77 problems

Nonnegativity conjecture for discrete pseudomanifolds.

Kühnel–Lutz conjecture. The triangulation of an $n$-manifold is tight if and only if it is strongly minimal.

A triangulated manifold is locally constructible (LC) if it can be obtained from a tree of simplices by recursively identifying boundary facets whose intersection has codimension o…

Lutz–Sulanke–Swartz conjecture. Every tight-neighborly triangulation is $\mathbb F$-tight.

Let $M$ be a $d$-dimensional complementary weak pseudomanifold, and let $\chi(M)$ denote its Euler characteristic. Euler characteristic conjecture. The Euler characteristic of $M$…

Alexander's conjecture. If $K$ and $L$ have a common subdivision, then they have a common stellar subdivision.

Let $K$ be one of the complexes described by Brehm and Kühnel: a triangulation of an $8$-manifold different from the sphere on $15$ vertices. Brehm–Kühnel's conjecture. These compl…

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…

Lickorish's conjecture. Any simplicial subdivision of a $d$-simplex is collapsible.

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…

A simplicial complex is a space obtained by gluing simplices along their faces, with its underlying topology determining the associated topological space. Hauptvermutung. The topol…

Flag complex conjecture.

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

Let a wedge of cycles be a graph obtained by gluing cycles at vertices, and equip it with an asymmetric initial configuration. Wedge-cycle variety conjecture. Wedges of more than t…

Let $C_n$ be the directed cycle on $n$ vertices with no sinks. Let $\mathbf{c}_0$ be a binary configuration with $|\mathbf{c}_0|>1$, and let $1\leq k\leq n$. For $1\leq i\leq n$, d…

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…

Let $G$) be a finite simple graph, let $P_n^2$ be the squared path on $[n]$, and for $k\ge 2$ and $n\ge k+3$ let … where $\Delta_k(P_n^2)$ is the $k$-cut complex of $P_n^2$. Fix…

A triangulation of the Poincare homology sphere is a simplicial complex homeomorphic to that manifold, and its number of vertices is the cardinality of its vertex set. Bjrnerer and…

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

Let $\sigma$ be a triangulation of $S^2$ with $v$ vertices, and let $\mathrm{tetvol}(\sigma)$ denote the minimum number of $3$-simplices in a simplicial triangulation of $B^3$ exte…

A polarization of an artinian monomial ideal is a square-free monomial ideal obtained from it by a polarization procedure. Generalized polarization conjecture. Any notion of polari…