Complexities of 3-manifolds from triangulations, Heegaard splittings, and surgery presentations

TL;DR

This paper investigates the relationships between different measures of complexity in 3-manifolds, establishing linear inequalities and optimal bounds, with applications to estimating invariants related to geometric group theory and knot theory.

Contribution

It introduces explicit geometric constructions linking triangulation, Heegaard splitting, and surgery complexities, and proves these inequalities are asymptotically optimal.

Findings

Linear inequalities relate different 3-manifold complexities

Explicit geometric constructions demonstrate these relationships

Inequalities are proven to be asymptotically optimal

Abstract

We study complexities of 3-manifolds defined from triangulations, Heegaard splittings, and surgery presentations. We show that these complexities are related by linear inequalities, by presenting explicit geometric constructions. We also show that our linear inequalities are asymptotically optimal. Our results are used in [arXiv:1405.1805] to estimate Cheeger-Gromov $L^2$ $\rho$-invariants in terms of geometric group theoretic and knot theoretic data.