One of the most influential conjectures in coarse geometry is the coarse Baum-Connes conjecture. It states that the so-called coarse assembly map

\(\mu\colon K{\mathcal X}_*(X)\to K_*(C^*X)\)

should be an isomorphism for metric spaces \(X\) of bounded geometry.

One way to investigate rational injectivity of the coarse assembly map is by investigating the coarse co-assembly map, since they are related by index pairings.

We want to investigate another relation between assembly and co-assembly, namely through the cap product. Concretely, one might hope to construct more counter-examples to the surjectivity-part of the coarse Baum-Connes conjecture by twisting a single counterexample with suitable elements from the domain of the co-assembly map.

Cap products may also be used to prove Poincaré duality in certain situations. Another goal of this project is to construct a secondary cap products on coarse (co-)homology and on coarse K-(co-)homology. Coarse Poincaré duality is then expected to hold true for open cones over compact spaces. This will be used to investigate visual hyperbolic spaces since coarsely they are cones over their geodesic boundaries.

## Publications

We show injectivity results for assembly maps using equivariant coarse homology theories with transfers. Our method is based on the descent principle and applies to a large class of linear groups or, more general, groups with finite decomposition complexity.

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

We enlarge the category of bornological coarse spaces by adding transfer morphisms and introduce the notion of an equivariant coarse homology theory with transfers. We then show that equivariant coarse algebraic K-homology and equivariant coarse ordinary homology can be extended to equivariant coarse homology theories with transfers. In the case of a finite group we observe that equivariant coarse homology theories with transfers provide Mackey functors. We express standard constructions with Mackey functors in terms of coarse geometry, and we demonstrate the usage of transfers in order to prove injectivity results about assembly maps.

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

This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the assumption of coherence and expandingness, attaching our corona to a Rips complex construction yields a contractible \(\sigma\)-compact space in which the corona sits as a \(\mathbb{Z}\)-set. This results in bijectivity of transgression maps, injectivity of the coarse assembly map and surjectivity of the coarse co-assembly map. For groups we get an estimate on the cohomological dimension of the corona in terms of the asymptotic dimension. Furthermore, if the group admits a finite model for its classifying space \(BG\), then our constructions yield a \(\mathbb{Z}\)-structure for the group.

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

## Team Members

**Dr. Alexander Engel**

Project leader

Universität Regensburg

Alexander.Engel(at)mathematik.uni-regensburg.de

**Dr. Christopher Wulff**

Project leader

Georg-August-Universität Göttingen

christopher.wulff(at)mathematik.uni-goettingen.de