3 problems

Let $k\geq 2$. A flag sphere is a simplicial sphere with no missing faces of dimension larger than one, and $g_0,g_1,\ldots,g_k$ are its $g$-numbers. Charney–Davis conjecture. Ever…

Let $S(j,d-1)$ denote the class of $(\mathbb{Z}/2\mathbb{Z})$-homology $(d-1)$-spheres without missing faces of dimension larger than $j$. Nevo's conjecture. For $1\leq j\leq d$, s…

Let $S(2,4)$ be the class of $(\mathbb{Z}/2\mathbb{Z})$-homology $4$-spheres without missing faces of dimension larger than two, and let $g_1=f_0-6$. Nevo's lower-bound conjecture.…