Heegaard splittings of the form $H+nK$

Suppose that a three-manifold $M$ contains infinitely many distinct strongly irreducible Heegaard splittings $H+nK$, obtained by Haken summing the surface $H$ with $n$ copies of the surface $K$. We show that $K$ is incompressible. All known examples, of manifolds containing infinitely many irreducible Heegaard splittings, are of this form. We also give new examples of such manifolds.

2010 Mathematics Subject Classification

Primary 57N10. Secondary 57M25.

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Published 1 January 2006