8 problems

Let $M^n$ be a closed $n$-manifold. A marked cubical decomposition of $M^n$ is a cubical decomposition together with the marking data used to distinguish its cubical cells, and two…

Let $M= obreakbackslash \mathbb{H}^n$ be a closed hyperbolic $n$-manifold with fundamental group $\Gamma$, where either $nleq 3$ or $M$ is arithmetic of simplest type. A **$\lambda$-q…

Cohomology approximation conjecture. If $d \geq 4(2k+1)^2$, then, as $h$ tends to infinity, the induced map $\tau^*$ on the $k$th cohomology groups is an isomorphism to…

Branched-cover conjecture. For every $k>0$ there is $d$ such that, whenever $Q$ is a finite cubical poset of dimension $k+1$ and every $k$-face is contained in at least $d$ facets,…

Let $\mathcal{C}$ be the boundary complex of a cubical polytope. A stable line shelling is a line shelling satisfying the stability condition defined through the realization and in…

Let $\Gamma_S$ be one of the lattices of finite characteristic not equal to $2$ constructed in the paper, let $\Gamma \subseteq \Gamma_S$ be a congruence subgroup, and let…

Near-prism homotopy theorem. Then $X$ is homotopy equivalent to a wedge of spheres, specifically

Let $P$ and $Q$ be two cubes, and let $f:V(P)\rightarrow V(Q)$. A cubical map is a map between the corresponding cubical face complexes that sends each cube into a cube. Cubical ap…