Prime decomposition of compact connected orientable three-manifolds

Fix the finite PL structure on $M$ supplied by [F1]. We give the compatibility operation needed below, rather than assume that an arbitrary TOP sphere was PL in this structure. For a finite disjoint locally flat sphere system, cut along its spheres using their local flat charts. The orientation of each sphere and of $M$ labels the two local sides consistently; the cut space is therefore a compact TOP manifold with two boundary copies of each sphere, and its old boundary unchanged. Assign the standard finite sphere PL parameter to one copy and the identical parameter under the cutting identification to the other. These finitely many new boundary components have collars by [F5]; keep also the old PL boundary collar of $M$, choosing all collars disjoint. Apply the relative assertion of [F1] to each cut component. On the smaller new boundary collars the cutting identification is the identity in the assigned sphere parameters, so gluing the collars gives an actual PL product across each seam. Finite product-cell subdivisions give a finite PL structure ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ on the original $M$ in which every sphere of the system is PL. It agrees with the original structure $\mathrm{\Sigma }$ near $\mathrm{\partial }M$. This operation also applies to any finite TOP connected-sum presentation: each gluing sphere receives matching parameters, and its cap balls have the cone parameters of [F4].

Two such finite PL structures $\mathrm{\Sigma },{\mathrm{\Sigma }}^{\mathrm{\prime }}$ on $M$ agreeing near its boundary are related by a homeomorphism $u:(M,\mathrm{\Sigma })\to (M,{\mathrm{\Sigma }}^{\mathrm{\prime }})$ TOP isotopic to the identity, fixed on a smaller boundary collar. Here is the needed relative comparison construction. Cover the compact remainder of that collar by ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ coordinate domains. Subdivide the finite $\mathrm{\Sigma }$ triangulation so finely that the closed thirty-two-fold star of each simplex meeting the remainder is in one of these domains; finite cover shrinking and the geometric mesh decrease give this subdivision. Form the vertex, edge, face and tetrahedron handles of [F1], Proof1.1, with all transverse extensions inside these stars. At each index straighten the current identity map in the chosen ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ coordinate chart using [F2]. Its source parameter is a $\mathrm{\Sigma }$ PL handle; for positive index its attaching band lies in the already straightened region. Protect smaller earlier core neighborhoods and trim the attaching bands exactly as in [F1], Proof2.1. Here the coordinate control is finite and explicit. Choose the transverse extensions inside the closed two-fold star of their core simplex; at a fixed index their nonidentity supports are disjoint, after trimming the protected attaching bands. A correction can therefore carry a point only between simplices at most four adjacency steps apart, and at most one correction of that index moves it. Before the last index there have been at most three such stages. Consequently the current image of a handle and its transverse extension stays in the closed thirty-two-fold star of its core simplex, inside its chosen target coordinate domain. A sufficiently fine subdivision has this property because each fixed finite iterate of the star has mesh tending to zero. Thus every coordinate embedding to which [F2] is applied really has its whole domain in the chosen chart; no PL compatibility of the original TOP transition maps is assumed. Each correction is the identity on its attaching boundary and outside its own compact twice-core, so extends to all of $M$. Subsequent widths are chosen inside the remaining PL neighborhoods. The index-zero through index-three cores and their protected bands cover the remainder, just as in [F1], Proof2.1. At the endpoint the map is locally PL everywhere, including the unchanged boundary collar. Its inverse is locally PL on the same finite image subdivisions, hence it is a PL homeomorphism. Concatenating the finitely many supported isotopies gives $u$ and its TOP ambient isotopy. In particular, pulling the system of step1.1 back by $u$ makes it a PL sphere system in the ORIGINAL structure by an actual TOP ambient isotopy; this statement does not assert a smooth isotopy.

