22 problems

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.

Lutz–Kühnel minimality conjecture. Tight triangulations of $M$ minimize every entry of the $f$-vector among all triangulations of $M$.

Kühn­el–Lassmann 3-torus conjecture. The Kühn­el–Lassmann triangulation of $T^3$ with $15$ vertices is vertex-minimal and is unique with

Kühn­el–Lassmann torus conjecture. The Kühn­el–Lassmann series of combinatorial $d$-tori $T^d$ with $2^{d+1}-1$ vertices is vertex-minimal.

Kühn­el's conjecture. For $j=0,\dots,\lfloor(d-1)/2\rfloor$,

Kühn­el–Kalai conjecture.

Neighborly sphere-product conjecture. There is a centrally $(\lfloor d/2\rfloor+1)$-neighborly combinatorial triangulation of every product…

Sparla's existence conjecture. There are centrally $(r+1)$-neighborly triangulations of $S^r\times S^r$ on $4r+4$ vertices.

Centrally symmetric cyclic upper bound conjecture. If $d$ is even, then the boundary complex $\partial C_d^{\Delta}$ of the $d$-dimensional crosspolytope on $n=2d$ vertices is the…

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…

A $k$-LC $d$-manifold is a triangulated $d$-manifold obtainable from a tree of $d$-simplices by recursively identifying boundary facets whose intersection has codimension at most…

For a fixed dimension, algorithmic recognizability asks whether there is an algorithm deciding whether a triangulated ball is a ball. Algorithmic unrecognizability conjecture for 4…

Billera–Lee link conjecture. Every vertex link $\operatorname{lk}_M(v)$ has the $\tau$-vector of its corresponding Billera–Lee sphere:

Fix a smooth Riemannian surface $(M,g)$. Let $(M_n,\mathcal{T}_n,\Phi_n,r_n)$ be a sequence of weighted triangulations of $M$ with initial circle packing metrics $r_n$, and let…

Let $\Delta$ be a connected triangulated $(d-1)$-manifold without boundary. Write $h_i(\Delta)$ for its $h$-numbers and $\beta_i(\Delta)$ for its $i$th Betti number. A triangulated…

Let $(K3)_{16}$ and $(K3)_{17}$ denote the 16-vertex and 17-vertex triangulations, respectively, of the K3 surface. K3 triangulation diffeomorphism conjecture. … The smooth type of…

Let $\Delta$ be a connected triangulated $(d-1)$-manifold without boundary, and let $h_r(\Delta)$ denote its $h$-numbers and $\beta_{j-1}(\Delta)$ its Betti numbers. Bagchi–Datta's…

Let $M$ be an $m$-vertex connected closed triangulated $d$-manifold, and let $\beta_i$ be its Betti numbers over some field. Let ${\cal K}_l^{\ast}(d)$ denote the class used in the…

Let $M$ be a triangulation of a connected closed $d$-manifold. Write $g_j(M)$ for its $g$-numbers, $\beta_i(M;\mathbb F)$ for Betti numbers over a field $\mathbb F$, and let…

Let $M$ be a compact three-dimensional manifold with triangulated boundary, and let $\vec I(M)$ be its invariant vector. A linear transformation $\mathcal A$ is constructed from a…

Let $d\geq 3$, and let $X$ be a connected, non-simply connected triangulated $d$-manifold. Lower bound conjecture for non-simply connected manifolds. Its face numbers satisfy … Equ…