We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.The pioneering work of Casson and Gordon [1] shows that a minimal genus Heegaard splitting of an irreducible, non-Haken 3-manifold is necessarily strongly irreducible; by contrast, Haken [2] showed that a minimal genus (indeed, any) Heegaard splitting of a composite 3manifold is necessarily reducible, and hence weakly reducible. The following question of Moriah [8] is therefore quite natural:
Question 1 ([8], Question 1.2). Can a 3-manifold M have both weakly reducible and strongly irreducible minimal genus Heegaard splittings?We answer this question affirmatively:Theorem 2. There exist infinitely many closed, orientable 3-manifolds of Heegaard genus 3, each admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings.