17 problems

HK conjecture. For $* = 0,1$, there is an isomorphism

Let $\Gamma$ be a countable group acting minimally and topologically freely on a space $X$ by … , and let $\Gamma\ltimes X$ be the associated transformation groupoid. Write…

Bass trace conjecture for groupoids. The image of $D_0$ is contained in the direct summand

Let $\mathcal G$ be an ample groupoid, let $H_0(\mathcal G)$ denote its $0$-th homology group, and let … be the standard homomorphism induced by the inclusion of the groupoid homol…

Let $\mathcal{G}$ be the groupoid under consideration, with $\widehat{\mathcal{G}}$ its blow-up groupoid and $H_{**}(\widehat{\mathcal{G}};\mathbb{C}_{\widehat{X}})$ the associated…

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…

Let $(X,phi)$ be a non-wandering Smale space whose unstable sets are contractible. Write $H^s_\bullet(X,\phi)$ for its stable homology and $H^{\bullet}(X)$ for the cohomology of…

For an ample groupoid, meaning an étale groupoid whose unit space is totally disconnected, let groupoid homology denote the homology of the groupoid and let $C^*(G)$ be its associa…

Rational HK-conjecture. One should have

HK-realization conjecture. There should exist a possibly different groupoid $\widetilde{\mathcal{G}}$ such that

Let $G$ be a principal ample groupoid. Write $H_k(G;\mathbb{Q})$ for groupoid homology with rational coefficients, and let $C_r^*(G)$ be the reduced groupoid $C^*$-algebra. Rationa…

AH conjecture. The index map $I$ should be surjective, and its kernel should be a quotient of

Let $E$ and $F$ be finite graphs, and let $R$ be a field. Write $\mathcal{G}_E$ and $\mathcal{G}_F$ for their graph groupoids, and let $H_0^{\operatorname{gr}}$ denote graded group…

AH conjecture. There exists an exact sequence

AH conjecture. There exists an exact sequence

HK conjecture. The even and odd homology groups of $\mathcal{G}$ satisfy