A Lower Bound for the Heegaard Genera of Annulus Sum
Received:November 27, 2009  Revised:April 27, 2010
Key Word: genus   distance   annulus.
Fund ProjectL:Supported by the Fundamental Research Funds for the Central Universities and the Key Grant of National Natural Science Foundation of China (Grant No.10931005).
 Author Name Affiliation Feng Ling LI School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China Feng Chun LEI School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China
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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 $M_2$.