17 problems

Kalai–Sarkaria conjecture. If $K$ admits an embedding into $S^{r}$, then

Kalai–Sarkaria face-number conjecture. The complex $K$ has at most as many $d$-simplices as $\partial C_{r+1}(n)$, and

Non-embeddability conjecture. The complex $L$ does not embed in $\mathbb{R}^{2d}$. This generalizes the example discussed in the source and is described there as joint work in prog…

Let $H$ and $K$ be simplicial complexes, and write $H<K$ when $H$ is a minor of $K$, obtained by admissible contractions and deletions. Let $m$ be a nonnegative integer. Minor mono…

A tree is a locally finite, acyclic simplicial $1$-complex, and a baseray is the ray-based object used in the results of Section; the source's preceding discussion concerns existen…

Integral embedding criterion for joinpowers. An embedding

Asymptotic mod 2 Kühnel conjecture. There is a constant $c_k>0$ such that, whenever $[n]^{*k+1}$ has a ${\mathbb Z}_2$-embedding into a closed $2k$-manifold $M$, one has

For a $k$-complex $X$, define its ${\mathbb Z}$-rank $r_{\mathbb Z}(X)$ to be the minimum of $\operatorname{rk}\cap_{N,{\mathbb Z}}$ over all $(k-1)$-connected PL $2k$-manifolds…

Let $K$ be a $k$-complex and let $M$ be a PL $2k$-manifold. A map $h:K\to M$ is a ${\mathbb Z}_2$-embedding if the mod-$2$ intersection number of the images of every pair of non-ad…

Universal exponent conjecture. For each $3$-graph $H$, there exists $C_H>0$ such that every $3$-graph $G$ on $n$ vertices with at least

Let $\Delta$ be a simplicial complex of dimension $d$. Assume that $\Delta$ admits a PL embedding into $\mathbb{R}^{2d}$. GrunbaumKalaiSarkaria conjecture. Then … This conjecture s…

Nevo–Wagner conjecture. Theorem holds for general simplicial spheres.

Sarkaria's non-embeddability conjecture. If

Let $K$ be a $d$-dimensional balanced complex, and let $\mathfrak{o}(K)$ denote its van Kampen obstruction to PL embeddability in $\mathbb{S}^{2d}$. Balanced van Kampen-obstruction…

Let $K$ be a $d$-dimensional balanced complex, and let $<$ be a $(2,\ldots,2)$-admissible order. Let $K^{b,<}$ denote the corresponding balanced shift. Balanced PL-embeddability co…

A finite thick building is a finite building in which every panel is contained in at least three chambers, and its dimension is denoted by $d$. Non-embeddability conjecture. No $d$…

For an integer $n$, consider $n$-dimensional cell complexes $K$ whose underlying spaces are $(n-1)$-connected and do not embed in $S^{2n}$. An $h$-minor is a cell complex obtained…