Additivity of Heegaard genera of bounded surface sums

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 $M^{\ast }$, where $M^i=M_i\setminus {\partial }_i\times I$, and $M^{\ast }={\partial }_1\times I{\cup }_F{\partial }_2\times I$. Furthermore, once both ${\partial }_i\setminus F$ are connected, then $g(M)=Min\{g(M_1)+g(M_2),\alpha \}$, where $\alpha =g(M_1)+g(M_2)+1/2(2\chi (F)+2-\chi ({\partial }_1)-\chi ({\partial }_2))-Max\{g({\partial }_1),g({\partial }_2)\}$; in particular $g(M)=g(M_1)+g(M_2)$ if and only if $\chi (F)\ge 1/2Max\{\chi ({\partial }_1),\chi ({\partial }_2)\}.$ The proofs rely on Scharlemann-Tomova's theorem.

References in corpus (1)

• On the Heegaard splittings of amalgamated 3-manifolds