Centrally symmetric manifolds with few vertices

A centrally symmetric $2d$-vertex combinatorial triangulation of the product of spheres $§^i\times§^{d-2-i}$ is constructed for all pairs of non-negative integers $i$ and $d$ with $0\leq i \leq d-2$. For the case of $i=d-2-i$, the existence of such a triangulation was conjectured by Sparla. The constructed complex admits a vertex-transitive action by a group of order $4d$. The crux of this construction is a definition of a certain full-dimensional subcomplex, $\B(i,d)$, of the boundary complex of the $d$-dimensional cross-polytope. This complex $\B(i,d)$ is a combinatorial manifold with boundary and its boundary provides a required triangulation of $§^i\times§^{d-i-2}$. Enumerative characteristics of $\B(i,d)$ and its boundary, and connections to another conjecture of Sparla are also discussed.

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