Secondary cup and cap products in coarse geometry

TL;DR

This paper develops secondary cup and cap products in coarse geometry, linking them to existing theories and demonstrating their compatibility with known products in coarse cohomology, K-theory, and Higson coronas.

Contribution

It introduces secondary products in coarse (co-)homology theories, extending the algebraic structures and connecting them with established products in K-theory and Higson corona frameworks.

Findings

Secondary products agree with Roe's secondary product on coarse cohomology.

Secondary products correspond to primary products on Higson coronas via transgression.

In coarse K-theory, secondary products relate to canonical products in K-theories of Higson corona and Roe algebra.

Abstract

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 $\mathrm{K}$-theory and -homology, the secondary products correspond to canonical primary products between the $\mathrm{K}$-theories of the stable Higson corona and the Roe algebra under assembly and co-assembly.