10 problems

Let $G$ be a topological group with a universal proper space $\mathcal E G$. Let $A$ be a $C^*$-algebra and let $B$ be a $G$-algebra. Define the naive topological $KK$-groups by ……

Let $\Gamma$ be a countable discrete group acting properly on a proper metric space $X$. Let $KX_*^{\Gamma}(X)$ denote the universal $K^{\Gamma}$-group assembling the equivariant i…

Let $\Gamma$ be a finitely generated discrete group acting on a metrizable finite-dimensional compact space $X$, and let ${\mathcal G}=X\rtimes\Gamma$. Write $RK_i(\Gamma,X)$ for t…

Let $\mathcal{G}$ be a second countable, étale groupoid, and let $\widehat{\mathcal{G}}$ be the blow-up groupoid with base-related notation $\widehat{X}$. Suppose that $c$-…

Let $\mathcal{G}$ be a second countable, locally compact, Hausdorff, étale groupoid, and let $\widehat{\mathcal{G}}$ be its blow-up at torsion elements of isotropy groups. Write…

Stable Gromov–Lawson–Rosenberg conjecture. A spin manifold stably admits a positive scalar curvature metric if and only if

The $\tau$-Baum--Connes conjecture with coefficients. The map $\mu_\tau$ is bijective. This is the $\tau$-localised form of the Baum--Connes conjecture with coefficients, asserting…

Real Baum–Connes conjecture. The assembly map $\mu_n$ is an isomorphism. This is the real, 8-periodic analogue of the Baum–Connes conjecture, with equivariant KO-homology on the le…

Let $Y\subseteq X$ be aspherical compact spaces, and let … be the maximal relative Baum–Connes map. If $j\colon \pi_1(Y)\to \pi_1(X)$ is injective, let … be the reduced relative Ba…

Let $\Gamma$ be a group. Let $K$ be the $C^*$-algebra of all compact operators on an infinite-dimensional separable Hilbert space, and let $C_r^*(\Gamma)$ be the reduced group…