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.
Highlights
The usefulness of multiplicative structures onhomology theories is undisputed in algebraic topology, their coarse geometric counterparts have been neglected for quite a long time and Roe’s secondary product on his coarse cohomology remained the only one of his kind for many years.Only recently, there has been more research on this topic, which was primarily motivated by applications to coarse index theory and thereby to positive scalar curvature
Our secondary cup product agrees with a secondary product defined by Roe
1 Introduction the usefulness of multiplicative structures onhomology theories is undisputed in algebraic topology, their coarse geometric counterparts have been neglected for quite a long time and Roe’s secondary product on his coarse cohomology remained the only one of his kind for many years
Summary
The usefulness of multiplicative structures on (co-)homology theories is undisputed in algebraic topology, their coarse geometric counterparts have been neglected for quite a long time and Roe’s secondary product on his coarse cohomology (cf. [14, Section 2.4]) remained the only one of his kind for many years. Cross products were the main multiplicative structure of interest, in particular the cross product between the analytic structure group and K-homology (cf [4,9,16,22,25,26], albeit not all of these references work in a truely coarse set-up) and the cross product between the K-theories of Roe algebras (cf [9]). All of these cross products were complemented by slant products in [9] and, the slant product between the K-theory of the Roe algebra and the K-theory of the stable Higson corona generalizes the pairing introduced in [7] to dualize the coarse assembly map. Further multiplicative structures are the ring and module multiplications between the K-theory of the stable Higson corona and the K-theory of the Roe algebra constructed in [18,20], which should be understood
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