A necessary condition for the tightness of odd-dimensional combinatorial manifolds

TL;DR

This paper establishes a necessary condition for the tightness of certain odd-dimensional combinatorial manifolds and identifies specific cases where tightness is possible, contributing to the understanding of manifold topology.

Contribution

It introduces a necessary condition for tightness in odd-dimensional combinatorial manifolds and characterizes the only known tight three-manifolds with low Betti numbers.

Findings

No tight combinatorial three-manifold with Betti number ≤ 2 exists except two known cases.

Provides a criterion for tightness in $( ext{ell}-1)$-connected $(2 ext{ell}+1)$-manifolds.

Advances classification of tight combinatorial manifolds.

Abstract

We present a necessary condition for $(\ell-1)$-connected combinatorial $(2\ell +1)$-manifolds to be tight. As a corollary, we show that there is no tight combinatorial three-manifold with Betti number at most two other than the boundary of the four-simplex and the nine-vertex triangulation of the three-dimensional Klein bottle.