  A centrally symmetric $2d$-vertex combinatorial triangulation of the product
of spheres $\S^i\times\S^{d-2-i}$ 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 $\S^i\times\S^{d-i-2}$. Enumerative
characteristics of $\B(i,d)$ and its boundary, and connections to another
conjecture of Sparla are also discussed.
