Embedding Heegaard Decompositions

TL;DR

This paper explores the relationship between embeddings of 3-manifolds in four-dimensional space and Heegaard splittings, identifying obstructions related to the curve complex that prevent certain embeddings.

Contribution

It introduces an obstruction based on the geometry of the curve complex that limits the realization of Heegaard splittings as embeddings in -dimensional space.

Findings

Obstructions from the curve complex affect embedding realizability.

Heegaard splittings can be obstructed by geometric properties.

The study links topology, geometry, and embeddings of 3-manifolds.

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 $M\hookrightarrow \mathbb{R}^4$.