Coronas for properly combable spaces

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 [Formula: see text]-compact space in which the corona sits as a [Formula: see text]-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 [Formula: see text], then our constructions yield a [Formula: see text]-structure for the group.

On the Assembly Map for Complex Semisimple Quantum Groups

We show that complex semisimple quantum groups, that is, Drinfeld doubles of $q$-deformations of compact semisimple Lie groups, satisfy a categorical version of the Baum–Connes conjecture with trivial coefficients. Our approach, based on homological algebra in triangulated categories, is compatible with the previously studied deformation picture of the assembly map and allows us to define an assembly map with arbitrary coefficients for these quantum groups.

The Baum–Connes conjecture localised at the unit element of a discrete group

We construct a Baum–Connes assembly map localised at the unit element of a discrete group $\Gamma$. This morphism, called $\mu _\tau$, is defined in $KK$-theory with coefficients in $\mathbb {R}$ by means of the action of the idempotent $[\tau ]\in KK_{\mathbin {{\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$ canonically associated to the group trace of $\Gamma$. We show that the corresponding $\tau$-Baum–Connes conjecture is weaker than the classical version, but still implies the strong Novikov conjecture. The right-hand side of $\mu _\tau$ is functorial with respect to the group $\Gamma$.

Cyclic homology for bornological coarse spaces

The goal of the paper is to define Hochschild and cyclic homology for bornological coarse spaces, i.e., lax symmetric monoidal functors $${{\,\mathrm{\mathcal {X}HH}\,}}_{}^G$$ X HH G and $${{\,\mathrm{\mathcal {X}HC}\,}}_{}^G$$ X HC G from the category $$G\mathbf {BornCoarse}$$ G BornCoarse of equivariant bornological coarse spaces to the cocomplete stable $$\infty $$ ∞ -category $$\mathbf {Ch}_\infty $$ Ch ∞ of chain complexes reminiscent of the classical Hochschild and cyclic homology. We investigate relations to coarse algebraic K-theory $$\mathcal {X}K^G_{}$$ X K G and to coarse ordinary homology $${{\,\mathrm{\mathcal {X}H}\,}}^G$$ X H G by constructing a trace-like natural transformation $$\mathcal {X}K_{}^G\rightarrow {{\,\mathrm{\mathcal {X}H}\,}}^G$$ X K G → X H G that factors through coarse Hochschild (and cyclic) homology. We further compare the forget-control map for $${{\,\mathrm{\mathcal {X}HH}\,}}_{}^G$$ X HH G with the associated generalized assembly map.

Topological 4-manifolds with right-angled Artin fundamental groups

We classify closed, spin[Formula: see text], topological [Formula: see text]-manifolds with fundamental group [Formula: see text] of cohomological dimension [Formula: see text] (up to [Formula: see text]-cobordism), after stabilization by connected sum with at most [Formula: see text] copies of [Formula: see text]. In general, we must also assume that [Formula: see text] satisfies certain [Formula: see text]-theory and assembly map conditions. Examples for which these conditions hold include the torsion-free fundamental groups of [Formula: see text]-manifolds and all right-angled Artin groupswhose defining graphs have no [Formula: see text]-cliques.

Assembly maps for topological cyclic homology of group algebras

We use assembly maps to study \mathbf{TC}(\mathbb{A}[G];p), the topological cyclic homology at a prime p of the group algebra of a discrete group G with coefficients in a connective ring spectrum \mathbb{A}. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism on homotopy groups. For infinite groups, we establish pro-isomorphism, (split) injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we show that the assembly map for the family of virtually cyclic subgroups is injective but in general not surjective.

Dynamical complexity and K-theory of Lp operator crossed products

We apply quantitative (or controlled) [Formula: see text]-theory to prove that a certain [Formula: see text] assembly map is an isomorphism for [Formula: see text] when an action of a countable discrete group [Formula: see text] on a compact Hausdorff space [Formula: see text] has finite dynamical complexity. When [Formula: see text], this is a model for the Baum–Connes assembly map for [Formula: see text] with coefficients in [Formula: see text], and was shown to be an isomorphism by Guentner et al.