36 problems

Let $K$ be a wnp map, meaning a polyhedral map attaining equality in the lower bound $n\geq (k-1)^2$ for an $n$-vertex $\{k,k\}$-equivelar polyhedral map. Nonexistence conjecture.…

Let $(E,\mathcal{L},g)$ be a simplicial affine oriented matroid, where $g$ is the distinguished element and $\mathcal{L}^{++}$ denotes its bounded complex. Simplicial bounded-compl…

Let $(E,\mathcal{L},g)$ be an affine oriented matroid, where $g$ is the distinguished element and $\mathcal{L}/g$ denotes contraction by $g$. The bounded complex is denoted by…

Let $\Gamma$ be a regular $(n+1)$-valent graph with a “good” $(\mathbb Z_2)^{n+1}$-coloring $\alpha$. A generalized $(n+1)$-skeletal expansion is an expansion denoted…

Kühn­el–Kalai conjecture.

A complementary weak pseudomanifold is a weak pseudomanifold whose simplicial complex satisfies complementarity. Let $M$ be a $d$-dimensional complementary weak pseudomanifold with…

Flag complex conjecture.

Wedge-of-spheres conjecture. The complex $\operatorname{PF}_t(G)$ is homotopy equivalent to a wedge of spheres.

Nonnegativity conjecture for discrete pseudomanifolds.

Let $P_n$ be the directed path on $n$ vertices, let $\beta_i$ denote the $i$-th Betti number of its avalanche complex, and let a binary configuration be an initial configuration wh…

Acyclicity conjecture. If $\Gamma$ is Gröbner smoothable over $K$, then $\Gamma$ is acyclic over $K$.

Let $d\geq 3$, and let $\mathcal{G}_M$ denote the balanced genus of a PL $d$-manifold $M$. Consider the product manifold $\mathbb{S}^{d-1}\times\mathbb{S}^1$. Balanced-genus conjec…

Minimality conjecture. The triangulated $n$-sphere constructed in Theorem is facet-minimal, and it is vertex-minimal as well.

Let $d\geq 1$, let $r\geq 2$ be a prime power, and set $n=(r-1)(d+1)$. Let $N>n$, and let $\Delta_N$ be the $N$-dimensional simplex. Topological Tverberg sunflower conjecture. For…

Let $X$ be a simplicial manifold. A fixing cycle is a $(3n-2)$-cycle $\phi\in Z_{3n-2}(Y,\pi^*({\mathcal D}))$ satisfying … where $n$ is the dimension of $X$, ${\mathcal D}$ is its…

Non-polytopality conjecture. The combinatorial spheres $L_2$ and $L_3$ are non-polytopal.

The distinguished-manifolds conjecture. The combinatorial manifolds $K_1$, $K_2$, $K_3$, and $K_4$ are PL homeomorphic to $\mathbb{OP}^2$.

Let $M$ be a smooth compact manifold smoothly embedded in euclidean space. Framed stratifications are dense in smooth embeddings. Any smooth embedding of $M$ into euclidean space h…

Let $M$ and $M'$ be smooth compact manifolds smoothly embedded in euclidean space, and let the embeddings define flat framed stratifications. Framed homeomorphism implies diffeomor…

For $p\geq 3$, consider the simplicial $2$-complexes d2n83torsion and d2n84torsion, which have $3$-torsion and $4$-torsion, respectively. Their $f$-vectors record the numbers of fa…

Let $M_1$ and $M_2$ be closed odd-dimensional triangulated manifolds, and let $M_1 \# M_2$ be any connected sum. Let $\Omega(M)$ denote Thompson's width. Odd-dimensional connected-…

Let $M_1$ and $M_2$ be $n$-dimensional simplicial manifolds, and choose a connected sum $M_1 \# M_2$. Here $\Omega(M)$ denotes Thompson's width of a simplicial manifold $M$. Connec…

Let $k\geq 2$. A balanced $k$-neighborly homology $(2k-1)$-sphere is a balanced simplicial homology sphere of dimension $2k-1$ in which every set of at most $k$ vertices with disti…

Let $d$ and $n$ be positive integers. A normal pure $(d-1)$-complex is a pure simplicial complex in which the link of every face of codimension at least two is connected. Its diame…

Let $K$ be a combinatoric, let $g$ be the genus of $\abs{K}$, and let $K^\ast$ be its dual complex. An admissible coloring of $K^\ast$ is a coloring satisfying the condition that e…