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Duality and the coarse assembly map

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

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):
10Duality and the coarse assembly map

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## Team Members

Dr. Christopher Wulff