Surface Sum of Heegaard Splittings

Suppose Mi = Vi ∪ Wi （i = 1,2） are Heegaard splittings. A homeomorphism f ： F1 → F2 produces an attached manifold M = M1 ∪F1=F2 M2, where Fi ∪→ δ_Wi. In this paper we define a surface sum of Heegaard splittings induced from the Heegaard splittings of M1 and M2, 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.

Suppose Mi = Vi ∪ Wi(i = 1,2) are Heegaard splittings.A homeomorphism f :F1 → F2 produces an attached manifold M = M1 ∪F1=F2 M2,where Fi ■-Wi.In this paper we define a surface sum of Heegaard splittings induced from the Heegaard splittings of M1 and M2,and give a suffcient 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.