On unstabilized Heegaard splittings of amalgamated 3-manifolds

Abstract

Let [Formula: see text] [Formula: see text] be a Heegaard splitting of compact orientable [Formula: see text]-manifold, and [Formula: see text] be an amalgamation along an essential surface [Formula: see text]. In this paper, we show that if there is an essential disk [Formula: see text] in [Formula: see text], such that [Formula: see text] cuts [Formula: see text] into two product [Formula: see text]-bundles, one of which is [Formula: see text], and [Formula: see text] [Formula: see text], then [Formula: see text] is unstabilized and [Formula: see text] is uncritical. We also give a sufficient condition for [Formula: see text] to be critical.