Duality and the coarse assembly map (II)

The coarse co-assembly map was introduced by Emerson and Meyer as a dual to the coarse assembly map. These two maps are mutually adjoint with respect to canonical pairings.

The main objective of our project is to further develop this coarse duality theory by studying analogues of multiplicative structures known from algebraic topology such as cup and cap products as well as external and slant products.

Many of these multiplicative structures have direct interpretations in terms of Dirac operators and vector bundles leading to new applications in index theory and geometric topology.

In addition, we will develop coarse versions of well-known results from algebraic topology involving multiplicative structures.

Most notably, we want to investigate a coarse version of Poincaré duality.

Furthermore, the question of whether coarse assembly and co-assembly are isomorphisms will also be examined in certain cases. We have a special focus on spaces which admit nice coarse compactifications constructed from coarse geometric versions of contractions or, more generally, deformation retractions. Such coarse deformation retractions are also key to the construction of secondary cup and cap products, establishing a close link between the two main aspects of our project.


We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we thereby give refined answers to questions on metric inequalities recently proposed by Gromov. This includes optimal estimates for Riemannian bands and for the long neck problem. In the case of bands over manifolds of non-vanishing \(\widehat{\mathrm{A}}\)-genus, we establish a rigidity result stating that any band attaining the predicted upper bound is isometric to a particular warped product over some spin manifold admitting a parallel spinor. Furthermore, we establish scalar- and mean curvature extremality results for certain log-concave warped products. The latter includes annuli in all simply-connected space forms. On a technical level, our proofs are based on new spectral estimates for the Dirac operator augmented by a Lipschitz potential together with local boundary conditions.


Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

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.


Related project(s):
10Duality and the coarse assembly map78Duality and the coarse assembly map II

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Team Members

Dr. Christopher Wulff
Project leader
Georg-August-Universität Göttingen

Dr. Rudolf Zeidler
Project leader
Westfälischen Wilhelms-Universität Münster

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