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

In previous definition of $\mathrm{E}$-theory, separability of the $\mathrm{C}^*$-algebras is needed either to construct the composition product or to prove the long exact sequences.

Considering the latter, the potential failure of the long exact sequences can be traced back to the fact that these $\mathrm{E}$-theory groups accommodate information about asymptotic processes in which one real parameter goes to infinity, but not about more complicated asymptotics parametrized by directed sets.

We propose a definition for $\mathrm{E}$-theory which also incorporates this additional information by generalizing the notion of asymptotic algebras. As a consequence, it not only has all desirable products but also all long exact sequences, even for non-separable $\mathrm{C}^*$-algebras.

More precisely, our construction yields equivariant $\mathrm{E}$-theory for $\mathbb{Z}_2$-graded $G$-$\mathrm{C}^*$-algebras for arbitrary discrete groups $G$.

We suspect that our model for $\mathrm{E}$-theory could be the right entity to investigate index theory on infinite dimensional manifolds.

**Related project(s):****10**Duality and the coarse assembly map**78**Duality and the coarse assembly map II

We present an Eilenberg--Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories with already existing or newly introduced equivariance fit into this setup.

Furthermore, a new and more flexible notion of coarse homotopy is given which is more in the spirit of topological homotopies. Some, but not all, coarse (co-)homology theories are even invariant under these new homotopies. They also led us to a meaningful concept of topological actions of locally compact groups on coarse spaces.

Journal | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) |

Volume | 18 (2022), 057 |

Pages | 62p |

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

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.

Journal | Research in the Mathematical Sciences |

Volume | 8, Article number: 36 |

Pages | 64p |

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**Related project(s):****10**Duality and the coarse assembly map**78**Duality and the coarse assembly map II

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.

Journal | Annales de l'Institut Fourier |

Volume | 71 (2021) no. 3 |

Pages | 913-1021 |

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**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.

Journal | Journal of Topology and Analysis |

Volume | online ready |

Pages | 83p |

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

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 |

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Link to published version |

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

We strengthen a result of Hanke-Schick about the strong Novikov conjecture for low degree cohomology by showing that their non-vanishing result for the maximal group C*-algebra even holds for the reduced group C*-algebra. To achieve this we provide a Fell absorption principle for certain exotic crossed product functors.

**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

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

## Team Members

**Jun.-Prof. Dr. Alexander Engel**

Project leader

Universität Greifswald

alexander.engel(at)uni-greifswald.de

**Dr. Christopher Wulff**

Project leader

Georg-August-Universität Göttingen

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