Towards Lower Bounds for Complexity of 3-Manifolds: a Program

TL;DR

This paper develops methods to estimate the minimal complexity of certain 3-manifolds, specifically torus bundles over S^1, by constructing special spines and studying theta-curves, advancing understanding of 3-manifold complexity.

Contribution

It introduces a new approach to lower bound estimation of 3-manifold complexity using theta-curves and constructs pseudominimal spines for torus bundles, proposing their minimality.

Findings

Constructed pseudominimal special spines for torus bundles.

Developed a method to estimate complexity from below using theta-curves.

Applied ideas to other 3-manifolds.

Abstract

For a 3-dimensional manifold $M^3$, its complexity $c(M^3)$, introduced by S.Matveev, is the minimal number of vertices of an almost simple spine of $M^3$; in many cases it is equal to the minimal number of tetrahedra in a singular triangulation of $M^3$. An approach to estimating $c(M^3)$ from below for total spaces of torus bundles over $S^1$, based on the study of theta-curves in the fibers, is developed, and pseudominimal special spines for these manifolds are constructed, which we conjecture to be their minimal spines. We also show how to apply some of these ideas to other 3-manifolds.