A Lower Bound for the Heegaard Genera of Annulus Sum

DOI：10.3770/j.issn:1000-341X.2011.04.002

 作者 单位 李风玲 大连理工大学数学科学学院, 辽宁 大连 116024 雷逢春 大连理工大学数学科学学院, 辽宁 大连 116024

设$M_{i}$, $i=1,2$, 为紧致可定向三维流形, 且$A_{i}$是$\partial M_i$的分支$F_i$上的不可压缩平环.设$A_{1}$在$F_{1}$上是分离的而$A_{2}$在$F_{2}$上是非分离的.令$M$为$M_1$和$M_2$沿$A_1$和$A_2$的平环和.本文在子流形$M_1$和$M_2$的Heegaard距离的条件下给出了平环和$M$的亏格的一个下界.

Let $M_{i}$, $i=1,2$, be a compact orientable 3-manifold, and $A_{i}$ an incompressible annulus on a component $F_i$ of $\partial M_i$. Suppose $A_{1}$ is separating on $F_{1}$ and $A_{2}$ is non-separating on $F_{2}$. Let $M$ be the annulus sum of $M_1$ and $M_2$ along $A_1$ and $A_2$. In the present paper, we give a lower bound for the genus of the annulus sum $M$ in the condition of the Heegaard distances of the submanifolds $M_1$ and $M_2$.