HakenManifold

A Haken 3-manifold, named after the German mathematician Wolfgang Haken, is a special type of 3-manifold with useful properties. A 3-manifold is a three-dimensional topological space that is locally homeomorphic to the Euclidean space $\mathbb{R}^3$. A Haken manifold has additional structure, defined as follows:

Definition of a Haken 3-manifold:

A Haken 3-manifold is a compact, orientable, irreducible 3-manifold that contains a properly embedded, two-sided incompressible surface other than a sphere or a disk.

1. Compact: A topological space is compact if every open cover has a finite subcover.

2. Orientable: A space is orientable if it has a consistent "direction". For example, a Möbius strip is not orientable because if you follow a path around the strip, you end up "flipped" compared to your original orientation.

3. Irreducible: A 3-manifold is irreducible if any embedded sphere bounds a 3-ball.

4. Properly embedded, two-sided incompressible surface: A surface $S$ is embedded in the manifold $M$ if it is a subset of $M$ and it is homeomorphic to a 2-manifold. It is two-sided if it has a neighborhood in $M$ that is homeomorphic to the product of $S$ with the interval $[-1, 1]$. It is incompressible if any loop in $S$ which bounds a disk in $M$ already bounds a disk in $S$.

A Seifert fibered space is a 3-manifold that is fibred over a 2-orbifold. The fibers are circles which can be thought of as "twisting" around the orbifold. More precisely, it is a circle bundle over an orbifold, except that it may have a finite number of exceptional fibers, which are circles that cover the orbifold multiple times.

The following theorem helps understand how Haken 3-manifolds and Seifert fibered spaces relate:

Every compact, orientable, irreducible 3-manifold with nonempty boundary is either a Seifert fibered space or contains a properly embedded, two-sided incompressible surface. In other words, such a 3-manifold is either a Seifert fibered space or a Haken 3-manifold.

Therefore, these two classes of 3-manifolds are mutually exclusive and together they cover all compact, orientable, irreducible 3-manifolds with nonempty boundary.

Haken's original definition of his manifolds included the condition that they be "sufficiently large", which means that they contain a properly embedded, two-sided incompressible surface. He then showed that many important classes of 3-manifolds are indeed Haken, including Seifert fibered spaces.