Duality and the coarse assembly map

• Applicants: Dr. Christopher Wulff; Privatdozent Dr. Rudolf Zeidler
• Subject Area: Mathematics
• Term: since 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 338480246

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity