Secondary cup and cap products in coarse geometry

We construct secondary cup and cap products on coarse (co-)homology theories from given cross and slant products. They are defined for coarse spaces relative to weak generalized controlled deformation retracts. On ordinary coarse cohomology, our secondary cup product agrees with a secondary product defined by Roe. For coarsifications of topological coarse (co-)homology theories, our secondary cup and cap products correspond to the primary cup and cap products on Higson dominated coronas via transgression maps. And in the case of coarse $$\mathrm {K}$$ K -theory and -homology, the secondary products correspond to canonical primary products between the $$\mathrm {K}$$ K -theories of the stable Higson corona and the Roe algebra under assembly and co-assembly.

Coarse indices of twisted operators

Several formulas for computing coarse indices of twisted Dirac type operators are introduced. One type of such formulas is by composition product in [Formula: see text]-theory. The other type is by module multiplications in [Formula: see text]-theory, which also yields an index theoretic interpretation of the duality between Roe algebra and stable Higson corona.

Coronae of relatively hyperbolic groups and coarse cohomologies

We construct a corona of a relatively hyperbolic group by blowing-up all parabolic points of its Bowditch boundary. We relate the [Formula: see text]-homology of the corona with the [Formula: see text]-theory of the Roe algebra, via the coarse assembly map. We also establish a dual theory, that is, we relate the [Formula: see text]-theory of the corona with the [Formula: see text]-theory of the reduced stable Higson corona via the coarse co-assembly map. For that purpose, we formulate generalized coarse cohomology theories. As an application, we give an explicit computation of the [Formula: see text]-theory of the Roe-algebra and that of the reduced stable Higson corona of the fundamental groups of closed 3-dimensional manifolds and of pinched negatively curved complete Riemannian manifolds with finite volume.

Metric Compactifications and Coarse Structures

Let TB be the category of totally bounded, locally compact metric spaces with the C0 coarse structures. We show that if X and Y are in TB, then X and Y are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories TB → K, where K is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space X induced by some metrizable compactification is determined only by the topology of the remainder .