Centrally symmetric manifolds with few vertices

Abstract

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 $0\le i\le d-2$. 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.

15 pages, 2 figures