Three-manifold

A three-manifold is a topological manifold locally modeled on ${\mathbb{R}}^3$, or on a half-space at its boundary.

An incompressible surface is a properly embedded two-sided topological surface with no compressing disk, subject to the usual exclusion of inessential spherical components. A compressing disk has boundary an essential closed curve on the topological surface and interior disjoint from it.

In an orientable irreducible Seifert fibered space, an essential two-sided incompressible surface that is also a boundary-incompressible surface can be isotoped to a horizontal surface in a Seifert fibered space or a vertical surface in a Seifert fibered space.

A properly embedded topological surface is boundary-incompressible when no disk in the ambient three-manifold joins an essential arc on the topological surface to an arc on the ambient boundary, with the disk's interior disjoint from both.

A Heegaard splitting divides a closed three-manifold into two handlebodies with common boundary. For a three-manifold with boundary, one uses compression bodies, which can retain negative boundary components.

A compression body is obtained from a product $F\times I$ by attaching one-handles to $F\times \{1\}$, allowing three-ball components as well. Its negative boundary is the retained copy of $F$ and its positive boundary is the other boundary component. A handlebody has empty negative boundary.

The Heegaard surface is the common positive boundary of the two pieces in a Heegaard splitting.

A Heegaard diagram marks the boundaries of compressing disks from the two sides of a Heegaard surface. Attaching two-handles along the two collections, and capping resulting spherical boundary components, reconstructs the three-manifold.

Thicken the planar graph underlying a knot diagram. Its boundary supports a Heegaard diagram with face disks on one side and crossing-tunnel disks on the other. The face generators and crossing curves give a Dehn presentation of a knot group.

A genus-$g$ handlebody is a three-ball with $g$ one-handles attached. Its boundary is a closed orientable topological surface of genus $g$.

Every connected closed orientable three-manifold with finite fundamental group admits spherical geometry. With cyclic fundamental group, it is a lens space.