Freedman’s theorem and topological sliceness

Section1: Finite contraction and cyclic-label cancellation. The following finite construction and its limiting steps use the supplied framed data.

A finite framed four-dimensional capped-grope neighborhood is supplied, with its usual product collars and self-plumbing charts. Its body is embedded; only cap interiors have generic double points. An integer labeling means a homomorphism from the fundamental group of this actual neighborhood to Z. Paths to individual cap sheets are fixed once. Conjugating a based loop does not change its integer label. No arbitrary locally flat surface is given a tubular neighborhood by this hypothesis: the actual grope's framed block neighborhood is input.

Contract one top surface stage. Designate a finite subset of cap double points to be pushed off at that contraction, and choose a first and second sheet at each designated point. Then the resulting shorter capped grope has the same lower body, attaching circle and framing. Every old intersection not designated has parallel descendants with its old label. No designated old intersection remains. Every new cap intersection is a descendant of two designated intersections removed across dual caps at this same contraction, and its label is a difference of their labels, up to overall sign. Consequently, if all designated labels lie in an interval [a,b], every new label has absolute value at most b−a.

Cut the top surface at each pair of symplectic attaching curves. Replace the cut part by two small parallel copies of each of its two caps and the small central square at their unique attaching-curve intersection. Cutting a genus-g one-boundary surface along its complete symplectic system leaves the disk region with paired edges; the four parallel cap sectors and this square fill each pair of edges. Thus this replaces the surface with its contracted disk. All replacements take place in the supplied framed block. Parallel copies are made in its chosen normal coordinate; the central square uses the untwisted product collar, so the external attaching framing is unchanged. The result may be immersed, which is allowed.

For a designated point, take an embedded path on its second cap from that point to its attaching boundary, with interior avoiding all other double-point preimages. Finitely many paths on one cap can be chosen with disjoint interiors by an embedded finite tree and small parallel offsets. The two parallel copies of that path, closed across the central square, bound a ribbon Whitney disk pairing the two intersections of the first sheet with the contracted cap. In a product rectangle around the path, this is just the rectangle between the two parallel copies, closed by the boundary square. Its interior is disjoint from the old body and all unmodified sheets. A generic sufficiently small choice of the finitely many product widths is possible because their unwanted incidences are finitely many closed disjoint compact sets. No appeal to a normal bundle over an arbitrary TOP surface occurs.

Push the first sheet across this ribbon. This removes precisely the two paired intersections. The only ribbons that can meet are ones running across dual cap sectors: their boundary paths cross in the central square. Resolving those crossings by the prescribed order of the pushes creates intersections between the pushed first sheets. Ribbons on the same cap have been chosen parallel and disjoint, and ribbons belonging to different handles have disjoint block neighborhoods. Therefore every new intersection has exactly the stated two designated ancestors. Repeating for all parallel descendants removes the designated old point altogether. This is a finite sequence of ordinary finger/Whitney operations in explicit smooth block charts; a small relative perturbation makes the resulting intersections transverse.

Here is the label calculation, including the path issue. Let the two designated points have based first-to-second loops g,h. A path to the new intersection follows the first pushed sheet along the ribbon to its second-sheet cap, crosses the common central square, and returns along the other ribbon in reverse. Cancel the common paths in the old body. The resulting word is g h^{-1}; changing starting path conjugates it and exchanging the two resulting sheets inverts it. This is a calculation in the original supplied neighborhood, since every ribbon and push lies there. Its Z label is consequently ±(phi(g)−phi(h)). An old point copied to parallel sheets has its old loop with a small contractible product strip added, so its label is unchanged. The construction, classification of new intersections and claimed bound follow.

Section2: The finite asymmetric bootstrap. The following finite construction and its limiting steps use the supplied framed data.

Start with one symplectic pair above the bottom surface: on the positive side a framed capped surface R of positive genus, on the negative side a framed immersed cap W. The same construction is repeated for the finitely many pairs; their body blocks are disjoint, and interactions between cap systems are allowed. All operations fix a neighborhood of the original bottom surface and its attaching region.

Take two small framed parallel copies of R, at negative and positive normal coordinate, and join their boundary circles by the normal annulus crossing the attaching curve of W once. The result is an embedded closed surface of genus twice that of R. Its caps are the corresponding parallel copies of the caps of R. This is a sphere-like capped surface F: compressing its symplectic system gives a sphere. Its body meets W in one point and is disjoint from R; its cap intersections with all the old caps are unrestricted. The annulus and the single crossing are read in the supplied symplectic handle block, not in a hypothetical neighborhood of an arbitrary locally flat surface.

Construct a small parallel copy of F and contract its top stage to an immersed framed sphere T. During contraction, push every other cap sheet off T by the ribbons of Section1, excluding the one prescribed body intersection T with W. This includes the original caps of F, the caps of R and the remaining parts of W. All intersections newly produced by those pushes are between those cap sheets. The sphere T is therefore disjoint from both R and the body/caps of F, and from the caps of R, and meets W at its one prescribed point only. Self-intersections of T are allowed. The ribbons have support near cap arcs and their central squares, away from the lower body, so they do not alter any of the displayed body intersections.

Use T as a Norman sphere to remove every intersection of W with a cap of R or F: for each such point, choose a path in W from that point to the unique point W meets T. Remove a small disk from the intersecting cap and connect it by an untwisted tube, along a parallel copy of that path, to T with a small disk removed. The second endpoint consumes the one intersection of that parallel T with W. The two disks removed and the tube are the boundary of a product neighborhood of the path, so the resulting cap misses W at the chosen point and has the same boundary and framing. Choose the finitely many paths as branches of a tree in the abstract disk W, with separate parallel approach paths at their common endpoint; their interiors avoid all other double-point preimages. Normal widths are chosen successively small. The added copies of T are framed immersed spheres; their self- and mutual intersections belong entirely to the modified CAPS. Thus they may create cap intersections but no body intersections. Their original intersections with R or the body of F are empty, and sufficiently small parallel copies retain that property. After this finite operation W is disjoint from every other part except its own self-intersections and the single point on the body of F. In particular F remains a proper capped surface transverse to W.

For each self-intersection of W, tube one of its sheets to a separate small parallel copy of the BODY of F, along the other sheet and then along a chosen arc in W to the transverse point. This is the same local Norman cancellation, now using a capped surface instead of the contracted sphere. The new body consists of W with that sheet disk deleted, the punctured closed body of F, and their connecting annulus. It is a one-boundary surface with additional genus, with the old W boundary/framing. The caps of the added F provide the symplectic caps for its new handles. Its body is embedded: before the tubing, distinct body copies were disjoint and met W only at their distinct transverse endpoints; the tube arcs avoid every other double-point preimage. Choose disjoint parallel branches of a finite tree in W and nested normal widths. The tube neighborhoods consequently miss all other body sheets. Every old self-intersection of W is removed once, and none is introduced into the new body. Its new caps may intersect each other or the old caps, but they miss every body: the prior operation made the old caps disjoint from W, and properness of F makes its caps disjoint from all the parallel body sheets. At the attaching collars this assertion is the defining product chart. Consequently W has become an actual proper framed capped surface, not merely a surface with unaccounted body intersections.

This changes the side heights (1,0) to (1,1), preserving the original bottom stage. All choices and moves were finite. The scorecard is: R/body(F) embedded and mutually disjoint throughout; T immersed auxiliary, intersecting W once; old caps arbitrary among themselves; after the Norman cancellations all old caps miss W; after the final tubing every new intersection lies among caps. The auxiliary T is discarded after its copies have been inserted into caps. No disjointness of its parallel copies is needed, because their intersections are allowed cap intersections.

Section3: Finite bootstrap to height three. The following finite construction and its limiting steps use the supplied framed data.

Here the two side heights count stages ABOVE the fixed bottom surface. Section2 gives (1,0) to (1,1). The following side extension, used only finitely often in this bootstrap, supplies (1,1) to (2,1) to (2,3). Let P be the side to be raised and Q its other side, of heights p,q at least one. Construct F transverse to the bottom surface of P by taking two normal parallels of Q and closing their rims with the normal annulus. It is sphere-like of height q. Its body is embedded, meets P only at the specified first-stage point, and misses all other body stages; intersections of its caps with the old caps are allowed.

First make a contracted parallel sphere T transverse to the bottom of P, using Section1 and pushing off all original cap sheets and the caps of F. Its only body intersection is that prescribed transverse point. To make the caps of F disjoint from the old P caps, push an F-cap sheet at each intersection down the old P cap to its attaching stage and then down successive P stages to its bottom. A finger across one attaching collar replaces one intersection with two on the immediately lower stage; this is two parallel paths around the two ends of that collar. After p stages there are finitely many points on the bottom of P. Tube them off over parallel copies of T. These changes affect only caps of F. They cannot reintroduce intersections with old P caps because T misses those caps and the finger paths on the old caps avoid the other finitely many double-point preimages. The new intersections are among caps of F and possibly the old Q caps, which are still caps. All old body stages are fixed.

Now for an intersection of an old P cap with a Q cap, push the Q-cap sheet down the P stages and tube it off over T. The Q cap is still an immersed disk; its boundary and framing are fixed. It may acquire self/intersections with other caps, which are allowed. It remains disjoint from the old P caps and all body stages: the punctured T used in the tubing has had its unique bottom-P intersection removed. Crucially T is also disjoint from the body of F, so the Q caps miss every body piece to be added in the next step.

Finally treat every self/mutual intersection among the old P caps by pushing one of its P sheets down the P stages and tubing it off over copies of the full F, rather than T. This replaces each old P cap by a capped grope: its first added stage is the old disk with punctured bodies of F attached, followed by the higher stages of those copies of F. Orient each added copy so its transverse intersection cancels the chosen point, and use the untwisted framed tubes. All old P-cap intersections have been removed; the added bodies are embedded and disjoint from each other because distinct body parallels of F are disjoint, and the tube paths avoid all other old preimages. The new caps are copies of F's caps. Those were made disjoint from ALL old P caps before this step, and properness makes them disjoint from F's parallel bodies. Consequently every new intersection is among caps, not on a new body. A cap with no intersections may be regarded as a genus-zero stage with no attached higher surface, as in the disk-like model. We have obtained side heights at least (p+q,q), with the previous body and its attaching framing preserved. The first and second steps are needed: merely asserting that tubing adds height would leave the old cap intersections on the new body.

All choices involve finite paths, parallels and normal widths in supplied framed blocks. Use the move first with p=q=1, then exchange the two sides and use p=1,q=2. Both sides then have height at least two, so the entire grope has height at least three. The induced integer labels on its finite plumbing generators are obtained by following their actual paths back to the original neighborhood; let B be the maximum of their absolute values. B exists without any growth estimate for this finite bootstrap.

Section4: Efficient symmetric raising and the word bound. The following finite construction and its limiting steps use the supplied framed data.

Once both sides have height at least two, there is an efficient variant which contracts only the TOP stage of a transverse grope. For the side P to be extended, form the transverse grope from the opposite side Q. Its height is that of Q. Contract its top stage and push the old caps off that contraction by Section1. The transverse grope now has height one less and its caps are disjoint from all old caps. Push each intersection on a P cap down the P stages, and at the bottom tube off over this transverse grope. The same body/cap scorecard as Section3 applies: new bodies consist of separate parallel copies of the old embedded opposite-side bodies, and all cap intersections have been removed from the new body before it is declared a body. This replaces each old P cap by a capped grope with the height of the contracted transverse grope. In terms of full path heights a,b INCLUDING the shared bottom, the updated path height is a+b−2. Alternate the sides. Starting with a_{−1}=a_0=3, the successive updated path heights obey a_n=a_{n−1}+a_{n−2}−2. The uniform height is the smaller of the two current path heights. Writing a_n−2 as a Fibonacci recursion gives, after 2k steps, uniform height at least 2^k: consecutive pairs increase the smaller Fibonacci term by at least a factor two. Additional initial height only improves the bound.

For clarity, the required estimate concerns WORDS, not numbers of double points. Base every old cap-point loop by a path in the old grope. Body-stage paths contribute no new plumbing generator. Contraction copies an old point without changing its generator. A push off a contraction creates the difference loop of Section1, a word of length at most two in the OLD cap-point generators. A new intersection among caps of the parallel transverse grope has length one, since its sheets are old parallel sheets. The only remaining type is a loop using a newly pushed-down finger: follow its first sheet across at most two old cap-point loops to the parallel transverse copy, cross its new cap intersection (one old generator), and return across at most two old loops on the other finger. The two bottom-stage paths cancel, as do the parallel collar paths. This is a concatenation of at most 2+1+2 old generators or inverses. These exhaust the new intersections: they occur on unchanged parallels, on a contraction push, or between two inserted fingers. Thus each step sends every NEW generator to a word of length at most five in the PRECEDING generators. Induction gives length at most 5^(2k) after 2k steps, and absolute Z label at most 5^(2k)B. No count of the possibly much larger number of points is substituted for this word bound.

Section5: The special relative disk-to-disk operation in standardness. The following finite construction and its limiting steps use the supplied framed data.

This is Freedman–Quinn, §4.7, Proposition 4.7.1, with the actual relative data made explicit. Let f:B^4→B^4 be continuous and onto, with a homeomorphic boundary restriction. Its point-inverse collection is null (only finitely many fibres have diameter at least any positive epsilon), its singular IMAGE is nowhere dense, and E is a closed subset of the target containing the whole boundary. Suppose f:f^−1(E)→E is a homeomorphism. Then there is a homeomorphism h:B^4→B^4 equal to f on f^−1(E). Here E CONTAINS the boundary; it is not merely a subset of that boundary. The latter misreading would make the relative assertion too weak for the collared-tower application.

Use the following intermediate configuration. W is compact metrizable; e,f:B^4→W and g:W→B^4 are onto. The composites ge,gf are identity on the boundary. The fibres of g,ge,gf are null; their relevant singular images Sigma_g, g(Sigma_e),g(Sigma_f) are nowhere dense and pairwise well separated: neither set meets the closure of another. A protected closed set E_0 containing the boundary is disjoint from all three, and the relation e^−1 f is fixed there. Nullity makes each singular image countable (group the nontrivial fibres by diameter at least1/m). We will decrease both e- and f-fibre diameters while controlling the relations e^−1 f and f^−1 e between the TWO fixed domain balls.

Here is the local replacement. Choose a closed round ball B in the TARGET of g, disjoint from E_0,Sigma_g,g(Sigma_f), and a smaller concentric ball B_0 whose boundary misses g(Sigma_e). The restrictions over B for g and over the sphere boundary of B_0 for ge are homeomorphisms: they are continuous bijections of compact spaces to Hausdorff spaces. We require no open neighborhood of that sphere free of e singular values, which may accumulate on it. Replace g^−1(int B_0) in W by (ge)^−1(B_0), gluing their boundaries by e. The result is W'. On the inserted piece define e' to be the identity and g' to be ge; outside use the old e,g. These maps match at the gluing and satisfy g'e'=ge. The map e' is a homeomorphism over the inserted piece, so every e-fibre with g-image in B_0 has been eliminated. This is a finite compact adjunction, not a claim that W' is already a manifold.

To transfer f, take a homeomorphism w:B→B^4 fixed on B_0. Such a map exists by radial extension from the center of these convex balls. On g^−1 B define H=g^−1 w^−1 g e w g,

where g^−1 is single-valued on B because B misses Sigma_g. Outside use the identity. On the boundary of B the definitions agree because ge is identity on the outer sphere. In the part whose image lies in g^−1 B_0, the first g^−1 w^−1 cancels the g before e; the remaining map e w g lifts to the inserted piece as w g. Thus H has a continuous lift H':W→W', and put f'=H'f. This gives f'=f outside the region with gf-image in B. On the inserted piece its formula is wgf. On the remaining annular region its fibres are the transported fibres of ge. Since gf is a homeomorphism over B, these new fibres are coordinate copies of a null family, so they remain null. The other fibres are unchanged. For g' the outside fibres are subsets of old g fibres, and the inside fibres are old ge fibres in the inserted compact piece. Both coordinate inclusions are uniformly continuous on their compact domains; a finite union of the corresponding null collections is null. For g'f', outside B nothing changes; in the inner piece it equals ge composed with the homeomorphism wgf, and in the annular piece it equals w^−1 ge wgf. Thus its nontrivial fibres are homeomorphic coordinate images of ge fibres. This is the actual nullity argument; composition of arbitrary null maps is not being asserted to preserve nullity.

The only possible failure of the separation condition is between the transported countable singular image and the retained image of ge. Choose w so these are well separated off B_0. This choice needs no unproved Baire assertion under the original countable-choice branch. Fix one radial reference map w_0:B→B^4. Enumerate the two countable singular sets. For each point condition require that its image avoid the CLOSED closure of the other singular set, and for each inverse-point condition require the analogous inverse avoidance. Each requirement is open and dense among homeomorphisms with the fixed inner/outer boundary data: a small coordinate ball around the free point lies off those fixed boundaries, and a compactly supported translation moves that point off the prescribed nowhere-dense closed set. Explicitly use x→x+rho(x)v with |v|Lip(rho)<1; the contraction inverse proves it is a homeomorphism. The required target complement is open dense, so a least satisfactory rational v in a fixed enumeration exists. At step m choose the translation and its inverse uniformly smaller than2^−m times every currently protected avoidance margin. Later changes then preserve all earlier requirements. Choose supports and rational radii from a fixed coordinate enumeration. The forward and inverse maps converge uniformly, and their limits are inverse by uniform continuity. The limit homeomorphism meets every avoidance requirement. This is a definable countable recursion with explicit finite rational searches, not dependent choice for arbitrary nonempty sets of homeomorphisms. It gives the required local configuration e',f',g'.

For the global iteration, first normalize the boundary: let e_0 be the radial cone on the given boundary homeomorphism, let W_0=B^4, g_0=e_0^−1 and f_0=f. Protect E_0=g_0(E). All conditions hold; the other singular sets are empty. Rescale the two fixed source metrics initially to diameter<1. Suppose all e_n- and f_n-fibres have diameter less than2^−n. There are only finitely many fibres of either map with diameter at least2^−n−1. Their g-images avoid Sigma_g; the separation condition makes the OTHER composite a local homeomorphism there. Around each large e-fibre choose a ball B, and then B_0, so that (ge)^−1 B has diameter less than2^−n while (gf)^−1 B has diameter less than2^−n−1. These choices follow from compactness: if inverse images of shrinking neighborhoods did not enter a prescribed neighborhood of a fibre, a convergent countersequence would give a point outside that fibre mapping to its center. Choose these finite balls disjoint and away from the protected set and the irrelevant singular images. Their boundary radii can avoid the countably many singular points.

Apply the local replacement. It kills the chosen large e-fibres, and all new f-fibres lie in the small (gf)^−1 B regions, hence have diameter less than2^−n−1. Reverse e and f and do the same to the remaining large f-fibres. The newly introduced e-fibres are now small by the analogous local-homeomorphism condition. The finite two-pass step therefore makes ALL fibres smaller than2^−n−1. It changes e^−1 f only within domains and ranges of diameter less than2^−n, and gives the same bound for the inverse relation. Make these bounds strictly summable, for instance use2^−n−3 after reindexing. On the protected set the relation stays e_0^−1 f_0 exactly. No source or target manifold classification is used in these replacements.

The relations R_n=e_n^−1 f_n and their inverses are thus uniformly Cauchy in the fibrewise Hausdorff distance. Their values have diameter tending to zero, so their limits are single-valued continuous maps R,S of the fixed balls. For completeness, continuity is not inferred from continuity of the set-valued R_n. Given x_j→x, fix n and select y_(n,j) in R_n(x_j) within the uniform tail bound of R(x_j). A subsequence of these selections converges to y_n in R_n(x), by compactness and the closed graph of R_n. The distance from y_n to R(x) is at most the tail bound plus diam R_n(x). Thus every subsequential limit of R(x_j) is within twice the tail bound plus diam R_n(x) of R(x). Letting n→infinity proves continuity. The identical argument applies to the inverse relations. Passing to the inverse graphs shows SR=RS=identity. Hence R is a homeomorphism. The fixed relation on the protected set yields R=e_0^−1 f on f^−1(E). Then h=e_0 R is the asserted homeomorphism agreeing with f there.

This proves the needed relative disk-to-disk replacement operation. It does NOT by itself prove that a tower is a disk: that application still needs the actual dual-handle/design/common-quotient construction, including the other approximation whose fibres are explicit disk-with-boundary-neighborhood pieces. No general4D cell-like approximation theorem is imported.

Section6: From a null grope to the first actual tower story. The following finite construction and its limiting steps use the supplied framed data.

This step has an exact input which the cyclic construction must supply: an actual properly immersed framed disk-like capped grope A of height at least three, whose entire image has trivial fundamental-group image in the smooth ambient four-manifold, together with framed geometric transverse spheres, each missing A except for its one prescribed bottom-stage point. It does not start with vanishing algebraic intersections alone. All modifications below take place in supplied plumbing charts and finitely many immersed disk charts; the ambient manifold is smooth at this stage.

First raise the height as far as needed by Section4, reserving one more full stage than the desired number of stages in the story. Because the raising maps factor through the previous grope neighborhood, its image remains group-null. Order the finitely many components of the extra top stage. Contract the first one, and push the cap sheets of every other component off that contraction by the ribbon Whitney disks of Section1. Contract the second and push all still uncontracted cap sheets off it; continue. The resulting grope caps are pairwise disjoint. They need not be embedded, but every self-intersection is part of the finger-pair model, with a specified Whitney disk. Here is the local pairing check: pushing a sheet off a contraction replaces its old cap intersection by the two parallel intersections with the opposite cap through the central square. Their signs are opposite, and the ribbon between the two parallel cap copies, continued along the chosen arc on the dual cap, is the Whitney disk. At an old cap self-intersection the four parallel-sheet intersections split into two such opposite-sign pairs. Order the pushes so each newly contracted component is protected from all later cap sheets. Distinct ribbon arcs are disjoint except at their prescribed central-square crossings; any interior crossings are among the disks which will be declared top caps, rather than intersections among the remaining grope caps. This is the actual disk-with-finger-pairs model, not the assertion that zero algebraic self-intersection supplies embedded Whitney disks.

For each finger pair choose an accessory circle: start at its chosen double point, follow one sheet to the Whitney arc and return on the other sheet. It is a loop in the image of the contracted grope. It is therefore nullhomotopic in the ambient manifold by the input nullness; no conclusion about the fundamental group of the complement is needed. A nullhomotopy relative to this fixed embedded boundary arc can be put in general position as an immersed disk with its prescribed boundary collar. The boundary collar is the accessory collar in the standard finger model. Its extending normal framing is corrected, if necessary, by interior twists in that disk together with boundary twists in the model collar; interior twists change the relative Euler number by two, and one boundary twist changes its parity. The latter creates an intersection with the grope cap, which is permitted temporarily. The boundary-circle germ itself and the original bottom attaching framing are fixed. The accessory disks and the previously specified ribbon Whitney disks are now the TOP caps. All finger pairs and their accessory boundary collars have the same finite plumbing model as an ordinary disk finger move followed by its Whitney and accessory disks. Thus the thickening is a one-story tower model before properness is imposed.

Only top-cap intersections with the lower grope remain to be removed. At such an intersection, modify the TOP-cap sheet, pushing it along the lower grope sheet through its successive attaching collars down to the first stage. At one attaching collar the two sides of the push produce two points on the preceding stage; after finitely many stages this produces finitely many points on the bottom. All paths avoid the other finitely many preimages. Remove each resulting bottom point by tubing the top cap over a separate parallel of the corresponding geometric transverse sphere, choosing its orientation to cancel the point. The punctured transverse sphere has no other intersection with the grope. Hence this operation changes only top caps, makes them disjoint from every grope stage including the grope caps, and may create intersections only among top caps. Tubing to a sphere preserves the top-cap disk topology. Its framing extends by the untwisted tube and the supplied sphere framing. Top caps are allowed to have mutual and self-plumbings. The bottom attaching circle, all its framing data, and every reserved lower stage are fixed.

We have obtained an ACTUAL proper one-story capped tower of the chosen finite grope height. Its model has an embedded lower body, grope caps with the prescribed finger-pair singularities, and arbitrary plumbings only among the top Whitney/accessory caps. The finite framed neighborhoods glue by their stipulated product collars, so the conclusion is a framed neighborhood of that tower, not only a map of its underlying two-complex. This is the first-story operation needed before replication and convergence. It has not constructed additional stories or identified any tower neighborhood with a standard two-handle.

Section7: Replication of a tower, with the body and top caps kept distinct. The following finite construction and its limiting steps use the supplied framed data.

Write G for a story's grope body, C for its grope caps with their paired finger points, and D for the top Whitney/accessory caps. Thus G union C is the story body; intersections among D are allowed, whereas D misses G union C. When another story replaces D, the new story bodies, and not their eventual top caps, must be disjoint and embedded in the stipulated plumbing model. This distinction is essential in the following replication.

First describe the transverse grope operation in the actual finite model. A surface attached at a positive curve of a symplectic pair has a sphere-like transverse capped grope formed from two parallels of the negative branch and the annulus joining their boundaries. Its bottom meets that positive surface once. Its caps are parallels of the original grope caps. At their intersections with the original grope caps, the two parallel sheets give the ribbon Whitney pairs of Section1. If a further story is present, the corresponding Whitney disk is the bottom of the next story's disk-like grope. Replace the affected band of the transverse grope by two parallels of that Whitney grope and the boundary band along its Whitney arc. This is the grope version of the ordinary Whitney move: cutting the band removes the two old intersection points, and all new intersections lie in the top caps of the inserted Whitney grope. Its body misses every lower story, since that next-story body does. Repeat through the finite list of stories. Stop with its caps parallel to the LAST GROPE caps C, so their possible intersections with the original tower are only with its top caps D. Do not take the additional contraction through the final Whitney/accessory caps: that would give geometric transverse gropes whose nullness is no longer guaranteed. The transverse grope then meets the original tower only at its single specified transverse body point and at the TOP caps D. Its body AND caps use only parallels of pieces of the original tower BODY strictly above the first surface stage.

The group-nullness used here is a direct model calculation. Collapse each surface stage to its symplectic attaching curves and connecting arcs, and each paired-finger grope cap to its double-point graph. The stage-curve generators bound the next stages after contraction. The double-point graph generators bound the specified accessory disks; the Whitney disk identifies the generator at the mate point with the same generator or its inverse according to the sheet orientation. Accordingly every loop in the body strictly above the first stage is null in the entire tower neighborhood. Only the plumbing generators among the TOP caps can remain in that neighborhood's fundamental group. Thus the transverse grope just constructed has trivial group image there. This calculation concerns its actual body and cap paths through the model; it makes no assertion that arbitrary top-cap plumbings are null.

Suppose now a proper tower has first-story grope height at least four. Apply the preceding operation to the second-stage surfaces, contract parallel transverse gropes, and push the old top-cap sheets off those contractions. This supplies framed geometric transverse spheres for the disk-like components remaining after the first surface stage and its fixed collar are removed. Work in that complement. The first-stage collar will be fixed throughout what follows.

Choose the positive side of the top story to be replicated. For its second-stage surfaces construct transverse gropes from the opposite, negative branches. Arrange their caps to meet only the negative-side top caps, as in the preceding Whitney-grope procedure. Contract one top surface stage of the transverse gropes and push the negative-side top caps off those contractions. The resulting transverse gropes and the changed negative cap sheets still have trivial group image whenever that assertion is made of the new BODY: their new body loops run in the old body neighborhood just analyzed, and contraction/push-off changes a loop only by a product of old body loops. Top-cap plumbing generators have not been declared trivial. The negative lower body is unchanged. The height-four hypothesis leaves enough height to raise these transverse gropes to any prescribed finite height by Section4.

Make the needed transverse gropes disjoint near-parallels by ordering their constructions. Contract a parallel, then push all still unused cap sheets off it, using separate ribbon arcs. Repeat for the finite number of copies needed. This is why merely taking many geometric parallels of an immersed capped grope would not suffice: their caps could intersect the old body to be added. The ordered contraction/push-off produces copies whose bodies are disjoint and whose caps miss those old body sheets. Each copy is still null by the model calculation.

At every mutual or self-intersection of POSITIVE top caps, push one sheet down that positive branch to its second-stage surface. A push through one stage doubles the number of points; there are finitely many stages, so only finitely many points result. At every resulting point tube over a separate full transverse grope, not its contraction. The punctured transverse bottom surface removes the point; its higher stages and caps become the higher stages and caps on the changed positive top sheet. The original positive top caps are now properly immersed capped gropes of the chosen height. Their new bodies are disjoint: all their former mutual/self-points were treated, the transverse bodies are disjoint near-parallels, and their caps were cleared off the former top sheets before the tubing. Intersections which remain are among the new caps. The negative branch body has stayed fixed.

There is also the necessary group check for the NEW body. A double-point loop in a changed cap comes from a single selected transverse-grope copy and from the fingers used for that one former top-cap point. The two paths down the original positive branch are parallel, so their whisker words cancel. What remains is a loop in that transverse copy, which is null by the preceding model calculation. Copies used for DIFFERENT former points are distinct near-parallels; their body pieces do not meet and cannot create additional body generators. Thus the new positive body has trivial group image in the complement being used. The retained negative body already has that property, since all its generators are killed by its old Whitney/accessory scaffold. No conclusion is taken about either collection's as-yet-unassigned top-cap plumbing loops.

Raise the new top grope by one extra stage, contract its top components successively and push all the later caps off each contraction. As in Section6, this gives mutually disjoint grope caps with canonical finger-pair Whitney disks. Their accessory loops lie in the just verified null BODY, so they admit immersed accessory disks in the working complement. Correct their framings and push their intersections with the lower body down to the lowest surfaces, then tube over the geometric transverse spheres constructed after deleting the first stage. This produces legitimate new top caps disjoint from the full body. The positive side now has one additional story. Interchange positive and negative and repeat. The original lower body and its framed attaching product are fixed, and both sides now have one additional story, with any prescribed finite height in the new top story.

This is a finite replication inside an arbitrarily small supplied neighborhood of the old tower together with the required accessory-disk images. The latter are not automatically small from the word estimate; localization is exactly the issue separated in the additional localization hypotheses for a controlled application. For the uncontrolled single-component standardness construction, the model nullhomotopies above lie in the tower neighborhood itself, so that neighborhood supplies their support. The source calculation checked here is the full FQ3.4–3.5, printed53–56. Its statement and the present operation require the actual framed tower model and first-story height at least four, not only a collection of algebraically unobstructed immersed disks.

Section8: Squeezing and convergence, retaining only a pinched neighborhood. The following finite construction and its limiting steps use the supplied framed data.

Replicate twice by Section7, obtaining three stories with as many stages as desired in the second story. Delete the top caps and contract each component of the third-story grope completely. The contractions supply caps for the second story. Distinct such caps are disjoint, since the distinct third-story BODIES were disjoint; the discarded top-cap intersections have not been retained. The old top caps give nullhomotopies of the generators of each upper-story body, in the complement of the retained first-story body. This is again the model generator calculation: each stage generator is killed by its cap and each finger generator by its accessory cap. All these killing disks lie above the retained first story. Thus the upper-story regular neighborhoods are group-null there. We need to move their finite spines, rather than assert that an arbitrary embedded two-complex can be shrunk by isotopy.

For each upper-story component collapse its regular neighborhood to a finite graph K. One can see the collapse directly: cut a surface stage to its attaching curves and arcs joining the curves; repeat upward; a cap with self-plumbings collapses to its double-point graph and connecting arcs. The disks of the preceding first story which enter this neighborhood consist of strips ending on K. More precisely, relative to their portions C outside the neighborhood, they are a mapping cylinder L times I whose inner ends are mapped into K. This description is obtained in the same finite surface cuts and normal disk blocks. It provides the regular neighborhood of the PAIR C union (L times I) union K, not a claim that the first-story cap disks are fixed pointwise while their attaching ends move.

Choose distinct small smooth coordinate balls, one for each upper-story component, in the complement of the retained first-story body. The null graph map is homotopic there to an embedding of the same abstract graph in the chosen ball: collapse a maximal tree and use the actual generator nullhomotopies, then reverse a contraction of a tiny standard graph in the ball. Perturb this graph homotopy relative to its initial end and its final small-ball end to a one-parameter family of embeddings. The dimension calculation is useful here: two nonincident moving graph points and their common time form a three-dimensional parameter space, less than four, so general position removes equal-image pairs at the SAME time. Preserve the finite vertex stars by translations in small vertex charts, and perturb edge interiors with the endpoints fixed. This gives an isotopy of the graph. Intersections of its TRACE at different times need not disappear; they do not obstruct a graph isotopy and are not being used as a Whitney-disk theorem.

The first-story grope body is avoided by the whole homotopy. Arrange its finitely many intersections with the remaining cap sheets C and the mapping-cylinder strips transversely. Push those sheet intersections along the homotopy toward its free final end and past the final graph. In a normal coordinate ball this is the ordinary finger push of a two-sheet off a terminal one-dimensional arc; its support is a narrow neighborhood of the chosen trace path. Each push modifies only a grope-cap sheet, not the retained grope body. Intersections it creates are among those cap sheets. At their finitely many crossings choose different narrow parallel paths, so the changes do not meet the initial graph or the retained body. The result is an isotopy of K disjoint from C and from the interiors of the mapping-cylinder strips, with their prescribed common inner endpoints allowed in the pair model. The outer ends of those strips on C are fixed. No claim of an embedded homotopy TRACE of a graph in four-space is required.

An isotopy of this finite smooth graph extends to an ambient isotopy of a sufficiently small regular neighborhood, fixed outside a neighborhood of its track and fixed on C and the retained grope body. Here is the elementary extension in the present setting. Near each moving vertex use its small translated coordinate ball. Subdivide time so the vertex balls remain disjoint and each moving edge segment lies in a fixed product chart. On the interior of such an edge, extend its velocity by a cutoff in the normal three-disk, agreeing with the chosen vertex translations at its ends. A finite partition of unity along the edge pieces gives a compactly supported smooth vector field agreeing with the velocity on the graph. Integrate for that short time interval; in the compact support the solution exists throughout the interval. Compose the finitely many intervals. The supports avoid the fixed pair pieces just cleared from the track. Transport the graph regular neighborhood by this isotopy; make its final normal widths small enough that the ENTIRE upper-story neighborhood, not just its spine, lies in the selected small ball. Transport the attaching mapping cylinders at the same time, retaining their outer ends. This proves the required squeezing without using a topological half-space unknotting assertion or a general TOP smoothing theorem.

