11 problems

Let $\mathcal{K}$ be a flag simplicial complex, and consider the grading on $H^*(\mathcal{Z}_\mathcal{K};\mathbb{Q})$ specified in the source's preceding theorem. Koszul formality…

Let $\mathcal{K}$ be a flag simplicial complex. The spaces $\mathrm{DJ}(\mathcal{K})$ and $\mathcal{Z}_\mathcal{K}$ are defined by the Davis–Januszkiewicz and moment-angle construc…

Let $\mathcal{K}$ be a simplicial complex and let $\mathbf{k}$ be a ring such that $H_*(\Omega\mathrm{DJ}(\mathcal{K});\mathbf{k})$ is a free $\mathbf{k}$-module. Recall that a Hop…

Minimal-realisation conjecture. The complex $\partial\Delta_w$ is the smallest complex realising $w$ for any iterated higher Whitehead product.

Golod–co-$H$-space conjecture. The complex $K$ is Golod if and only if $\mathcal Z_K$ is a co-$H$-space.

A simplicial sphere $K$ is strongly $B$-rigid if, whenever a simplicial sphere $L$ satisfies … then $K$ and $L$ are combinatorially equivalent. Strong B-rigidity conjecture. Every…

Indecomposability conjecture. This graded ring is indecomposable, and consequently $\mathcal {Z}_{K}$ is a prime manifold. The analogous assertion is proved in the paper for flag…

Let $K_1$ and $K_2$ be generalized homology spheres of dimension $n-1$, with $m_1$ and $m_2$ vertices, respectively. Let … The operation $\mathcal{G}$ on manifolds is the gyration,…

Golod characterization. The moment-angle complex $(D^2,S^1)^K$ is a co-$H$-space if and only if $K$ is Golod.

Pyramid characterization conjecture. Pyramids are the only polytopes whose Buchstaber number is equal to $1$.

Let $K$ be a shifted simplicial complex, let $(CX_i,X_i,x_i)$ be based pairs indexed by the vertices of $K$, and write $Z(K;(\underline{CX},\underline{X}))$ for the associated poly…