Coarse assembly maps

Abstract

For every strong coarse homology theory we construct a coarse assembly map as a natural transformation between coarse homology theories. We provide various conditions implying that this assembly map is an equivalence. These results generalize known results for the analytic coarse assembly map for K -homology to general coarse homology theories. Furthermore, we calculate the domain of the coarse assembly map explicitly in terms of locally finite homology theory.

Paschke duality and assembly maps

“…If X is in GUBC, then the cone O ∞ (X) is the G-set R × X with a certain G-bornological coarse structure described in [BEKW20a, Sec. 9] and [BE20a,Section 8]. It contains the underlying G-bornological coarse space of X as the subspace {0} × X.…”

The coarse Pimsner-Voiculescu sequence

“…9.7] for the non-equivariant coarse homology theory E Z X (− ⊗ Z min,min ) which is strong since E Z was assumed to be strong. Since Res Z (Z can,min ) has finite asymptotic dimension this coarse Baum-Connes assembly map is an equivalence by [BE20a,Thm. 10.4].…”

“…We consider this note as a possibility to demonstrate the usage of results of coarse homotopy homotopy as developed in [BE20b], [BE20a], [BEKW20b], [BEb] and [BELa].…”

Coarse homotopy theory and boundary value problems

“…Its equivariant generalization is developed [BEDW]. As explained in [BE17a] (in the non-equivariant case) one can interpret the basic objects of index theory, namely the symbol of a Dirac operator and its index, as coarse indices of Dirac operators on appropriatly defined equivariant bornological coarse spaces. We will recall this coarse translation of index theory in Section 4.…”

“…This abstract version has no relation with Dirac operators at all. Following the philosophy explained in [BE17a], by just replacing equivariant coarse K-homology theory by an arbitrary equivariant coarse homology theory, we can define the notions of symbol and index classes, and the analytic assembly map sending symbols to indices. Furthermore, in the geometric situation of a boundary value problem, we define the notion of a boundary condition for a symbol and derive the corresponding index of the resulting abstract boundary value problem.…”