Now iterate within the resulting upper-story neighborhoods. At stage n choose disjoint closed coordinate four-balls B_(n,j), of diameter<2^−n, containing precisely the next replacement components; choose their closures in the interiors of their parent balls. The old body and its marked attaching product lie outside their interiors. All subsequent modifications for a descendant are required to lie in that same ball. Section7 supplies the next finite story and the preceding squeezing supplies such balls of the next diameter, so the induction uses only the finite operations already specified. Heights can be prescribed in advance at every story; in particular the first story retains height at least four, sufficient for the standardness construction.

We spell out why this gives an embedded LIMIT OBJECT and why it gives less than an ordinary open neighborhood at its ends. Away from the nested balls every point is fixed after finitely many stages, so its map is the finite-stage framed embedding there. An infinite descending branch of balls has a unique intersection point: the balls are nested nonempty compact sets in a compact first-stage neighborhood and their diameters tend to zero. Distinct branches separate in disjoint balls at some finite level, so their points are distinct. Such a limit point cannot equal an old finite-stage body point: a deeper descendant closure lies in the interior of its parent, whereas that finite body point is excluded from all sufficiently deep replacement interiors. The topology at a branch is given by the descendant balls. These observations show injectivity and continuity of the natural map from the end-compactified model to the ambient space. Its domain is compact: an open cover first covers the compact set of branches by finitely many cylinder neighborhoods in the finite-branching tree, then covers the remaining finite-stage compact body by finitely many sets. A continuous injection from that compact domain to the Hausdorff ambient manifold is a homeomorphism onto its image.

For the NEIGHBORHOOD make the same construction with the framed four-dimensional model blocks, not just their spines. At each stage modify the previous block so it meets each chosen ball in the ordinary neighborhood of the disk being replaced; later replacement blocks are contained in that ball. Repeat the preceding compactness and shrinking-ball argument for these blocks and their ends. The result is an embedded PINCHED regular neighborhood, with the prescribed attaching solid torus and framing unchanged. At an end it may have no ambient interior and no locally flat boundary. Nothing in the argument identifies it with a two-handle or supplies a TOP normal bundle there. That identification is exactly the separate parameterization obligation. The construction can be supported in any prescribed open neighborhood containing the actual input tower and the finite nullhomotopies used in the replication, by choosing all balls and normal widths with closures in that open set. This is the support guarantee needed by the additional localization hypotheses for a controlled application once its localized inputs are supplied.

Section9: The explicit radial collapse used on the model side. The following finite construction and its limiting steps use the supplied framed data.

The model-side shrinking needs more than the assertion that a compact set is contractible. Its nontrivial fibers have supplied neighborhood coordinates of two particular kinds: a closed ball, or an embedded framed disk together with a regular neighborhood of its boundary circle. In the second case that regular neighborhood overlaps the disk in an ANNULAR COLLAR. Treating the overlap as just the core circle would give the wrong quotient model. We describe the collapse in those coordinates.

First let E be a closed starlike subset in the interior of a Euclidean ball, with center0. Its radial extent e(u)=max{r:ru is in E} is upper semicontinuous: if u_j→u and r_j≤e(u_j) converge to r, closedness gives ru in E. Given an open neighborhood U of E, compactness permits a continuous positive radial function v(u)>e(u) whose radial closed ball V lies in U. To construct v, take for each direction a small angular neighborhood on which e is below a chosen strict radial bound; take finitely many of these neighborhoods and a partition of unity, and use their strict upper bounds. Refine the angular cover enough that the resulting radial set stays in U. A sequence of such V_i can therefore be chosen with V_(i+1) in int V_i and intersection E.

Let {T_j} be a null collection of closed sets disjoint from E, and fix delta>0. Choose V_0 so any T_j meeting V_0 has diameter<delta/2; only finitely many exceptions must be avoided. Choose positive radii b_i decreasing to0, with b_0 the radius of the ambient ball and b_i−b_(i+1)<delta/8. Inductively choose V_i in int V_(i−1), within distance2^−i of E, so any T_j meeting V_i misses the complement of V_(i−1). This is possible: any T_j meeting arbitrarily close to E and also outside the fixed V_(i−1) has diameter bounded below by half the positive distance from E to that complement. There are only finitely many such T_j, all disjoint from E. Avoid them in V_i. Put W_i=V_i union the centered ball B_(b_i). These remain starlike with continuous positive radius functions, are nested, and intersect in E.

On each ray map the interval W_i minus int W_(i+1) affinely to the interval B_(b_i) minus int B_(b_(i+1)), and send E to0. Adjacent formulas agree at their endpoints; continuous radial functions give continuity away from E, and b_i→0 gives continuity at E. The result k is identity on the outer boundary, collapses exactly E, and is a homeomorphism off E. One must check its effect on the OTHER T_j, not merely the distinguished collapse. If x is outside V_i in the shell between W_i and W_(i+1), then k moves x by at most b_i−b_(i+1)<delta/8. Any T_j meeting V_i is contained in V_(i−1), so the whole part which may be moved a large distance lies at most one such radial level apart. If both x,y in T_j lie outside all relevant V levels, their displacements are<delta/8. Otherwise take the deepest level met by T_j and project x,y radially to the corresponding small sphere. Radial projection from points outside that sphere does not increase their distance; the two interpolation intervals change their distances by at most2delta/8 each. Hence, when diam T_j<delta/2, diam k(T_j)<delta. A T_j not meeting V_0 is unchanged except for the small radial displacements. Thus for every j either all its points move by<delta, or its entire image has diameter<delta. This is the control required for later null fibers.

Here is a concrete verification for the second fiber type. In its supplied framed disk chart write x=r exp(i theta) for the disk coordinate and y=s exp(i phi) for its normal coordinate. After a small radial adjustment of the boundary tube, the pair is D={0≤r≤1,s=0}, E=D union {(r−1)^2+s^2≤a^2}, 0<a<1,

with the angular identifications at r=0 and s=0. The neighborhood is rotational in both angles. Its two-dimensional orbit quadrant has a boundary interval [0,1] on s=0 representing D, and a half-disk about (1,0) representing the boundary tube. Collapse [0,1]. The quotient quadrant is again a quadrant, and the tube half-disk becomes a half-disk meeting the corner: this is an elementary planar construction, not a four-dimensional quotient theorem. To see it explicitly, first replace the two circular arcs by polygonal arcs using radial interpolation in disjoint annular neighborhoods. Triangulate the finite region between the arcs and the quadrant boundary. Collapsing the boundary interval merges its consecutive boundary vertices; the remaining region has the same cyclic boundary order as the quadrant with a half-disk based at its corner. Match these boundary orders, cut the remaining region along an arc joining its two boundary components, and extend the boundary maps by coning the resulting polygons from interior points. The maps agree on the two copies of the cut. Use the identity outside the chosen orbit neighborhood. This is a homeomorphism of the QUOTIENT quadrant to the ordinary quadrant, preserving its two distinguished axes.

Choose its target tube half-disk to be {(r−1)^2+s^2≤1,r≥0,s≥0}. Lift the orbit map by keeping theta and phi. Preservation of the axes ensures continuity across the angular collapses; its inverse has the same property. It is therefore an ambient homeomorphism from the four-dimensional disk quotient by D to the standard disk, supported in the supplied normal chart. The image of E is E'={ (x,y): |x|^2+|y|^2≤2|x| }.

It is starlike about0: for 0≤t≤1, the displayed left side at (tx,ty) is t^2 times the old left side, whereas the right side is t times the old right side. The image contains0 and has precisely the intended overlapping-boundary-tube quotient topology. Thus E is EVENTUALLY starlike by two explicit collapses: first the framed disk D, then E'. This uses the supplied disk and boundary-tube coordinates, not a normal-bundle existence theorem for an arbitrary locally flat disk.

Apply the preceding starlike collapse first to D and then to E'. The other disjoint null sets still form a null collection after the first collapse, by uniform continuity on the compact source; the quotient images remain disjoint from the distinguished set. Choose the first collapse's control small enough for the second step's coordinate uniform-continuity tolerances. Finite sets whose coordinate diameters exceed the chosen tolerance can be excluded from the collapse neighborhoods because they are disjoint from the distinguished set. Pulling the radial operations back through the supplied coordinate maps proves the same movement-or-small-image conclusion in the original metric. The maps are identity on the boundary of the local ball throughout.

Finally suppose alpha:B^4→Q is an onto quotient with a null collection of nontrivial fibers, each of one of these two SUPPLIED coordinate types, Q compact metric, and alpha a homeomorphism on the outer boundary. We give the iteration with the correct inverse CONTROL MAP, rather than claiming the raw inverse relations converge to a function. A raw inverse fiber may remain large permanently when that original alpha fiber is collapsed.

Start with f_0 the identity. Maintain a continuous onto f_n:B^4→B^4 whose only nontrivial fibers are finitely many original alpha fibers, and such that every set f_n(alpha^−1(q)) has diameter<epsilon_n, where epsilon_n decreases summably to0. Because alpha is constant on every f_n fiber, it descends to a continuous map u_n:B^4→Q with u_n f_n=alpha. Continuity follows from the compact quotient property of f_n. Every current image E_q=f_n(alpha^−1(q)) is therefore a set on which u_n is constant. The uncollapsed E_q retain their original supplied neighborhood type, since f_n is a homeomorphism near each such compact set, disjoint from the finite already collapsed fibers.

There are finitely many current E_q of diameter at least epsilon_(n+1)/2: nullity follows from uniform continuity of f_n on the original null family. Choose mutually disjoint small coordinate neighborhoods U_q of these E_q so diam U_q<2epsilon_n and diam u_n(U_q)<epsilon_n in the chosen metric of Q. The first bound follows from diam E_q<epsilon_n and a small neighborhood; the second follows from continuity of u_n and its constancy on E_q. Choose a closed coordinate ball in each U_q carrying the prescribed fiber model. If there are N selected sets, apply each local collapse with the movement-or-small-image tolerance d=epsilon_(n+1)/(4N+4), treating the other current E sets as the null family. Other selected sets are outside its support. Every unselected set starts with diameter<epsilon_(n+1)/2. Over the N collapses its diameter either increases by at most2Nd, or, after the last small-image outcome, is at most d+2Nd. Both bounds are strictly below epsilon_(n+1). Thus no untreated current fiber image crosses the next size threshold. Compose these finitely many collapses to get g_n and put f_(n+1)=g_n f_n. Its newly nontrivial fibers are exactly the selected entire original alpha fibers, so u_n is constant on each g_n fiber. Hence it descends to a continuous u_(n+1) with u_(n+1)g_n=u_n and u_(n+1)f_(n+1)=alpha.

The changes of f_n occur in the coordinate U_q, so their uniform size is<2epsilon_n. For the inverse control maps, if y=g_n(z) is in such a neighborhood, then u_(n+1)(y)=u_n(z) and both z,y lie in U_q. Consequently d_Q(u_(n+1)(y),u_n(y))<epsilon_n. Outside those neighborhoods there is no change. Thus both f_n and u_n are uniformly Cauchy; write their continuous limits f_infinity and u. Passing to u_n f_n=alpha gives u f_infinity=alpha. Uniform convergence also implies that f_infinity is constant on every alpha fiber, since diam f_n(alpha^−1(q))→0.

