19 problems

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

Let $M_d$ be the transfer matrix associated with the McMullen correspondence, with entries … A matrix is totally nonnegative if every minor is nonnegative. Bjbfrner's conjecture. A…

Let $K$ be a flag complex, and let $r$ be the number of vertices in a largest face of $K$. The face vector of $K$ is the vector of face numbers of $K$. Kalai–Eckhoff conjecture. If…

McMullen's conjecture. The face vector should satisfy $g_i\geq 0$ for $1\leq i\leq\lfloor k/2\rfloor$, and, if

Let $A$ be a vertex decomposable flag simplicial complex, and let $h(A)$ denote its $h$-vector. A face-vector conjecture. The vector $h(A)$ should be the face vector of some flag s…

Let $Q$ be a cubical $d$-polytope, meaning that every proper face of $Q$ is combinatorially a cube. Its cubical $h$-polynomial is written as … and its cubical $g$-vector is…

Let $\Sigma$ be a simplicial sphere of dimension $d-1$, with $f$-vector $(f_{-1},f_0,\ldots,f_{d-1})$ and associated $g$-vector defined by $g_k(\Sigma)=h_k-h_{k-1}$ for $k\leq d/2$…

The $g$-conjecture. The same characterization should be valid for the $f$-vectors of simplicial spheres.

Let $P$ be a polytope, and let its $f$-vector be the vector $(f_0,\ldots,f_{d-1})$, where $f_i$ is the number of $i$-dimensional proper faces of a $d$-polytope. A vector…

Eckhoff–Kalai conjecture. For every flag complex, there exists a balanced complex with the same face numbers.

Scaled face-vector conjecture. There is an integer $q$ such that $(1,qc_1,q^2c_2,\dots,q^dc_d)$ is the face vector of a flag complex.

Continuous face-vector representation conjecture. There are such numbers satisfying, for every $1\leq j\leq d$,

Let $f$ be a finite sequence of positive integers. A flag complex is a simplicial complex whose minimal non-faces are edges; its face vector records the numbers of faces in each di…

Let $\Delta$ be a $(d-1)$-dimensional simplicial complex that is doubly Cohen–Macaulay over $k$. Write ${\mathfrak h}_i(\Delta)$ for its ${\mathfrak h}$-vector and…

A simplicial sphere is a triangulation of a topological sphere, and its ${\mathfrak f}$-vector records the numbers of faces in each dimension. The ${\mathfrak g}$-theorem gives a s…

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…

Let $d$ and $k$ be integers with $d\geq 2k+1$, and let $X$ be a triangulated $d$-sphere with face-vector $(f_0,\dots,f_d)$. Generalized lower bound conjecture. For every $j$, the f…

Let $\Delta$ be a homology manifold, let $h'_i$ denote its h-prime vector, and let $\beta_{i-1}$ denote its $(i-1)$st Betti number. Manifold g-conjecture. For every…

McMullen's conjecture. The sequence $(h_0,h_1,\dots,h_d)$ is the h-vector of the boundary of a simplicial $d$-polytope if and only if $h_0=1$,…