Unstabilized dual Heegaard splittings of 3-manifolds

Abstract

Let [Formula: see text] be a [Formula: see text]-manifold, [Formula: see text] be an essential planar surface which cuts [Formula: see text] into two 3-manifolds [Formula: see text] and [Formula: see text]. Suppose [Formula: see text] [Formula: see text] is a Heegaard splitting of [Formula: see text], [Formula: see text] is the dual Heegaard splitting of [Formula: see text] and [Formula: see text] along [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, we give a condition of unstabilized dual Heegaard splittings of [Formula: see text]-manifolds by using Hempel’s distance and the method of proof of Gordon’s Conjecture. Also, we give a counterexample for stabilized dual Heegaard splittings of [Formula: see text]-manifolds for any Hempel’s distance.