10 problems

Let $(W,S)$ be a Coxeter system with nerve $L$, let $\Sigma$ be its associated complex, and suppose that $L$ is a generalized homology sphere of dimension $n-1$, written…

Flag-complex partial-link conjecture.

Partial-link Singer conjecture. If $X$ has spherical links in codimensions at most $2l+1$, where $2l+1\le n$, then

Finite flag subcomplex vanishing conjecture.

Let $G$ be a torsion-free countable group, and let ${\mathcal{D}}(\mathbb{C}G\subset {\mathcal{U}}G)$ denote the ring appearing in the Atiyah-conjecture construction. Atiyah conjec…

Let $M$ be a closed aspherical $n$-manifold. Singer's conjecture. The $ell^2$-Betti numbers satisfy … for all $p\neq n/2$. The Singer conjecture is a major unresolved problem conce…

Let $(M^{2k+1},F^{2k})$ be a tidy pair, with $i:F\hookrightarrow M$ the inclusion and $i_*:L^2H_k(F,\partial F)\to L^2H_k(M,\partial M)$ the induced map. Tidy-pair vanishing conjec…

Let $(M,\partial M)$ be an $n$-manifold with contractible components that admits a geometric group action, and let $L^2H_i(M)$ denote its $L^2$-homology. Manifold cocompact action…

For a discrete group $G$, let $L^2H_i(G)$ denote the $L^2$-homology of a contractible $G$-space, and let $\operatorname{cadim}(G)$ be the least dimension of a contractible $G$-mani…

For a discrete group $G$, let $L^2H_i(G)$ denote the $L^2$-homology of a contractible $G$-space, and let $\operatorname{actdim}(G)$ be the least dimension of a contractible manifol…