Three-manifold hierarchy

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 $1-\chi (P)$.