Embedding Heegaard Decompositions

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