Consequently PL primeness and PL irreducibility for a finite PL manifold agree with their TOP tame versions. One recognition detail is needed in the reverse direction: if a closed finite PL manifold is TOP homeomorphic to $S^3$, pull its PL structure to standard $S^3$ and apply step2.1 (there is no boundary collar condition) to obtain a PL homeomorphism to standard PL $S^3$. If a finite PL manifold with sphere boundary is a TOP ball, its PL double is TOP $S^3$; the preceding recognition and [F4] make each half a PL ball. Thus neither a TOP sphere factor nor a TOP ball side is silently being recognized as PL without the comparison construction. Indeed a TOP splitting sphere can be moved to a PL sphere by steps1.1 and2.1, and the ambient homeomorphism identifies its two cut components. If its PL side is a parametrized ball, its TOP side is a parametrized TOP ball; conversely every PL splitting is an admitted TOP splitting. The same argument compares every finite TOP connected-sum presentation with a finite PL presentation. Capping choices do not affect the homeomorphism types: conjugate any boundary identification to a sphere map and extend by the cone formula $[x,t]\mapsto [f(x),t]$, which is a homeomorphism even when $f$ is only TOP. No assertion that a TOP ball parameter was PL in the old structure is used. Now apply [F3] to the fixed finite triangulation of $M$. It gives a finite connected sum of factors prime in the full tame TOP sense, with boundary allowed and every original boundary mark retained. This proves existence. It also proves the asserted prime-but-not-irreducible exception: [F3] proves $S^1\times S^2$ PL prime, hence TOP prime by this step, and $\{\ast \}\times S^2$ is nonseparating and cannot bound a ball. Every other prime is irreducible by [F3] and this same compatibility operation.

We prove uniqueness first for finite PL presentations in one fixed PL manifold. A system $S$ will be said to reduce $M$ to a list $P_1,\dots ,P_k$ if cutting along $S$ and capping its cut faces gives precisely these manifolds together with some standard three-spheres having no original boundary marks. All original boundary components remain assigned to their cut component. Each $P_i$ is irreducible and is not $S^3$; balls and other factors with original boundary are retained. Given a prime presentation, take its separating connected-sum spheres and add one standard nonseparating sphere in each $S^1\times S^2$ summand. Their finite collars and the marked model in [F3], Proof3.1, show that they reduce $M$ to its irreducible nonspherical prime factors: cutting and capping the added sphere turns that summand into $S^3$. The cap-ball avoidance construction [F3], Proof1.1, realizes all these spheres disjointly in the original manifold. Conversely, if $S$ reduces to this list, adding any finite PL spheres disjoint from $S$ leaves the list unchanged. In a capped irreducible component each new sphere bounds a ball. Its cut-and-cap operation leaves that component and an unmarked $S^3$; in a capped $S^3$ this follows from [F4]. Move cap balls off the sphere as in [F3], Proof1.1, to carry out the cutting, or equivalently keep track of the cap balls as punctures on their respective sides. They introduce only extra punctures and do not erase an original boundary component. Induction over the finite added spheres proves the assertion.

Let $S$ and $T$ be systems reducing $M$ to the irreducible lists from two prime presentations. Make their intersection transverse by a finite PL perturbation of one system. To see the finiteness and the local condition, take finite common subdivisions of their surface charts and ambient stars, and move their vertices by sufficiently small generic displacements in the finite linear charts. Avoid the finitely many affine incidences giving coincident vertices, tangencies or edge/face degeneracies; choose them one at a time outside these proper affine subsets. Finite compact disjoint simplices retain their positive separation, and the transverse link disks retain their embedding under such small moves. Use the boundary-fixed face-cone extensions in the ambient little stars to realize the moves, or just keep the parametrized perturbed surfaces; no factor changes. The intersection is now a compact one-manifold with a finite triangulation, hence finitely many disjoint polygonal circles. If there are no circles, keep the two systems. Otherwise choose a circle innermost on a component of $T$, bounding a PL disk $D$ whose interior misses $S$. The disk exists because a polygonal circle on a PL sphere bounds two parametrized disks by [F4]'s sphere boundary-disk construction.

