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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.

Apply the boundary-degree argument to each source cut component. It maps to one target cut component and is a homeomorphism on each of its boundary components onto distinct target boundary components. Its relative degree must be $\pm 1$, so it meets the whole boundary of that target component. Since the global cut boundary map is one-to-one, two source components cannot map to the same target component; since it is onto, every target component is accounted for. Thus the cut components correspond bijectively, and each paired map satisfies the original hypotheses.

Induct on the remaining number of cuts in the chosen target three-manifold hierarchy, keeping fixed all boundary pieces already straightened. At the final stage the target components are three-balls. Each corresponding source component is irreducible with a single spherical boundary, and is therefore a three-ball by the inward-pushed-boundary argument above. Extend its prescribed boundary homeomorphism over the three-ball by coning. The original three-ball map and this homeomorphism are homotopic relative to the boundary: after identifying the target with a convex three-ball, take their straight-line homotopy. This establishes the induction base.

Reverse the cuts. On both copies of every cutting surface the two constructed maps agree with the fixed product homeomorphism of its collar, so the component homeomorphisms glue. Their homotopies also glue, because all cut boundaries have stayed fixed. Repeating through the three-manifold hierarchy givesThe proof derives the three-dimensional rigidity from the three-manifold hierarchy and the permitted surface-map straightening; it does not use the desired three-dimensional homeomorphism assertion as a quoted theorem.

Choose a connected proper incompressible surface $S$ with nonempty boundary as the first cut in a three-manifold hierarchy of $N$. Homotope $f$ relative to $\partial M$ into transverse position and let $F=f^{-1}(S)$.

We first make $F$ incompressible. If $D$ is a compression disk for $F$, its boundary maps to a loop in $S$ which is null-homotopic in $N$. Since $S$ is fundamental-group-injective, this loop is null-homotopic in $S$. Fill it in $S$ and alter $f$ in a small neighbourhood of $D$ so that its inverse image is obtained by compressing $F$ along $D$. The alteration is a homotopy: the filling and the old disk give a sphere map in $N$, which is null-homotopic by the asphericity proved above. A product collar of $S$ specifies the two sides of the modification. The integer ${\sum }_j\max (0,1-\chi (F_j))$ decreases at every essential compression; sphere components can be removed separately. A resulting sphere component of $F$ bounds a three-ball in $M$ by three-manifold irreducibility; move the map of that three-ball to one side of $S$ to remove it. The required extension and homotopy exist because the higher homotopy groups of $N$ vanish. Continuing leaves an incompressible $F$ with no spherical components.

For each component $F_j$, the map on fundamental groups to $S$ is injective. Indeed, its composition with the injection ${\pi }_1(S)\to {\pi }_1(N)$ equals the composition of ${\pi }_1(F_j)\to {\pi }_1(M)$ with the isomorphism $f_{\ast }$. Both maps in the latter composition are injective. A closed nonspherical orientable $F_j$ cannot occur: its fundamental group would inject into ${\pi }_1(S)$, a free group, whereas every subgroup of a free group is free and a closed positive-genus surface group is not free. Thus every component of $F$ has boundary.

Apply the permitted surface-map homotopy result to $F_j\to S$, fixing its boundary. A fundamental-group-injective map of compact surfaces is homotopic either to a covering or, in the annulus case here, to a map into one target boundary circle. The exceptional annulus case is impossible: its two distinct boundary circles would map onto the same circle of $\partial S$, contradicting the injectivity of the original boundary map. If $S$ is a disk, injectivity makes each $F_j$ a disk, and the unique boundary circle directly gives a single disk homeomorphism. Otherwise each component is homotopic to a cover of $S$.

Every point of $\partial S$ has exactly one preimage on $\partial F$, since $f{|}_{\partial M}$ is a homeomorphism. Every cover component contributes its positive degree to the number of preimages on every target boundary circle. Thus there is exactly one component and its covering degree is one. We have proved that $F\to S$ can be homotoped to a homeomorphism, relative to its boundary.

Extend this surface homotopy to $M$, and use the two-sided collars to make $f$ a product homeomorphism on a neighbourhood of $F$. The transverse normal direction is a degree-one local crossing, so the collars on its two sides match the two sides of $S$; any superfluous folds can be removed in the normal interval. We may now cut $M$ along $F$ and $N$ along $S$, obtaining a map whose restriction to the entire cut boundary is a homeomorphism. The other hypotheses also survive: the cut manifolds are irreducible, have nonempty boundary, and the inclusion of every cut component into its uncut manifold is fundamental-group-injective by the normal form of the van Kampen theorem. A loop killed by the cut map is consequently killed by $f_{\ast }$ and hence is trivial in its source cut component.

For the example, take a once-punctured torus $P$, an essential nonseparating simple closed curve $c\subset P$, andThe inclusion of the annulus $A$ induces the injection $⟨c⟩\hookrightarrow {\pi }_1(P)$, so $A$ is an incompressible surface. Cutting gives $(P\text{ cut along }c)\times I$, a pair of pants times an interval. A compact connected surface with boundary times an interval is a handlebody of genus $1-\chi (P)$: cut the surface to a disk by proper arcs and reverse those cuts as one-handle attachments. Both the punctured torus and the pair of pants have Euler characteristic $-1$. ThereforeThis self-reproducing incompressible annulus cut also shows why arbitrary incompressible cutting choices need not terminate.

A three-manifold hierarchy is a finite sequence of cuts along properly embedded two-sided incompressible surfaces, with no spherical components, whose final manifold is a disjoint union of three-balls. Disks are permitted. In an orientable ambient manifold two-sided surfaces are orientable; for an ambient manifold which is not orientable we permit two-sided surfaces without assuming their orientability as well.

