Surface Sum of Heegaard Splittings
Received:March 16, 2007  Revised:May 26, 2007
Key Words: Heegaard splitting   stabilized   amalgamation.
Fund Project:the Specialized Research Fund for the Doctoral Program of Higher Education (No.200801411069).
 Author Name Affiliation ZHANG Ming Xing School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, China
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Suppose $M_i=V_i\cup W_i$~($i=1,2$) are Heegaard splittings. A homeomorphism $f:F_1\rightarrow F_2$ produces an attached manifold $M=M_1\cup_{F_1=F_2}M_2$, where $F_i \subset \partial_-W_i$. In this paper we define a surface sum of Heegaard splittings induced from the Heegaard splittings of $M_1$ and $M_2$, and give a sufficient condition when the surface sum of Heegaard splitting is stabilized. We also give examples showing that the surface sum of Heegaard splittings can be unstabilized. This indicates that the surface sum of Heegaard splittings and the amalgamation of Heegaard splittings can give different Heegaard structures.