Centrally symmetric manifolds with few vertices

TL;DR

This paper constructs centrally symmetric triangulations of products of spheres with few vertices for all relevant dimensions, confirming a conjecture for specific cases and introducing a new combinatorial manifold structure.

Contribution

It provides a general construction of minimal vertex triangulations of sphere products, including a proof of a conjecture by Sparla for certain cases.

Findings

Constructed vertex-transitive triangulations for all pairs (i,d) with 0≤i≤d-2.

Introduced the complex B(i,d) as a combinatorial manifold with boundary.

Connected the construction to Sparla's conjecture and enumerative properties.

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