To prove u injective, let u(y)=q and choose x_n with f_n(x_n)=y. Then alpha(x_n)=u_n(y)→q. A subsequence converges to x in the compact original ball. Uniform convergence gives f_infinity(x)=y and alpha(x)=q. Therefore EVERY y with u(y)=q lies in the singleton f_infinity(alpha^−1(q)). Conversely this singleton maps to q by u f_infinity=alpha. Thus u is bijective. It is a homeomorphism from compact B^4 to Hausdorff Q, agreeing with alpha on the outer boundary. This proves the model-side approximation without any assertion that the raw f_n inverse fibers shrink or that a uniform limit of set-valued inverses is automatically a homeomorphism.

Section10: Why the collared singular model is an actual ball. The following finite construction and its limiting steps use the supplied framed data.

There is a necessary reduction before Section5 can be applied to the tower model: its DOMAIN must be a standard ball. A map beta:C'→B^4 by itself does not prove that hypothesis. The special singular parameterization supplies extra structure which gives the reduction explicitly.

Suppose a surjection q:B^4→C' has exactly the following nontrivial fibers: for a closed K in S^3 they are the full intervals c({s} times [0,1]), s in K, in a supplied embedded collar c:S^3 times [0,1]→B^4 with c(s,1)=s. Every other fiber is a point. The inner collar boundary may be wild on its inner side; the formula below requires only the GIVEN collar embedding. Suppose q on the OUTER boundary S^3 is a homeomorphism to the specified boundary of C'. Attach an exterior boundary collar S^3 times [1,2] to both domain and quotient, using that boundary homeomorphism. The enlarged domain is still the literal standard ball with an exterior product collar.

Choose j:S^3→[0,1] with zero set K, for example j(s)=min(1,dist(s,K)), with the empty-K case j=1. On the combined collar S^3 times [0,2] define F(s,t)=(s,j(s)t) for 0≤t≤1, F(s,t)=(s,j(s)+(t−1)(2−j(s))) for 1≤t≤2.

On the complement of that collar use the identity, identifying its t=0 boundary by c. The formulas agree at t=1, fix t=0 and t=2, and vary continuously with s. If j(s)>0 they give a strictly increasing interval homeomorphism. If j(s)=0 their only nontrivial point inverse is [0,1], sent to0; the remainder maps strictly increasingly to[0,2]. Thus F is onto the same standard enlarged ball and has EXACTLY the fibers of the quotient q with the added collar. The induced bijection from C' union its added collar to that enlarged ball is a homeomorphism by compactness and the Hausdorff target. In particular this is not an invocation of a four-dimensional Schoenflies theorem or a conclusion that the wild inner boundary is locally flat.

Now assume the actual common quotient construction has supplied beta:C'→Q with null fibers and nowhere-dense singular image, and Section9 has given a homeomorphism a:B^4→Q matching alpha on the boundary. Extend a^−1 beta over the added collar by its boundary product to the ordinary B^4 union exterior collar. Use the explicit homeomorphism just constructed to identify the DOMAIN with a standard ball. The extended map is an onto ball-to-ball map with null fibers and nowhere-dense singular image. Its restriction over the ENTIRE target exterior collar is a homeomorphism. Apply Section5 with that closed collar as protected E. The resulting homeomorphism agrees with the extended map on the whole preimage collar, so it takes its complementary inner C' exactly onto the complementary inner standard B^4, with the required boundary map. Its attaching solid torus inherits the specified original framing because that boundary map was protected, not subsequently adjusted by an arbitrary sphere homeomorphism.

This is the full reduction made in Freedman–Quinn, §4.7, printed pp.78–80, including its paragraph between the proposition and the proof. That paragraph is indispensable: applying the ball-to-ball proposition directly to an unparameterized pinched neighborhood C' would be circular. It uses the special wild-COLLAR arc fibers, not merely arbitrary arc fibers. The actual dual-handle construction must therefore provide that precise singular parameterization, in addition to providing beta and its common quotient; those are still the geometric design obligations before the present operation can be used on the original disk problem.

Section11: Dual model and the singular collar parameterization. The following finite construction and its limiting steps use the supplied framed data.

Fix a FINITE model tower, before its extra plumbings are mapped into the ambient manifold. The entire disk-like capped model has regular neighborhood D^2 times D^2. Deleting the caps instead leaves the complement of disjoint dual two-handles H_m=(D^2 times D^2,D^2 times S^1). The latter convention for their attaching solid torus is the dual one; it is not the original attaching S^1 times D^2. This statement is checked at one elementary surface move: replace the disk by its self-tube along a short arc. Its complement acquires a one-handle, and the two caps supply its cancelling dual two-handle pair. Slice the tube by a normal three-disk. The two cap dual disks run on opposite sides of that slice; outside the tube they are two parallel strips, while inside it their strips pass once on opposite normal sides. Gluing the slice back gives the two linked solid tori in the original D^2 times S^1. Replacing one cap by another capped surface repeats exactly this operation INSIDE its own dual handle. Multiple handles on a surface give disjoint stacks of this pair. Finger moves on a cap, and its added Whitney/accessory disks, give further dual handles contained in those already present. Thus successive finite stories give nested finite families H_(n,m), and the neighborhood of their body is (D^2 times D^2) minus union_m int H_(n,m).

The elementary pair of linked tori just described is the Bing pair in this argument; its specified strips, rather than a link name alone, determine the relevant pattern.

Here is the finite shortening operation on that pattern. Cut the longitude of the parent solid torus at a small box. After the first doubling each core has a long forward-and-return pair of strands, monotone in the longitude coordinate, with the remaining clasp confined to the cut box. A 'monotone loop of length r' means one strand runs monotonically through an interval of length r and the other returns through that same interval; the rest has longitude diameter<a. If r>a, place a doubling clasp a short distance a along the forward strand and its mate a short distance a from the far turn on the return strand. On either child the two middle opposite parallel runs bound an empty ribbon: they are the two sides of the same normal strip, with its clasp endpoints outside that ribbon. Cancel these two runs by pushing across this ribbon. The remaining child has one forward-and-return pair of longitude length at most r−a; its two short joining pieces and clasp have longitude diameter<a. Perform the cancellation for the two children in different normal levels of the parent strip. Their supports are disjoint, and the old single linked passage between the children stays in their short clasp boxes. This is an isotopy of the doubled pair inside the parent torus: the moving ribbon is in its product normal disk, its endpoints are fixed on the two clasp boxes, and at each intermediate time the strands remain in their prescribed separate normal levels. It does not cut a linked component or pass it through the other one.

Iterate the shortening. The initial forward-and-return length is less than the circumference of the longitude circle, say2pi. After more than2pi/a steps every child core has longitude diameter<a. Compress the transverse disk widths as well, with an isotopy in the solid-torus product coordinates. Hence sufficiently many grope stages put every boundary solid torus of the dual handles in a ball of arbitrarily small prescribed diameter about a point of the longitude core. Further Whitney/accessory handles lie inside the preceding dual handles, so also satisfy that bound. All moves are in the finite MODEL, with its actual normal strips; no assertion is made that an arbitrary linked family can be made small componentwise without that iterated pattern.

Apply this to the convergent tower of Section8. At the n-th step use replication and squeezing to add two stories: choose the grope height of the first high enough for the preceding dual-boundary shortening to the next required model tolerance, and put the ambient next-story pieces in balls of diameter<2^−n. For each dual handle H_(n,m):D^2 times D^2→D^2 times D^2, choose its collar on the attaching D^2 times S^1. The precise two model bounds are diam H_(n,m)(D^2 times S^1)<2^−n and diam H_(n,m)(D^2 times ((1/2+2^−n)D^2))<2^−n. The former follows from the boundary shortening; the latter is arranged by compressing the radial disk factor toward the center, leaving the attaching collar fixed. Arrange the new collars to agree with the old collar on the outer interval [2^−n,1]; this is accomplished by product interpolation inside the old dual handle. The changes at step n are supported over the part of the old collar below height2^−n. All attaching parts and their ambient images have the same shrinking bounds.

For precision, let d:(D^2 times S^1) times [0,1]→D^2 times D^2 be the initial radial collar. Let c_n be its successive modifications. Require: c_(n+1)=c_n outside the new dual handles times [0,2^−n]; inside them it has the just specified product attaching collar; its complete descendant collar lies inside the parent handle; and each modified handle's central region H_(n,m)(D^2 times ((1/2+2^−n)D^2)) has diameter<2^−n. Nested dual-handle branches then give a limit collar c. If a point of the collar parameter lies in only finitely many modified pieces, its image stabilizes in a finite embedded collar chart. If it lies in an infinite branch, the corresponding nested transverse regions have diameters tending to zero, so define its unique limit image. Distinct branches are disjoint at a finite level; their limit points cannot coincide. A branch point and a finite-stage point also separate at a finite level. This proves injectivity. The same diameter bounds give continuity at branch points and local finite-chart continuity elsewhere. Compactness gives an embedding of the whole collar.

The complement of the body in the finite dual model is the union of the interiors of the H_(n,m). Its endpoint compactification is obtained by collapsing each connected component of their nested intersection to a point. For a fixed descending handle branch the attaching tori have diameters tending to zero, so their intersection is a unique z in the boundary collar parameter space. The descendant-collar containment puts the full c({z} times [0,1]) in every handle of the branch. Conversely each handle is covered by its attaching radial collar and its specified central region. Points of its collar approach c({z} times [0,1]) by uniform continuity of the embedded limit collar as its attaching torus shrinks; points in its central region approach c(z,0) by that region's diameter bound. Thus any point in the nested handle intersection lies on that full fiber. That component is EXACTLY c({z} times [0,1]), for z in the closed intersection K of the nested attaching solid tori. Conversely every such fiber is one component of the intersection. The finite model identifications with the actual convergent tower neighborhoods, followed by the nested-ball limiting identification in Section8, therefore give an onto parameterization q:D^2 times D^2 → C'

whose nontrivial fibers are precisely those wild-collar intervals. It is identity on the marked S^1 times D^2, and its restriction on the complementary boundary D^2 times S^1 is a homeomorphism to the frontier of C'. Its boundary collar is the one needed in Section10. It does not assert q itself is a homeomorphism; distinct points along a nontrivial collar interval have deliberately been identified.

Section12: Separating the collar fibers and forming the common quotient. The following finite construction and its limiting steps use the supplied framed data.

The singular intervals in Section11 must not simply be declared cellular decomposition fibers. Instead separate their points in a partially defined map, using the actual spare stories between cofinal finite levels.

Let words s have letters0,2; their closed Cantor intervals I_s have ternary prefix s, and J_s is the closed middle third of I_s. Choose cofinal story levels n(i), increasing by at least two, so that every dual handle at level n(i) has both boundary-solid-torus diameter and ambient image diameter<2^−i. Initially use only words2...2. Let T_s be the boundary solid tori of those handles. Retain the upper thirds of their collar blocks, remove the middle thirds T_s times int J_s, and remove the fixed interior model ball B complementary to the two boundary collars. The restriction of the Section11 singular parameterization to this retained set is injective: along each singular collar interval the retained all-upper-thirds set meets that interval only at its last endpoint.

For the induction assume tori and an injective partial map are defined for words containing at most m zeros. Between two consecutive cofinal levels of an all-2 tail are at least two complete stories. Their difference is a neighborhood of an uncapped tower. The old parameterization is a homeomorphism on that difference, since it is outside the nested collapsed-interval intersection. Contract the upper story to get a capped tower there. Section8 and Section11 put a new convergent tower and singular parameterization inside that actual difference, fixed on its attaching product. Repeat the preceding all-upper-thirds restriction inside it. Index its tori by adding one0 followed by an arbitrary all-2 tail. Thus the new partial blocks lie strictly INSIDE the previously omitted gaps; their images are disjoint from all earlier retained blocks and from each other. Choose their cofinal levels far enough out that every word-s block and its whole containing ambient gap region have diameter<2^−|s|. These choices are finite at each depth, and the same nested-ball construction supplies the full containment, not merely a bound on the gap boundary.

