Simplifying 3-manifolds in R^4

TL;DR

This paper demonstrates a method to isotopically simplify smooth embeddings of 3-manifolds in S^3 x R into Heegaard positions, and explores implications for the Schoenflies conjecture and embedding conditions for connected sums.

Contribution

It introduces a new isotopy technique for 3-manifold embeddings in R^4 and links the problem to the mapping class group and Schoenflies conjecture.

Findings

Embeddings can be isotoped to Heegaard positions under certain conditions.

Provides a criterion for embedding connected sums of 3-manifolds in R^4.

Connects embedding uniqueness to the mapping class group and Schoenflies conjecture.

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.