证明
Lemma 15 ([Complementary fillings]). For every such stabilized chamber there is…Proposition 14. The manifold Y has two Δ -periodic systems of sphere pairs,…观点
First project the outer immersed systems through the pinch and put them in general position. The deleted and retained blocks were orthogonal, since their classes agree with the correspondingly labelled standard pairs. The retained forms and quadratic self-intersection data are therefore still hyperbolic over π_1(Y_R) . Choose framed immersed representatives and cancel the algebraic intersection pairs. The framing corrections and, where necessary, local cusp changes can be chosen from finitely many local drawings. The group in question is a free product of free chamber groups, hence is torsion-free; there is no order-two ambiguity in these quadratic data. We justify carefully the uniformity of the Whitney discs required at this and subsequent finite stages. The initial lifted data have finitely many types under a cocompact group of translations of Y . A gallery ball of fixed radius contains a uniformly bounded number of chambers and of pieces of those data. There are thus finitely many local templates, even after specifying which of the standard neighborhoods meeting that ball have been peeled. Paths on participating spheres are taken inside their bounded carriers. Common long whiskers conjugate an intersection label but need not be included in the geometric Whitney circuit. If a circuit enters a replacement ball, move its portions there to chosen arcs on the boundary three-sphere. This gives a circuit in the unpeeled space. Nullity is unchanged, since both the deleted punctured S^2×S^2 and the replacement ball are simply connected and have simply connected boundary. For every null circuit among the finitely many resulting templates, choose one compact filling in Y . There is a finite maximum of their propagation bounds. Translate the chosen filling to the location of the circuit and apply the actual global pinch map. This constructs a filling in Y_R . A filling may encounter additional peeled neighborhoods. No further choice of fillings is then needed: the global pinch is already defined on them, and each such neighborhood has the same bounded carrier. Consequently this procedure is uniform in R . It is a finite-template argument for the specified circuits, not a bound for arbitrary null loops. Repeating a fixed finite number of stages preserves the argument: enlarge the carrier radius and refine the finite template list at each stage. Now apply the geometric Casson lemma to the two halves of the sphere system, removing the paired mixed intersections by regular homotopies. There is no iterative repair cascade here. Use the local construction of with the two surface arguments taken to be the entire possibly disconnected families. Preselect all framed pairing Whitney discs with disjoint boundaries; their interiors miss the other Whitney boundary arcs by 2+1<4 general position. First push the first family off all these interiors by fingers along arcs in the Whitney discs. Choose the arcs away from their finitely many intersections with other Whitney discs and the second family. The finger supports then leave the other Whitney discs and attaching boundaries fixed. Next Whitney-move the second family using the same discs. The new intersections in each phase are only within that phase's family; intersections between Whitney interiors cause only allowed second-family intersections. Thus the whole batch stays in the original bounded Whitney neighborhoods. No later disc is transported or chosen again through an expanding sequence of carriers. Local finiteness permits these bounded batches on the unbounded Y_R . This produces the geometric duality in (1), while permitting self-intersections within either half. The remaining double points of the first half admit framed Whitney discs. Tube their interiors off the first-half spheres using the geometric duals. All the Whitney circuits involved are the null circuits just discussed. For completeness, we verify the algebraic input to the grope preparation in the complement M_0 of small plumbed neighborhoods of the a_i . The punctured geometric dual b_i bounds an a_i -meridian in M_0 . Since these meridians normally generate the kernel, inclusion induces π_1M_0≅π_1Y_R . Thus the following calculation retains all intersection labels in the required complement group ring. At one point of every paired double point take its Clifford torus. Cap its two basis curves by standard meridional discs and tube each meridional disc off its incident a_i -sheet into a punctured parallel of b_i , with the product framing. Denote these caps by C_ℓ , and let i(ℓ) be the index of the dual used. Their equivariant intersection calculation is the Clifford-cap calculation in the proof of . The local meridional pieces have disjoint interiors. Choose the thin tubing tracks away from the finitely many other designated dual points; local changes of tracks introduce only oppositely signed points with the same group label. The remaining intersections are those between the corresponding based parallels of b_ i(ℓ) . Hence their sums are obtained from the pairings of the b_i by the basing units and orientation signs. Likewise, a cap's self-intersection sum is that of its framed b_i parallel, together with cancelling local terms. Since the mixed Casson regular homotopies preserved λ(b_i,b_j)=0 and μ(b_i)=0 , this proves λ(C_ℓ,C_m)=0, μ(C_ℓ)=0 every ℓ,m, including distinct parallel caps using the same dual sphere. The Clifford bodies lie in a boundary collar of M_0 . Following , contract temporary parallel copies and push the Whitney discs off those contractions before using them as geometric dual spheres. At the remaining intersections with Whitney discs, tube the Clifford caps into parallel copies of these temporary duals. Lemma~4.1 of now applies with its full cap-pairing hypothesis. Its two separation batches and height raising provide the prepared Clifford copies, whose entire capped surfaces are geometrically dual in the sense of . These prepared copies, including all their caps, are the protected surfaces used from this point onward. The preparation in produces height-two Whitney gropes, separates their caps from the protected Clifford caps, and raises the height to h . The collar levels and the full dual-cap disjointness are part of this preparation, not an assertion made after the caps have been changed. There are exactly two cap-family separation batches before height raising. Each uses the same no-new-mixed-pairs invariant, with its protected bodies at fixed collar levels. Each of the operations before height raising involves only a fixed number of kinds of tubing, push-off, Whitney-disc choice and regular homotopy. Their simultaneous use does not require traversing the whole list of sphere components. At each fixed stage, bounded carriers and finite drawings bound the number of other carriers meeting a given one. Divide the operations into finitely many classes with disjoint enlarged carriers and perform one class at a time. This gives a number of local stages independent of R . New null circuits are handled by the finite-template procedure above. For height raising there is also an explicit size estimate: , in the January 2013 numbering, bounds the increase of component-image diameter by a factor of seven at each step. A fixed requested height therefore has a fixed propagation bound. One can apply this estimate to the gallery map into the Davis complex: extend that map over each replacement ball by coning its boundary image inside a metric ball of uniformly bounded radius. This is possible by CAT(0) convexity. A point in a chamber is mapped within a fixed distance of its label, and properness of the Coxeter action turns a fixed metric bound into a fixed word-distance bound. The construction uses parallel transverse gropes and neighborhood operations; it preserves the protected Clifford capped surfaces. All these operations can be made locally finite in the unbounded development. Finally, outside a fixed buffer of the peeling region the input is the original system in an individual chamber. Its intersection cancellations hold already over that chamber's group ring. The same preparation, including the null discs, can consequently be chosen in that chamber. Only a bounded buffer of the switch between the peeled and unpeeled data requires the preceding translated templates. This proves the last assertion and permits later truncation without cutting off an unfinished preparation.