Continue over the number of zeros. The maps agree on their attaching product faces. The union is injective, and the stated diameters give its continuous extension to its closure E. An added limit point corresponds to an infinite word with infinitely many zeros; words with only finitely many zeros were already included at a finite induction stage. To check injectivity at new points, take the first zero at which two words enter different gap components. Their images lie in distinct nested ambient gap regions. If their zero positions differ, one enters an inserted gap while the other stays on its retained complementary face, also disjoint. Repeating after any common finite prefix separates every pair of different infinite words. The shrinking diameters then distinguish a limit point from every finite-stage point. This proves the extension q:E→C' is a homeomorphism onto its image, by compactness. It remains identity on the original attaching solid torus. This is the part of the construction that really uses additional STORIES; extra stages alone would not give the uncapped gap tower in which the new parameterizations are inserted.

For the common quotient we also need the finite disk data on each model block. The boundary tori T_(s,m) are regular neighborhoods of boundaries of framed immersed disks D_(s,m) in the parent solid torus. They come from the actual dual-disk strips of Section11, separated at the next cofinal story levels. Their double curves are disjoint arcs with no triple points: in one elementary dual strip block the two sheets meet in one strip arc, and stacking uses disjoint blocks; later fingers are placed in distinct normal strips. This verifies that property for THESE disks, without inferring it for an arbitrary link of immersed disks in a three-manifold.

Assign a height function e_(s,m) to each such disk. On its collar inside its OWN T_(s,m) use the middle height of J_s. On its intersections with earlier tori use constant nonendpoint Cantor heights. Inductively choose the remaining height interval (a,b) inside the permitted parent Cantor interval, avoiding the finitely many earlier constant heights. The new disks lie inside those parent tori, so this separates their graphs from all earlier graphs. At the present finite level, the double arcs have disjoint preimage neighborhoods and no triple points. On one preimage strip of each double arc add a small positive height bump and on the other add a negative one, supported off the fixed boundary collars. The finitely many arcs have disjoint strip neighborhoods, so these changes are compatible and can be made in (a,b). The graphs are embedded and mutually disjoint. Their boundary collars still lie in their own solid-torus blocks, and all graphs avoid every other collapsed block: on an intersected descendant torus their height is the chosen nonendpoint Cantor value, never a middle-third height. This is why arbitrary constant-height spanning disks would not have worked.

In the model ball collapse the closure of B and, for every s,m, the compact set A_(s,m)=(T_(s,m) times J_s) union graph(e_(s,m))

to a point. The sets are pairwise disjoint by the preceding height choices. Each is the framed embedded disk plus its boundary solid-torus neighborhood of Section9; the overlap is its boundary annular collar. Their diameters tend to zero with depth, since their disks lie in the preceding small parent torus and their heights lie in its small Cantor interval. There are finitely many at each depth. Thus this is a null family. Let alpha be the quotient map to Q.

We verify that this quotient has the topological properties actually needed by Section5 and Section9. Its equivalence relation is closed: for converging pairs of equivalent points, either a single closed collapsed set occurs on a subsequence, or infinitely many distinct sets occur, in which case their diameters tend to zero and the two limits coincide. A closed equivalence relation on a compact metric space has Hausdorff quotient: separate two compact equivalence classes by disjoint neighborhoods, then shrink to saturated neighborhoods using closedness of the relation's projections. It also has a countable basis here. If a class lies in a saturated open set, choose a finite union V of rational basic balls containing that compact class with closure in the saturated open set. The open quotient set Q minus alpha(B^4 minus V) contains the class and lies in the required open set. Countably many finite unions V suffice. Hence the compact Hausdorff quotient is metrizable. Nullity of its nontrivial alpha fibers is exactly the construction, and their supplied local shapes are exactly those of Section9.

On q(E) define beta=alpha q^−1. Each component of the complement of q(E) is one of the nested omitted gap regions or the initial interior region. Its frontier corresponds to the boundary of B or of one T_(s,m) times J_s, together with the spanning graph collar inserted there; alpha is constant on that whole frontier. Extend beta constantly across the closure of the component. The extensions agree on their frontiers. Away from a branch they are finite compact gluings and hence continuous. At a branch both the containing model block and its entire ambient gap region have diameters tending to zero, giving continuity by the neighborhood criterion. They produce no additional unintended identifications, since distinct gap frontiers correspond to distinct collapsed sets, and different branches separate in disjoint finite-level blocks. Thus beta:C'→Q is continuous and onto. Its nontrivial fibers are a null collection: beyond any fixed depth they lie in the specified small ambient gap regions, and only finitely many shallower fibers remain. No particular manifold type is required of those beta fibers.

The singular images are nowhere dense, not just countable. Every neighborhood of a branch image contains a sufficiently deep parent block. Within that block retain a collar slab between its next child tori and away from the finite disk graphs. This is a nonempty open four-dimensional piece missed by all collapsed sets. At a finite collapsed-set image the same construction is made in an arbitrarily small neighboring deeper block; at an uncollapsed point it is immediate from the retained open pieces. Therefore every open subset of Q meets an open set of singleton fibers, for both alpha and beta. Both maps are homeomorphisms on the protected outer boundary, with the same prescribed attaching product.

Section9 now gives a boundary-matching homeomorphism from the model ball to Q. Section10 identifies the collared C' with a literal standard ball by the explicit singular-collar interval formula. The extended beta then satisfies Section5 with the exterior collar protected. Its resulting homeomorphism restricts to C'→B^4 with the prescribed attaching solid torus/framing. Pulling back the standard core gives the locally flat disk with its OWN product normal chart, which is exactly the standard-handle output stipulated in Section11. The construction remains in the prescribed component neighborhood: every descendant tower, gap insertion and limiting block was chosen there, and all standardness maps are maps of that embedded pinched neighborhood rather than changes of the ambient manifold outside it.

Section13: The good-group input and the geometric-dual correction. The following finite construction and its limiting steps use the supplied framed data.

There is a real additional issue in applying the old tower argument to controlled handle cancellation. The arbitrary accessory nullhomotopies in Section6 may meet the old geometric dual spheres. Pushing those accessories down and tubing over the duals clears the BODY but does not clear their intersections with the duals themselves. Thus Sections6–12, by themselves, give standard disk cores but do not prove that the original dual spheres remain geometrically dual to the completed towers. The following collar-controlled preparation supplies the missing operation and also uses the ORIGINAL good-group Definition directly, without replacing arbitrary good groups by cyclic groups.

The smooth relative filling used in these finite operations has a concrete justification under Countable Choice. A continuous disk filling with its supplied smooth boundary collar first admits a smooth approximation. To keep the collar fixed, use the Euclidean embedding and smooth neighborhood retraction constructed in the actual Whitney-approximation proof: interpolate in the ambient Euclidean coordinates between the old map and its smooth approximation with a cutoff which is zero on a smaller boundary collar and one outside the region where the old map is already smooth, then apply that retraction. Its pointwise approximation bound keeps the whole interpolation inside the retraction neighborhood. This gives a smooth filling and homotopy fixed on the smaller collar. In a target with boundary, first push any unwanted interior boundary contacts into the target collar's positive side, fixed on the prescribed source collar; double the target only for the approximation, and choose its error smaller than half the positive target-boundary distance on the remaining interior. Thus the final filling stays in the intended manifold with its prescribed proper collar.

For genericity, cover the remaining compact disk interior by finitely many coordinate patches and add small affine coordinate polynomials times cutoffs. Their parameters independently vary values and first derivatives on smaller patches. The evaluation into first-jet coordinates is therefore submersive, for parameters in a sufficiently small common ball. The rank-one derivative stratum has codimension(2−1)(4−1)=3 and the rank-zero stratum codimension8, both exceeding the domain dimension2. Parametric transversality consequently makes the perturbed map an immersion. Compactness gives a common local-injectivity scale for this immersion and for sufficiently small subsequent C1 perturbations. Off that scale the pair configurations form a compact set. At two distinct points choose disjoint bump patches, so the two-value evaluation is submersive. Transversality to the diagonal in V times V makes its double locus zero-dimensional and compact, hence finite. At three distinct points use three disjoint bumps; the three-value diagonal has codimension8>6, so triple points are absent. Interior-point/boundary-point coincidences likewise have expected dimension2+1−4<0 and are removed while the boundary collar is fixed. The finite products of parameter balls have dense sets of good parameters by the actual parametric-transversality theorem. No assertion that ordinary pointwise transversality alone proves a jet statement is being used: the independent affine-jet parameters supply the required submersion explicitly.

These finite operations also apply to Whitney and accessory fillings with their fixed local corner models: perform the perturbations off those closed collars and keep their prescribed normal directions. The same finite multi-evaluation argument gives pairwise transversality of the different new disks and the old sheets.

The given maps and collars here are smooth in the supplied neighborhood. The TOP-generic interpretation of the broader disk-data Definition requires a later relative smoothing bridge and is not inferred from ambient smoothness. First record the finite geometric Casson move. Let two immersed surfaces F,G have an opposite-sign pair p,q with a framed Whitney disk W. For every interior point of G intersection W choose an arc in W from its preimage to the interior of the G Whitney boundary arc. Choose these arcs away from the finite preimages of F intersection W and from the double-point preimages of W, and use separate parallel widths where they share an endpoint region. Push the G sheet along that arc and past the G boundary arc. This removes the selected G intersection W and introduces only G self-intersections near that boundary arc. It misses F, since the interior of the G Whitney arc misses F and all arcs avoided the F preimages. After the finite pushes, the interior of W misses G. Perform the immersed Whitney move on F across W. It removes p,q; its only new intersections are F self-intersections coming from W self-points or F intersection W. No F–G points remain from the operation. It is a regular homotopy supported in a neighborhood of W, so all algebraic intersection and self-intersection numbers and boundary framings are preserved. For a finite family, exclude the other prescribed body sheets from the support by first clearing W using their geometric duals whenever that is stipulated below. This is the actual finite separation move, not a smooth embedded-Whitney-disk assertion.

Apply it to the input disks f_i and framed spheres g_i. The excess f_i–g_j points can be paired by framed immersed Whitney disks because their group-ring coefficients cancel. The required boundary circle is null: equal group labels, with the sheet choices matched, are exactly the equality making that Whitney loop trivial. A relative smooth generic filling supplies the disk; boundary twisting corrects its Whitney framing. Use the preceding move on the two collections, so it creates only self-intersections within either collection and removes all excess mutual points. This makes g_i geometrically dual to f_i. Since the original f_i already have zero mutual lambda and mu, no spherical tubing which changes their homotopy classes is required. They have only undergone regular homotopies.

The inclusion M'=M minus an open framed plumbing neighborhood of the f_i into M induces a pi1 isomorphism. Surjectivity follows by moving loops off a finite two-dimensional image. For injectivity, put a nullhomotopy in general position with respect to that image; after deleting small disks at its finitely many intersections, its extra boundary circles are meridians of the f_i sheets. Each meridian is null in the complement using the punctured immersed geometric dual g_i. This fills all punctures and gives the nullhomotopy in M'. The same argument handles both sheets at a double point. Hence M' has the given good fundamental group.