Suppose $\mathrm{\partial }D\subset S_j$. Surger $S_j$ along $D$: its two complementary disks capped by two close parallel copies of $D$ give spheres $S_j^{\mathrm{\prime }},S_j^{\mathrm{\prime }\mathrm{\prime }}$. Small product pushes make both disjoint from $S_j$ and from all the other spheres of $S$. These are actual finite PL surfaces: in the finite disk and circle collar charts take two levels, triangulate the product strips, and flatten their quadrant corners by finite face-cone maps. The three spheres $S_j,S_j^{\mathrm{\prime }},S_j^{\mathrm{\prime }\mathrm{\prime }}$ bound a parametrized three-punctured sphere $P$. For its parameter, take the collar of $S_j$ on the $D$ side and the narrow disk prism thickening $D$; it is the standard collar with a disk two-handle on one end. Cut the sphere parameter along the common circle into its two parametrized disks. The two disk prisms and their circle strip give, by the marked window gluing of [F4], a ball with two disjoint parametrized interior balls removed. This verifies the three-punctured-sphere description with its three specified boundary components, not just its homology. Add both new spheres to $S$. By step4.1 this enlarged system reduces to the same list. Then delete $S_j$. This attaches $P$ to the component on the other side of $S_j$ along exactly that one boundary sphere; the other two boundary spheres stay cut. After capping these two faces, the operation just fills the old $S_j$ face by a parametrized ball. Thus the capped component is unchanged, and all other capped components are unchanged. This remains true if the two sides belonged to the same component before adding the new spheres, since $P$ is an isolated component AFTER the two new cuts. The new system therefore reduces to the same list.

The surgery reduces the number of intersections with $T$. Choose its disk levels on opposite sides of $D\subset T$ and the circle collar so that the two cap disks miss $T$. Every remaining old intersection circle lies in one of the two complementary disks of $S_j$ and is merely carried to the corresponding new sphere. The chosen circle disappears, and no new one is created; all other members of $S$ are unchanged. Repeat step6.1 until the finite nonnegative intersection count is zero. The resulting $S^{\ast }$ is disjoint from $T$ and reduces to precisely the original $P_i$ list. Their union is now an enlargement both of $S^{\ast }$ and of $T$. Step4.1 says its capped components consist of the $P_i$ together with unmarked $S^3$'s and also of the $Q_i$ together with unmarked $S^3$'s. Hence the nonspherical irreducible factors occur with identical multiplicity and homeomorphism types, up to permutation. Original boundary marks may be retained throughout, though the Statement does not require that stronger marking.

The remaining number of $S^1\times S^2$ factors is also determined. Work over $\mathbb{Q}$. The finite triangulation and [F6] make $b_1(M)=\dim H_1(M;\mathbb{Q})$ finite. For a separating connected-sum sphere, use its collar to form an open two-set cover; the intersection retracts to $S^2$. This sphere has $H_1=0$ and connected $H_0=\mathbb{Q}$, and its map to the two connected sides' $H_0$ groups is $a\mapsto (a,-a)$, injective. Exactness in [F6] gives $H_1(M;\mathbb{Q})\cong H_1(P;\mathbb{Q})\oplus H_1(Q;\mathbb{Q})$; filling the boundary spheres has no effect on $H_1$ by the same cover sequence with a ball. For $S^1\times S^2$ use two proper overlapping open circle arcs whose intersection has two interval components, and multiply by $S^2$. Each of the two open sets retracts to $S^2$ and the overlap retracts to two copies of $S^2$. The sphere has zero $H_1$ by its finite boundary-of-a-tetrahedron chain complex in [F6]. The degree-zero map ${\mathbb{Q}}^2\to {\mathbb{Q}}^2$ sends both component generators to $(1,-1)$, so its kernel is one-dimensional. The same exact sequence gives $b_1(S^1\times S^2)=1$. Thus a presentation with irreducible list $P_i$ and $\mathrm{\ell }$ such factors satisfies $b_1(M)={\sum }_i{b}_1(P_i)+\mathrm{\ell }$. Step7.1 identifies the sum for the two presentations, so their numbers $\mathrm{\ell }$ are equal.

Finally take any two admitted finite tame TOP prime presentations of the original $M$. By steps1.1 and2.1 move each of their finite cutting systems to a PL system in the fixed structure. Their capped factors are homeomorphic to the original factors by step3.1. TOP prime factors become PL prime, so [F3]'s prime-versus-irreducible assertion applies; the explicit nonseparating handle model identifies the possible $S^1\times S^2$ factors. Steps4.1–8.1 therefore compare these two presentations and give precisely the required equality of prime factors up to order and insertion or deletion of $S^3$. A single-factor presentation and $M=S^3$ are included: the reducing system may be empty and its only capped component may be an unmarked $S^3$; the same refinement argument forces every nonspherical irreducible list to be empty, and step8.1 forces zero handle factors. Existence and the exceptional prime factor were proved in step3.1. All boundary cuts and cap maps used actual parameters, the only TOP-to-PL compatibility used the explicit relative operation of steps1.1 and2.1, and full AC entered through [A1], [F1] and [F2] exactly as declared.