Surface Sum of Heegaard Splittings

Abstract:

      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.