Choose framed Whitney disks D_j for all intersections within and among the f_i, and tube their interiors over separate parallels of g_i to remove their intersections with every f_i. Thus D_j are immersed proper disks in M'. At one paired double point take its Clifford torus T_j in the local product normal chart. Its two meridional caps each intersect exactly one f sheet; tube each cap at that point to a punctured parallel g_i. This gives capped tori T_j^c entirely in M'. Their bodies lie arbitrarily close to the corresponding component of the boundary of the plumbing neighborhood, and are geometrically dual to D_j. Their caps have zero mutual lambda and mu: the local meridional interiors are disjoint, their attaching collars meet only in their prescribed boundary crossing, and the remaining points are exactly those among the framed g_i parallels, with the same cancelling group labels. The input zero Wall numbers therefore give zero coefficients before augmentation. Contract separate parallel capped tori to get geometric dual spheres for D_j, push D_j off these contractions, and tube intersections of the ORIGINAL torus caps with D_j into those parallel spheres. The resulting T_j^c are genuinely dual capped surfaces, meaning their caps also miss D_j. This cleanup preserves the cap Wall hypotheses: the contraction sphere is built from two oppositely oriented parallels of each cap and the torus body, so its intersection with any old cap is a signed sum of the old cap–cap coefficients, hence zero; its self-number is a sum of the same zero self/cross coefficients. The sesquilinear and quadratic sphere-sum formulas therefore leave all cap lambda/mu numbers zero when a cap is tubed to those spheres. The bodies stay in their chosen boundary collar.

Here is the collar reservation which prevents the old gap. Parameterize the actual boundary collar by t in [0,1], with boundary t=0, and put the initial T_j bodies in5/6<t<1. Only attaching collars of D_j and of the torus caps enter this collar otherwise. Tube all D_j mutual/self-points over full parallels of T_j^c in this outermost slab. The resulting bottom surfaces D'_j are embedded, mutually disjoint, and have the same framed boundaries; their caps C' are the two appropriate torus-cap parallels at every tube. If C denotes the torus caps, lambda(C,C')=mu(C)=0 by the preceding coefficient calculation and parallel-copy identities. Take a fresh family of capped-torus parallels with bodies in4/6<t<5/6 and contract it, pushing nothing off, to obtain spheres S_j geometrically dual to D'_j.

Pair the C–C' points by framed immersed Whitney disks. Their boundaries lie above t=4/6, and their nullhomotopies can be chosen there: deleting t<4/6 from a boundary collar is a homotopy equivalence with an explicit collar-push inverse. Tube these disks over S_j to make their interiors disjoint from all D'_j. The tube paths and spheres stay above the fresh lower collar slabs. Apply the geometric Casson move to C and C'. Its supports miss D'_j and all unused torus BODIES below4/6. Thus the new caps miss each other as the two collections, while arbitrary self/intersections within each cap collection are still allowed and remain algebraically cancelling.

Push the C' self/mutual points down D'_j and tube full capped-torus parallels with bodies in3/6<t<4/6. This adds the second surface stages, producing height-two capped gropes D''_j. Their body is embedded by the same scorecard as Section3. Separate their new caps C'' from C exactly as before, now using contracted parallels in2/6<t<3/6 and Whitney nullhomotopies outside t<2/6. All remaining torus bodies may now be taken in0<t<2/6. They and their caps are geometrically dual to the entire height-two gropes, including their caps. Raise the latter to any required finite height, keeping their first two stages, and successively contract one extra top stage to make their caps pairwise disjoint, with their self-points in canonical cancelling pairs. These neighborhood operations miss the dual capped surfaces because those were made disjoint from the whole starting grope except at the bottom transverse point.

Now the actual good-group Definition applies in its exact place. Choose grope height n+2.5. Each disjoint top height-1.5 component is a supplied disk-like framed grope. Apply goodness to its homomorphism into pi1(M') to obtain the framed null disk in that canonical model. Section14 smooths it within a neighborhood with trivial group image while retaining its attaching frame; thus it supplies the smooth immersed disk required here, with double-point loops null in M'. Distinct replacement disks lie in disjoint component neighborhoods. The retained grope has height n+1. Contract one top stage symmetrically: the new cap double loops are parallel copies or differences of the null loops, hence null; the cap self-points come in algebraically cancelling finger pairs. The first n stages have been kept EXACTLY. Taking n≥4 is sufficient for Sections7–12.

Fill those double-point loops by framed immersed accessory disks. Their boundaries miss t<2/6, so their fillings may be pushed off that collar. Hence they miss the torus BODIES reserved there, although they may intersect its caps and the grope arbitrarily. Contract fresh dual capped surfaces with bodies in1/6<t<2/6 to get spheres S''_j. Correct the accessory framings, push their body intersections down, and tube over S''_j to clear the whole grope body. The resulting accessories are the proper tower top caps. Finally take the LAST capped-torus parallels with bodies in0<t<1/6 and contract them, pushing every tower-cap sheet off the contraction. Their bodies already miss the accessories by the collar choice. The resulting spheres R_j therefore miss the ENTIRE tower except for their prescribed bottom transverse point. This final contraction/push-off is the step which the uncontrolled use of the original spheres in Section6 could not provide.

The R_j are nullhomotopic in the ORIGINAL M. Indeed symmetric contraction of a fixed capped surface is independent up to homotopy of its caps: it is regularly homotopic to surgery on only the first cap of each pair; switch the unused second caps; it is also homotopic to surgery on only the second caps; then switch the first caps. Thus in M replace the caps built from g_i by the original small meridional caps. The contracted Clifford torus is now contained in the original double-point four-ball, and is null there. This proves [R_j]=0 in pi2(M), despite its being a useful nonzero class in pi2(M').

Remove intersections of the original g_i with the towers by pushing them down the tower to the bottom and tubing over R_j. They still meet the f_i only at their one original dual point, since all these changes lie in M'. They now miss the entire tower. The changes of g_i are homotopies and additions of whiskered R_j, which are null in M, so preserve their original homotopy classes. Perform Sections7–12 inside neighborhoods of the towers disjoint from these g_i, with their one dual intersection moved into the protected attaching collar. The resulting embedded framed Whitney disks are disjoint from every g_i. Their Whitney moves on the f_i give disjoint properly locally flat embedded disks, regularly homotopic rel boundary/framing to the original f_i, and with the g_i still geometrically dual. They also provide the product normal charts required in the final surface-compression application: on the untouched pieces keep the old smooth disk frames; on each Whitney replacement use the pulled-back standard two-handle frame, glued by the Whitney framing along its two boundary arcs. This is a finite construction of a normal disk bundle over the resulting disk, trivialized by those compatible frames. Compactness permits sufficiently small widths so different patches overlap only by their prescribed bundle transitions and the resulting total-space map is injective. Thus the constructed disks have their OWN product tubes, without using the later general TOP normal-bundle theorem. Thus the original disk theorem's homotopy conclusion is supplied by WHITNEY MOVES, rather than assumed from an arbitrary standard-core choice.

Section14: Smoothing a framed null disk inside a canonical grope neighborhood. The following construction makes the relevant framing and foundation inputs explicit.

This operation addresses the category of the disk obtained from GOODNESS, whose boundary is the canonical smooth attaching region of the framed model grope. It does not smooth an arbitrary original boundary knot or a TOP-generic disk in an unrelated ambient manifold. Its input is a generic immersed disk with the supplied extending framing/product germs which make its boundary the model attaching region. If 'framed boundary' were read as an arbitrary external integer without an extending framed immersion, the following parity argument would not apply; that is not the framed-disk convention used here.

Let Q be the smooth framed model neighborhood, phi:pi1(Q)→Gamma, and d:D^2→Q the supplied generic framed disk with phi-null double-point loops. Its compact image A is homeomorphic to a finite polyhedron P: subdivide the parameter disk with the finitely many paired double-point preimages as vertices and with their stars disjoint, identify those vertex pairs, and subdivide once more. The resulting finite quotient complex maps continuously bijectively onto A and hence homeomorphically, by compactness and the Hausdorff target. Its fundamental group is generated by the finitely many double-point loops, since the disk before identifying the pairs is simply connected. Thus phi kills the entire image of pi1(A).

Here is an explicit neighborhood-retraction argument, avoiding a general normal-bundle theorem. Realize P as a finite subcomplex of a standard simplex, subdivide barycentrically, and note that its subdivision is FULL: a simplex whose chain of face-vertices all belong to P belongs to P. On the open star of P in that subdivided simplex, delete the barycentric coordinates at vertices outside P and normalize the remaining positive sum. Fullness makes this a continuous retraction onto P. A small neighborhood in Euclidean space first retracts to the simplex, by taking nonnegative coordinates and normalizing their sum. Composing gives a Euclidean neighborhood retraction onto P.

Extend the finitely many coordinates of the inverse homeomorphism A→P continuously to Q as follows. For a residual continuous scalar function r_n on compact A with sup norm at most c_n, choose the least positive integer L_n such that |r_n(a)−r_n(b)| ≤ c_n/2 + L_n dist(a,b) for all a,b in A.

Uniform continuity and boundedness give such an integer. Put h_n(x)=inf_(a in A)(r_n(a)+L_n dist(x,a)), clamped to[−c_n,c_n]. It is bounded and L_n-Lipschitz; on A its error from r_n lies between0 and c_n/2. Define r_(n+1)=r_n−h_n|A and c_(n+1)=c_n/2. The uniformly convergent sum of the h_n extends the original coordinate function exactly. All constants are suprema/infima or least integers, so this is a definable recursion. For a zero starting function use the zero extension. The finite coordinate extensions give a map Psi:Q→R^N. On the inverse image of the preceding Euclidean retraction neighborhood, compose with its retraction and P→A. This gives a continuous retraction r:U→A from an actual open neighborhood U of A.

Form the smooth double $Q^{\mathrm{d}\mathrm{b}\mathrm{l}}$ from the two copies of the supplied smooth grope neighborhood, using its given product boundary collar, as in Section13. Use the proper Euclidean embedding j of this BOUNDARYLESS double and its smooth neighborhood retraction R supplied by the actual Whitney-approximation proof. There is a continuous fold c:Q^{dbl}→Q: it is the identity on the first copy, identifies the second copy with the first by the doubling identification, and in the shared collar is (x,t)→(x,|t|). For x in a sufficiently small relatively open neighborhood U' of compact A in U⊂Q, the segment between j(x) and j(r(x)) stays in the domain of R. Indeed r fixes A, and compact j(A) has positive distance from the closed complement of that domain. The formula c(R((1−t)j(x)+t j(r(x)))) is a continuous homotopy IN Q from the inclusion U'→Q to r|U', fixed on A. Hence every loop in U' has phi-trivial image in Q. No smooth retraction onto the boundary of Q is used, and no smoothness of the fold is claimed.

Carry out the smooth relative approximation and finite jet perturbation of d in the DOUBLE, fixing its supplied smooth attaching collar pointwise. Choose a smaller closed source boundary collar inside the collar where d is already smooth; keep a neighborhood of that smaller collar fixed by the cutoff. The complement of that neighborhood in the compact parameter disk is a compact set mapped into IntQ, since d is proper. Its image therefore has positive distance from the boundary of Q in the doubled metric. It also has a positive tolerance for remaining in U', by relative openness and compactness. Choose the approximation and subsequent finite perturbations below both tolerances. On the fixed collar the map remains d; everywhere else it stays in IntQ∩U'. Thus the resulting map is a smooth proper generic disk IN Q, with the original attaching collar and framing, and all its double-point loops lie in U'. The continuous fold used in1.127 is NOT applied to this smooth approximation. For the framing, take the supplied framed parallel of the original d with its disjoint boundary parallel. Its ordinary relative intersection with d is twice the signed self-point count, and is therefore even; the normal-zero term is absent because that frame extends. The relative intersection number is unchanged under homotopy fixing these boundary parallels. For the smooth approximation it equals2mu+e, by the local parallel-sheet count and the zeros of a generic normal section. Thus its relative Euler number e is even. Interior cusps change e by2, so finitely many restore e=0 and the required frame. Their new loops are trivial in the same small coordinate balls, so phi-nullness survives. The resulting SMOOTH framed generic null disk has the original attaching region.

