# Members & Guests

## Dr. Christopher Wulff

Georg-August-Universität Göttingen

E-mail: christopher.wulff(at)mathematik.uni-goettingen.de
Telephone: +49 551 39-7751
Homepage: http://www.uni-math.gwdg.de/cwulff/

## Project

10Duality and the coarse assembly map

## Publications within SPP2026

Several formulas for computing coarse indices of twisted Dirac type operators are introduced. One type of such formulas is by composition product in $$E$$-theory. The other type is by module multiplications in $$K$$-theory, which also yields an index theoretic interpretation of the duality between Roe algebra and stable Higson corona.

 Journal Journal of Topology and Analysis Publisher World Scientific Publishing Volume 11(4) Pages 823-873 Link to preprint version Link to published version

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

We construct a slant product on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer, which is the domain of the co-assembly map. In fact, we obtain such products simultaneously on the entire Higson-Roe sequence. The existence of these products implies injectivity results for external product maps on each term of the Higson-Roe sequence. Our results apply in particular to taking products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the general class of manifolds where this method applies, we introduce the notion of Higson-essentialness. A complete spin-c manifold is Higson-essential if its fundamental class is detected by the stable Higson corona via the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. Finally, we draw conclusions for positive scalar curvature metrics on product spaces, in particular on non-compact manifolds. We also construct suitable equivariant versions of these slant products and discuss related problems of exactness and amenability of the stable Higson corona.

Related project(s):
10Duality 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.