3 problems

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…

The centrally symmetric lower bound conjecture. If $\Delta$ is a simplicial sphere, a connected simplicial manifold, or a normal pseudomanifold, then