15 problems

A cubical complex is locally $5$--large when the link of every vertex is $5$--large, meaning that every simplicial cycle of length less than $5$ has a diagonal. The phrase “contain…

Local Euler characteristic conjecture. Each local Euler characteristic should satisfy

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…

A cubical sphere is a cube complex homeomorphic to a sphere, and its faces include faces of every dimension. The boundary of a neighbourly cubical polytope is the cubical sphere ob…

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 ${\cal C}(n,k)$ be the family of $(k+1)$-cubes whose boundaries are ${\cal B}(n,k)$, and let ${\cal C}'(n,k)$ be the corresponding family for the simplicial complex…

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…

Let $Q^d=[0,1]^d$ be partitioned into $n^d$ small cubes, and color the small cubes with $m+1$ colors. A monochromatic connected component is a connected component consisting of sma…

A CAT(0) cubical complex has an interval between any two cubes, and this interval is a CAT(0) cubical complex corresponding to an ordinary distributive lattice. For a positive inte…

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

Let $Q_n$ be the $n$-dimensional cube, and let $x_i,y_i,q_i$ be commuting indeterminates. For a cubical complex, write…

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…