Unstabilized dual Heegaard splittings of -manifolds.

Let be a -manifold, be an essential planar surface which cuts into two 3-manifolds and . Suppose is a Heegaard splitting of , is the dual Heegaard splitting of and along , where and . In this paper, we give a condition of unstabilized dual Heegaard splittings of -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 -manifolds for any Hempel's distance. [ABSTRACT FROM AUTHOR]