Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds

TL;DR

This paper investigates how complex gluings of 3-manifolds affect their Heegaard splittings, providing counterexamples to existing conjectures and insights into their uniqueness and stabilization properties.

Contribution

It demonstrates that sufficiently complicated gluings can both stabilize and destabilize Heegaard splittings, challenging previous assumptions and resolving longstanding conjectures.

Findings

Counter-examples to the Stabilization Conjecture

Resolution of a higher genus Gordon conjecture

Results on the uniqueness of Heegaard splitting amalgamations

Abstract

Let M_1 and M_2 be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism $\phi:\bdy M_1 \to \bdy M_2$. We analyze the relationship between the sets of low genus Heegaard splittings of M_1, M_2, and M, assuming the map \phi is "sufficiently complicated." This analysis yields counter-examples to the Stabilization Conjecture, a resolution of the higher genus analogue of a conjecture of Gordon, and a result about the uniqueness of expressions of Heegaard splittings as amalgamations.