Additivity of Heegaard genera of bounded surface sums

TL;DR

This paper investigates how the Heegaard genus behaves under surface sums of 3-manifolds, establishing conditions for additivity based on the properties of the boundary surfaces and the complexity of the splitting.

Contribution

It provides a new formula for the Heegaard genus of a surface sum of 3-manifolds, extending understanding of genus additivity in the presence of high distance Heegaard splittings.

Findings

Heegaard genus is additive under certain conditions involving boundary surface connectivity.

Derived a formula relating genus of the sum to the genera of the summands and boundary surface characteristics.

Proved that genus equality holds when the Euler characteristic of the surface exceeds a specific bound.

Abstract

Let $M$ be a surface sum of 3-manifolds $M_1$ and $M_2$ along a bounded connected surface $F$ and $\partial_i$ be the component of $\partial M_i$ containing $F$. If $M_i$ has a high distance Heegaard splitting, then any minimal Heegaard splitting of $M$ is the amalgamation of those of $M^1, M^2$ and $M^*$, where $M^i=M_i\setminus\partial_i\times I$, and $M^{*}=\partial_1\times I\cup_{F} \partial_2\times I$. Furthermore, once both $\partial_i\setminus F$ are connected, then $g(M) = Min\bigl\{g(M_1)+g(M_2), \alpha\bigr\}$, where $\alpha = g(M_1) + g(M_2) + 1/2(2\chi(F) + 2 - \chi(\partial_1) - \chi(\partial_2)) - Max\bigl\{g(\partial_1), g(\partial_2)\bigl\}$; in particular $g(M)=g(M_1)+g(M_2)$ if and only if $\chi(F)\geq 1/2Max\bigl\{\chi(\partial_1), \chi(\partial_2)\bigr\}.$ The proofs rely on Scharlemann-Tomova's theorem.