Heegaard surfaces and the distance of amalgamation

TL;DR

This paper investigates how complex boundary gluings of 3-manifolds influence their Heegaard splittings, showing that sufficiently complicated gluings produce manifolds with predictable genus and standard splittings.

Contribution

It establishes a relationship between the complexity of boundary gluings and the Heegaard genus and structure of the resulting 3-manifold, extending to manifolds with multiple boundaries.

Findings

Sufficiently complicated gluings prevent the manifold from being $S^3$.

Small-genus Heegaard splittings are standard under complex gluings.

The Heegaard genus of the resulting manifold is additive minus boundary genus.

Abstract

Let $M_1$ and $M_2$ be orientable irreducible 3--manifolds with connected boundary and suppose $\partial M_1\cong\partial M_2$. Let $M$ be a closed 3--manifold obtained by gluing $M_1$ to $M_2$ along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then $M$ is not homeomorphic to $S^3$ and all small-genus Heegaard splittings of $M$ are standard in a certain sense. In particular, $g(M)=g(M_1)+g(M_2)-g(\partial M_i)$, where $g(M)$ denotes the Heegaard genus of $M$. This theorem is also true for certain manifolds with multiple boundary components.