For $S^1\times F$, first cut along $\{p\}\times F$, obtaining $I\times F$. This surface is two-sided and its fundamental group injects as a direct factor. Next choose essential two-sided curves in $F$ until the cut surface has boundary, and cut along their products with $I$. For orientable $F$ one nonseparating curve suffices. For $F$ which is not orientable and has genus at least two, the boundary of an embedded Möbius band is a two-sided essential curve; neither side is a disk, so its fundamental group injects. Cutting this curve leaves pieces with boundary, including the Möbius band. For each resulting compact surface choose a finite system of proper arcs cutting it into disks. The corresponding vertical disks give the remaining three-manifold hierarchy cuts. Essential annuli have injective fundamental group, and disks have trivial fundamental group, so all cuts are incompressible. The terminal products are $D^2\times I$, hence three-balls. This construction covers the possibilities without orientability in the stated range as well as the orientable ones.

Here is a terminating construction for a compact connected orientable irreducible three-manifold $M$ with nonempty boundary. We use the permitted general results in the following precise forms: a boundary-incompressible non-ball component has a connected two-sided nonseparating incompressible surface with nonempty boundary; and a sufficiently large collection of disjoint closed incompressible surfaces has two parallel components with no intervening member. These are respectively the existence and finiteness results for incompressible surfaces.

First perform essential boundary disk compressions as long as possible. Their number is finite: each nonseparating compression decreases boundary genus, while a separating compression divides a boundary component into two of smaller genus. More explicitly, the sum of $\max (0,2g-1)$ over the closed boundary components decreases at every essential compression. Each cut preserves three-manifold irreducibility. To see this for any incompressible cutter, a sphere in the cut manifold bounds a three-ball in the uncut manifold. A closed nonspherical incompressible component of the cutter cannot lie in this three-ball, since its fundamental group could not inject into the trivial group. A component with boundary cannot lie in an interior three-ball either. Thus the three-ball misses the cutter and survives in the cut manifold. Fundamental-group injectivity of two-sided incompressible surfaces follows from the Loop theorem, applied to their two boundary copies after cutting.

Push the boundary of each non-ball residual component slightly into the original $M$ to make a closed frontier $F_1$, omitting spherical components. This frontier is incompressible on its residual side, since all boundary compressions there have been exhausted. It is also incompressible on the side already cut off. Indeed, a hypothetical compression disk on that side can be made transverse to the earlier cutting surfaces. An innermost intersection circle bounds a disk on its cutting surface by incompressibility; replacing the corresponding subdisk removes the circle, without introducing intersections with earlier cutters. Repeat in the order of the cuts. The disk would finally lie in a product collar of the frontier, where its boundary is fundamental-group-injective, a contradiction. Thus $F_1$ is incompressible in $M$.

If a residual component is not a three-ball, choose the permitted nonseparating incompressible surface with boundary, cut it, and compress the new boundary fully again. Push that boundary in to obtain a new closed frontier $F_2$ disjoint from $F_1$. Continue. Three-manifold irreducibility persists, and the same innermost-circle proof makes every nonspherical frontier incompressible in the original $M$.

We explain why the parallel frontiers forced by the finiteness theorem cannot occur in this construction. The required product fact is: a connected nonseparating incompressible surface in $G\times I$ cannot be proper and disjoint from $G\times \{1\}$. To prove it, intersect the surface with vertical annuli $c\times I$ for essential curves $c$ in a closed $G$. Inessential intersection circles can be removed using incompressibility and the three-balls supplied by three-manifold irreducibility. An outermost intersection arc has both endpoints on the lower boundary; boundary compression along its annular bigon reduces the intersections. At least one resulting component stays nonseparating, since the mod-two relative class of the original surface is the sum of the resulting classes. Essential intersection circles can likewise be removed by surgery along the annulus running to its lower boundary. Retain a nonseparating incompressible component after each surgery. These operations reduce the intersection number and cut the base surface to one with boundary. Now intersect with the vertical disks of a proper arc system cutting that base to a disk. The same innermost-circle and outermost-arc operations make the surface disjoint from these disks. It then lies in a three-ball $D^2\times I$, where any proper two-sided surface separates. This is the contradiction proving the product fact.

If two frontiers in the construction were parallel with no other frontier between, their product region would contain one of the intervening nonseparating cutters, disjoint from the later frontier. The product fact forbids this. Consequently the closed-frontier construction has only finitely many stages. Once a residual irreducible component has only spherical boundary, push one boundary sphere inward. The three-ball it bounds cannot be on the collar side, which contains a boundary component of the manifold. The three-ball and the collar therefore exhaust that component, making it a three-ball. We have constructed a finite three-manifold hierarchy, and can choose all its cutting surfaces to have nonempty boundary.

We will use one further consequence in the map argument: these manifolds are aspherical. Work backwards through the three-manifold hierarchy. A three-ball is contractible. At one gluing stage, the van Kampen theorem normal form identifies the lifts of the cut pieces and cutting surface in the universal cover with a tree of spaces. The lifted pieces are contractible by induction, and the lifted cutting surfaces are contractible because the surfaces have boundary and are fundamental-group-injective. A finite subtree is contractible by removing terminal vertices; every sphere map meets only finitely many pieces. All higher homotopy groups therefore vanish at this stage, and induction proves asphericity from a three-manifold hierarchy. Apply this tree argument separately at each gluing stage.

The following three entries complete a single proof of the relative-boundary homeomorphism assertion.