# Incompressible surface

In mathematics, an **incompressible surface**, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold.

For a precise definition, suppose that *S* is a compact surface properly embedded in a 3-manifold *M*. Suppose that *D* is a disk, also embedded in *M*, with

Suppose finally that the curve in *S* does not bound a disk inside of *S*. Then *D* is called a **compressing disk** for *S* and we also call *S* a **compressible surface** in *M*. If no such disk exists and *S* is not the 2-sphere, then we call *S* **incompressible** (or **geometrically incompressible**).

Note that we must exclude the 2-sphere to get any interesting consequences for the 3-manifold. Every 3-manifold has many embedded 2-spheres, and a 2-sphere embedded in a 3-manifold never has a compressing disc.

Sometimes one defines an **incompressible sphere** to be a 2-sphere in a 3-manifold that does not bound a 3-ball. Thus, such a sphere either does not separate the 3-manifold or gives a nontrivial connected sum decomposition. Since this notion of incompressibility for a sphere is quite different from the above definition for surfaces, often an incompressible sphere is instead referred to as an **essential sphere** or **reducing sphere**.

There is also an algebraic version of incompressibility: Suppose is a proper embedding of a compact surface. Then *S* is **-injective** (or **algebraically incompressible**) if the induced map on fundamental groups

is injective.

In general, every -injective surface is incompressible, but the reverse implication is not always true. For instance, the Lens space contains an incompressible Klein bottle that is not -injective. However, if is a two-sided properly embedded, compact surface (not a 2-sphere), the loop theorem implies is incompressible if and only if it is -injective.

## See also

- Compression (3-manifold)
- Haken manifold
- Virtually Haken conjecture
- Thurston norm
- Boundary-incompressible surface

## References

- W. Jaco,
*Lectures on Three-Manifold Topology*, volume 43 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, R.I., 1980.