Incompressible surface

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.

A hierarchy is a finite sequence of cuts along properly embedded two-sided incompressible surfaces, without spherical components, ending in disjoint three-balls. Disks are allowed. For an ambient manifold which is not orientable the two-sided cutting surfaces need not themselves be orientable. The existence of a hierarchy does not imply that every choice of incompressible cuts terminates.

A map between compact connected orientable irreducible 3-manifolds with nonempty boundaries, injective on the boundary and on the fundamental group, is homotopic to a homeomorphism relative to the boundary. The boundary determines degree $\pm 1$, and the cover corresponding to the image of the fundamental group consequently has one sheet. Straighten the preimage of an incompressible hierarchy surface, cut both manifolds, and repeat. At the terminal balls, extend the prescribed boundary homeomorphisms by coning. The homotopies and homeomorphisms glue across the straightened surface collars.

In an orientable irreducible three-manifold, an incompressible cut is fundamental group-injective by the Loop theorem. Lift a three-manifold hierarchy to the universal cover. The terminal ball pieces are contractible and their gluing surfaces have contractible universal covers. The incidence graph is a tree, by the van Kampen theorem normal form for successive cuts. Finite subtrees are contractible by successively attaching a terminal piece along a contractible intersection. Every sphere map has compact image contained in a finite subtree, so all higher homotopy groups of the universal cover vanish.

Let $P$ be a once-punctured torus and let $c$ be a nonseparating essential curve. The annulus $c\times [0,1]$ is an incompressible surface in $P\times [0,1]$. Cutting gives a pair of pants times an interval. Both products are genus-two handlebodies, since a compact connected surface with boundary times an interval is a handlebody of genus $1-\chi (P)$.

A compression disk for a two-sided embedded surface $F$ in a three-manifold is an embedded disk meeting $F$ exactly in its boundary, with boundary essential in $F$. Replacing an annular neighborhood of this boundary by two parallel copies of the disk reduces the surface's genus or separates it into simpler pieces.

A properly embedded topological surface is boundary-parallel if it cobounds a product region with a topological surface in the ambient boundary, with the remaining boundary of that product contained in the ambient boundary. An annulus parallel into one boundary component has zero net oriented boundary class on that component.

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.