Simplifying 3-manifolds in R^4

Abstract

We show that a smooth embedding of a closed 3-manifold in S^3 x R can be isotoped so that every generic level divides S^3 x t into two handlebodies (i.e., is Heegaard) provided the original embedding has a unique local maximum with respect to the R coordinate. This allows uniqueness of embeddings to be studied via the mapping class group of surfaces and the Schoenflies conjecture is considered in this light. We also give a necessary and sufficient condition that a 3-manifold connected summed with arbitrarily many copies of S^1 x S^2 embeds in R^4.