Minimal Balanced Triangulations of Sphere Bundles over the Circle

TL;DR

This paper establishes the minimal number of vertices required for balanced triangulations of sphere bundles over the circle, revealing precise formulas depending on the bundle's orientability and dimension.

Contribution

It provides exact vertex bounds for balanced triangulations of sphere bundles over the circle, extending to all balanced complexes triangulating certain homology manifolds.

Findings

Minimum vertices for orientable odd-dimensional bundles is 3d

Minimum vertices for non-orientable even-dimensional bundles is 3d+2

Results apply to balanced complexes with specific Betti number conditions

Abstract

We determine the minimum number of vertices needed to provide balanced triangulations of $\mathbb S^{d-2}$-bundles over $\mathbb S^1$. If $d$ is odd and the bundle is orientable, or $d$ is even and the bundle is non-orientable, the minimum number of vertices is $3d$; otherwise, it is $3d+2$. Similar results apply to all balanced simplicial complexes that triangulate homology manifolds with $\beta_1\neq 0$ and $\beta_2=0$, where $\beta_i$'s are the Betti numbers, computed with coefficients in $\mathbb Q$.