Enumerative properties of triangulations of spherical bundles over S^1

TL;DR

This paper characterizes all possible pairs of vertices and edges in simplicial triangulations of spherical bundles over S^1, highlighting the uniqueness of certain known triangulations among homology manifolds with specific Betti number conditions.

Contribution

It provides a complete characterization of (v,e) pairs for triangulations of S^k-bundles over S^1, identifying the uniqueness of Kuhnel's triangulations under certain topological constraints.

Findings

Complete (v,e) characterization for triangulations of S^k-bundles over S^1.

Uniqueness of Kuhnel's triangulations among specific homology manifolds.

Identification of conditions under which these triangulations are unique.

Abstract

We give a complete characterization of all possible pairs (v,e), where v is the number of vertices and e is the number of edges, of any simplicial triangulation of an S^k-bundle over S^1. The main point is that Kuhnel's triangulations of S^{2k+1} x S^1 and the nonorientable S^{2k}-bundle over S^1 are unique among all triangulations of (n-1)-dimensional homology manifolds with first Betti number nonzero, vanishing second Betti number, and 2n+1 vertices.