Secondary cup and cap products in coarse geometry

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 $$\mathrm {K}$$ 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.

Equivariant Coarse (Co-)Homology Theories

“…The claim is then equivalent to the fact that the two ways of performing the double suspension E Γ * +1 Σ 2 A ∼ = E Γ * (ΣA) ∼ = E Γ * −1 (A) (suspending the first copy of (0, 1) first and then the second copy versus the other way around) coincide up to the sign −1, where the sign arises from the reflection of the square (0, 1) 2 along the diagonal. How the proof works can be seen in the follow-up article [45,Lemma 4.6.3], where a more complicated analogous statement is shown for suspension in coarse (co-)homology. ■…”

“…Our primary motivation for expanding further on the theory -and also for our specific choice of contents -was to lay the foundations for the author's subsequent publication [45] about secondary cup and cap products on coarse (co-)homology theories, which was written completely in the framework presented here, albeit without equivariance. For this reason, we focus mainly on the material relevant therein instead of attempting to give a complete encyclopedic account on all the aspects of coarse (co-)homology which have been discussed in the literature so far.…”