Embedding Heegaard Decompositions

Abstract

A smooth embedding of a closed $3$-manifold $M$ in ${\mathbb{R}}^4$ may generically be composed with projection to the fourth coordinate to determine a Morse function on $M$ and hence a Heegaard splitting $M=X{\cup }_{\Sigma }Y$. However, starting with a Heegaard splitting, we find an obstruction coming from the geometry of the curve complex $C(\Sigma )$ to realizing a corresponding embedding $M\hookrightarrow {\mathbb{R}}^4$.

6 pages