Heegaard splitting

Remarks

Every closed connected orientable $3$-manifold has a Heegaard splitting. Take a triangulation and let $V$ be a regular neighbourhood of its $1$-skeleton. The closure $W$ of the complement is a regular neighbourhood of the dual $1$-skeleton. The Heegaard genus of $M$ is the least genus of a Heegaard splitting.

Example: $S^3 = \set{(z, w) \in \C^2 : \abs{z}^2 + \abs{w}^2 = 1}$ is the union of the solid tori $V = \set{\abs{z} \le \abs{w}}$ and $W = \set{\abs{z} \ge \abs{w}}$, which meet in the torus $\abs{z} = \abs{w} = 1/\sqrt{2}$; this is a splitting of genus $1$. Lens spaces also have splittings of genus $1$.

Non-example: let $N$ be a closed tubular neighbourhood of a trefoil knot in $S^3$ and $E$ the closure of its complement. Then $S^3 = N \cup E$ with $N \cap E = \partial N = \partial E$ a torus, and $N$ is a solid torus, but $E$ is not a handlebody. Indeed the fundamental group of $E$ is $\langle x, y \mid x^2 = y^3 \rangle$, which is not free, while handlebodies have free fundamental groups. So clause 2 fails.