Normalizing Heegaard-scharlemann-thompson Splittings

We define a Heegaard-Scharlemann-Thompson (HST) splitting of a 3-manifold M to be a sequence of pairwise-disjoint, embedded surfaces, {Fi}, such that for each odd value of i, Fi is a Heegaard splitting of the submanifold of M cobounded by Fi−1 and Fi+1. Our main result is the following: Suppose M (6= B or S) is an irreducible submanifold of a triangulated 3-manifold, bounded by a normal or almost normal surface, and containing at most one maximal normal 2sphere. If {Fi} is a strongly irreducible HST splitting of M then we may isotope it so that for each even value of i the surface Fi is normal and for each odd value of i the surface Fi is almost normal. We then show how various theorems of Rubinstein, Thompson, Stocking and Schleimer follow from this result. We also show how our results imply the following: (1) a manifold that contains a non-separating surface contains an almost normal one, and (2) if a manifold contains a normal Heegaard surface then it contains two almost normal ones that are topologically parallel to it.