The simplicial volume of 3-manifolds with boundary

TL;DR

This paper establishes sharp bounds for the simplicial volume of 3-manifolds with boundary, computes exact values for specific classes, and introduces new triangulation results, advancing understanding of 3-manifold topology.

Contribution

It provides the first exact computation of simplicial volume for manifolds with boundary having positive boundary volume and relates boundary and interior volumes.

Findings

Sharp lower bounds for simplicial volume in terms of boundary volume

Exact simplicial volume calculations for handlebodies and surface products

Minimal tetrahedra count in triangulations of surface products

Abstract

We provide sharp lower bounds for the simplicial volume of compact $3$-manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of $3$-manifolds, including handlebodies and products of surfaces with the interval. Our results provide the first exact computation of the simplicial volume of a compact manifold whose boundary has positive simplicial volume. We also compute the minimal number of tetrahedra in a (loose) triangulation of the product of a surface with the interval.