Decompositions of the 3-sphere and lens spaces with three handlebodies

TL;DR

This paper classifies and analyzes decompositions of the 3-sphere and lens spaces into three handlebodies, focusing on small genus cases and the stabilization properties of these decompositions.

Contribution

It provides a classification of three-handlebody decompositions for specific 3-manifolds and investigates their stabilization status, advancing understanding of 3-manifold decompositions.

Findings

Classified small genus decompositions of the 3-sphere and lens spaces

Identified which decompositions are stabilized

Analyzed stabilization operations for these decompositions

Abstract

In this paper, we consider decompositions of 3-manifolds with three handlebodies. We classify such decompositions of the 3-sphere and lens spaces with small genera. These decompositions admit operations called stabilizations. We also determine whether these decompositions are stabilized.