coarse Baum--Connes conjecture is true for a wide variety of spaces, and in particular, spaces of finite asymptotic dimension. Mitchener formulated a version of the coarse Baum--Connes conjecture for a general class of coarse invariants, not just the K-theory of the Roe C*-algebra. This generalisation has applications to geometry and algebra beyond those of the original conjecture and these generalisations are explored in this thesis.
The underlying idea is that Wright's proof of the coarse Baum--Connes conjecture for finite asymptotic dimension does not rely on the precise definition of the Roe C*-algebra, and depends only on the coarse geometry.
In this thesis, we construct a generalised coarse assembly map from a coarsely excisive functor. To show that this map is an isomorphism for spaces of finite asymptotic dimension, we begin by constructing a sequence of coarsening spaces which approximate the original space of finite asymptotic dimension. We then relate each side of the assembly map to the direct limit of homology groups of these coarsening spaces, equipped with the C_0 coarse structure for the domain and the bounded coarse structure for the codomain. It is then shown that these direct limits agree under certain conditions, allowing us to conclude the main result. This result is then applied to C*-category and algebraic K-theory, as well as equivariant versions of these theories.
