Heegaard splitting of a closed oriented three-manifold

For $g\in {\mathbb{Z}}_{\ge 0}$, an orientable handlebody of genus $g$ is a compact connected orientable smooth $3$-manifold obtained from a $3$-ball by attaching $g$ smooth $1$-handles along orientation-compatible pairs of disjoint boundary disks and rounding the corners. A $1$-handle here is a copy of $D^2\times [-1,1]$ attached by its two end disks to those boundary disks. Its boundary is a connected closed orientable surface of genus $g$. A Heegaard splitting of a closed connected oriented smooth $3$-manifold $M$ is a decomposition

$$M=H_{-}{\cup }_{\phi }{H}_{+}$$

in which $H_{-}$ and $H_{+}$ are orientable handlebodies of the same genus, their common boundary is a connected closed surface $\mathrm{\Sigma }=\mathrm{\partial }{H}_{-}=\mathrm{\partial }{H}_{+}$, and $\phi :\mathrm{\partial }{H}_{-}\to \mathrm{\partial }{H}_{+}$ is an orientation-reversing diffeomorphism whose gluing gives $M$. The surface $\mathrm{\Sigma }$ is the Heegaard surface of the splitting.

Smooth orientations on the manifold and surface

Smooth embeddings of attaching disks and handlebodies

Danny Calegari, Notes on 3-Manifolds, Definitions 3.20–3.21