This operation shows why the goodness step can be consumed in the canonical smooth grope model with the framed-disk convention, without using arbitrary TOP neighborhood smoothing, stable5 integration, or the full boundary/collar smoothing needed for an unrelated original TOP-generic input. That boundary/collar bootstrap is not claimed here.

Section15: Countable Choice versus the successive tower choices. The following construction makes the relevant framing and foundation inputs explicit.

Countable Choice is not being identified with Dependent Choice. Saying 'choose a next tower at each stage' would not justify this foundation claim. The geometric selections in this particular smooth-model construction can instead be made by a single-valued least-code rule, after fixing the Euclidean embedding/retraction of the smooth neighborhood once.

For a new disk with a specified smooth boundary collar, list pairs consisting of a dyadic collar width and a finite tuple of rational polynomials on the standard parameter disk. Use its supplied boundary-collar map b and a fixed smooth radial cutoff alpha, equal to1 on a smaller collar and0 off that collar. Form the Euclidean candidate alpha j(b)+(1−alpha)P and apply the fixed neighborhood retraction R. The term alpha j(b) is extended by zero only where alpha already vanishes; it is smooth there. Candidates are retained only when the Euclidean values lie in the retraction domain and the resulting map has the required proper boundary collar, genericity, support and finite intersection conditions. Framed candidates are listed simultaneously: project rational polynomial vector fields to the smooth normal plane of the candidate disk, blend them with the prescribed boundary frame on the same collar, and retain those with oriented linearly independent normal vectors everywhere. Orthogonal projection is computed from the ambient embedding metric and the derivative matrix, so is part of the fixed formula.

Whenever an admissible smooth framed filling exists, some listed candidate exists. Choose the dyadic collar inside a collar on which that filling agrees with b. Rational polynomials approximate its Euclidean coordinates and its normal frame uniformly with the required first derivatives. Here is the needed C1 density calculation. Extend each smooth coordinate function slightly past the compact disk, using its finitely many local extensions and a finite smooth partition, and restrict to a containing square rescaled to[0,1]^2. Its Bernstein polynomial is the finite sum of f(k/n,l/n) times the two binomial basis polynomials. Uniform convergence follows from uniform continuity: the binomial weights are probabilities, and the weight outside a fixed distance eta of(x,y) is at most1/(2n eta^2) by the sum of their variances. Its x derivative is the analogous degree(n−1,n) sum of n(f((k+1)/n,l/n)−f(k/n,l/n)), each an average of the x derivative on an interval of length1/n; uniform continuity of that derivative and the same variance bound give uniform derivative convergence. The y derivative is identical. Approximate the finitely many coefficients by rationals with error smaller than the desired C1 tolerance divided by4n: the basis sums to1 and the derivative difference has bound2n times that error. Restrict back to the disk. This proves the claimed rational-polynomial C1 density rather than assuming an analytic power series for an arbitrary smooth function. The values stay in the open retraction domain; properness away from the fixed collar, the normal-frame determinant and the finite transversality angles have positive compact lower bounds. Thus sufficiently close candidates preserve them. Near a generic immersion the double points and their labels persist in disjoint coordinate balls and no new double points occur outside them, by compactness off the local-injectivity diagonal. For a group-null filling use Section14's open neighborhood with null group image: sufficiently close candidates lie there, so retain all null-loop conditions. Candidates which are required to lie in a prescribed relative homotopy class do so for sufficiently close approximations, by the straight retraction homotopy fixed on the boundary collar.

The least natural-number code satisfying these actual predicates therefore specifies the disk and its frame uniquely. Decidability of the predicates is not asserted or needed for set-theoretic definability. This is not a choice function on arbitrary subsets of a mapping space: it is minimization in a specified COUNTABLE list which has just been proved nonempty for each geometric step.

The other choices have the same form. Paths and finger arcs are finite polygonal/rational smooth approximations in the already specified charts, with endpoint templates and finite disjointness conditions. Parallel widths and modification balls use the first dyadic radius meeting the strict compact separation conditions. Finite orderings of double points can be obtained lexicographically from their distinct parameter preimage coordinate tuples, and every finite tree or interval subdivision is chosen in that ordering. The local shrinking Baire step of Section5 has its own explicit countable rational bump rule, preserving every earlier finite avoidance margin. The successive approximations of Sections9–12 use least cofinal story levels and least integer tolerances. No choice is made over an uncountable family of possible future branches.

Thus each next finite stage is a uniquely defined function of the previous finite stage, the original input and the fixed embedding/retraction. Ordinary recursion on the natural numbers constructs the entire sequence. The single fixed smooth embedding/retraction and the actual transversality/approximation suppliers use their stated Countable Choice hypothesis; once those are fixed, no additional Dependent Choice principle is introduced by the tower or shrinking recursion. This is the precise foundation claim of the current smooth-input candidate. It does not turn the open unrestricted problem into a theorem and does not supply a TOP smoothing theorem.

A standard open $2$-handle is $D^2\times {\mathbb{R}}^2$, and its core $D^2\times \{0\}$ is a locally flat properly embedded disk whose boundary is the core attaching circle with the standard normal-framing class.

Write $h_i:(T_i,A_i)\to (D^2\times {\mathbb{R}}^2,S^1\times {\mathbb{R}}^2)$ for the supplied homeomorphism, and set ${\bar{f}}_i(z)=h_i^{-1}(z,0)$. Since $T_i$ is relatively open in $X$ and $T_i\cap \mathrm{\partial }X=A_i$, the inverse of $h_i$, followed by inclusion in $X$, identifies the standard handle with a relatively open ambient neighborhood, including its boundary. Restricting its product charts along the core gives charts of $(X,{\bar{f}}_i)$ modeled on $({\mathbb{R}}^4,{\mathbb{R}}^2)$ at interior points and $({\mathbb{R}}_{+}^4,{\mathbb{R}}_{+}^2)$ at boundary points. Moreover ${\bar{f}}_i^{-1}(\mathrm{\partial }X)=S^1$, because $h_i^{-1}(S^1\times {\mathbb{R}}^2)=A_i$. Thus these are locally flat proper disks in the ambient pair, not merely intrinsically in the towers.

The disks lie in disjoint $T_i$, so are pairwise disjoint. The framing-preserving condition on $h_i$ transports the standard boundary frame to the prescribed boundary normal-framing class. The disks therefore have all the required boundary data.

Section1: Precise input. A finite framed four-dimensional capped-grope neighborhood is supplied, with its usual product collars and self-plumbing charts. Its body is embedded; only cap interiors have generic double points. An integer labeling means a homomorphism from the fundamental group of this actual neighborhood to Z. Paths to individual cap sheets are fixed once. Conjugating a based loop does not change its integer label. No arbitrary locally flat surface is given a tubular neighborhood by this hypothesis: the actual grope's framed block neighborhood is input.

Operation. Contract one top surface stage. Designate a finite subset of cap double points to be pushed off at that contraction, and choose a first and second sheet at each designated point. Then the resulting shorter capped grope has the same lower body, attaching circle and framing. Every old intersection not designated has parallel descendants with its old label. No designated old intersection remains. Every new cap intersection is a descendant of two designated intersections removed across dual caps at this same contraction, and its label is a difference of their labels, up to overall sign. Consequently, if all designated labels lie in an interval [a,b], every new label has absolute value at most b−a.

Proof of the contraction operation. Cut the top surface at each pair of symplectic attaching curves. Replace the cut part by two small parallel copies of each of its two caps and the small central square at their unique attaching-curve intersection. Cutting a genus-g one-boundary surface along its complete symplectic system leaves the disk region with paired edges; the four parallel cap sectors and this square fill each pair of edges. Thus this replaces the surface with its contracted disk. All replacements take place in the supplied framed block. Parallel copies are made in its chosen normal coordinate; the central square uses the untwisted product collar, so the external attaching framing is unchanged. The result may be immersed, which is allowed.

For a designated point, take an embedded path on its second cap from that point to its attaching boundary, with interior avoiding all other double-point preimages. Finitely many paths on one cap can be chosen with disjoint interiors by an embedded finite tree and small parallel offsets. The two parallel copies of that path, closed across the central square, bound a ribbon Whitney disk pairing the two intersections of the first sheet with the contracted cap. In a product rectangle around the path, this is just the rectangle between the two parallel copies, closed by the boundary square. Its interior is disjoint from the old body and all unmodified sheets. A generic sufficiently small choice of the finitely many product widths is possible because their unwanted incidences are finitely many closed disjoint compact sets. No appeal to a normal bundle over an arbitrary TOP surface occurs.

Push the first sheet across this ribbon. This removes precisely the two paired intersections. The only ribbons that can meet are ones running across dual cap sectors: their boundary paths cross in the central square. Resolving those crossings by the prescribed order of the pushes creates intersections between the pushed first sheets. Ribbons on the same cap have been chosen parallel and disjoint, and ribbons belonging to different handles have disjoint block neighborhoods. Therefore every new intersection has exactly the stated two designated ancestors. Repeating for all parallel descendants removes the designated old point altogether. This is a finite sequence of ordinary finger/Whitney operations in explicit smooth block charts; a small relative perturbation makes the resulting intersections transverse.

Here is the label calculation, including the path issue. Let the two designated points have based first-to-second loops g,h. A path to the new intersection follows the first pushed sheet along the ribbon to its second-sheet cap, crosses the common central square, and returns along the other ribbon in reverse. Cancel the common paths in the old body. The resulting word is g h^{-1}; changing starting path conjugates it and exchanging the two resulting sheets inverts it. This is a calculation in the original supplied neighborhood, since every ribbon and push lies there. Its Z label is consequently ±(phi(g)−phi(h)). An old point copied to parallel sheets has its old loop with a small contractible product strip added, so its label is unchanged. The construction, classification of new intersections and claimed bound follow.

Section2: Halving and the finite stopping bound. Suppose there are h top surface stages available for contraction and the absolute labels of all the actual cap-point generators are at most an integer N. Contract the top stage, designating all points whose labels are in [N/2,N]. Their new labels have absolute value at most N/2 by Section1. The unmodified old positive labels are less than N/2; old negative labels may still range to −N. Contract the next top stage and designate the labels in [−N,−N/2]. Again new labels have absolute value at most N/2. Old labels outside this interval already have absolute value at most N/2. Thus two contractions reduce the bound to N/2. The same construction works when N is odd, since inequalities refer to real bounds and labels remain integers.

After q pairs of contractions the bound is N/2^q. Choose the least integer q with 2^q>N; if N=0 take q=0. This is a finite integer algorithm. At that point every cap-point generator has label zero. The neighborhood group is generated by these loops: the abstract disk-like capped grope is simply connected (its symplectic basis loops are killed by its attached higher stages/caps), and each self-plumbing identifies two small simply connected sheet neighborhoods, adding its one double-point loop. Cutting the finite plumbing neighborhoods restores the abstract source. Van Kampen, adding them back one at a time, gives precisely these generators, with no assertion that an arbitrary ambient group is free. Hence the entire neighborhood image has zero Z label.

Contract the remaining body stages to a disk. Every operation stays in this neighborhood, so all double-point loops of the final framed immersed disk have zero image under the original homomorphism. Its boundary and framing are unchanged. This proves null-disk production when sufficiently many actual stages have been constructed; it does not yet manufacture those stages from height1.5.

In the symmetric height-raising construction, the precise quantitative target can be stated without the harmless factor-of-two compression in FQ2.9. If 2k raising steps give height at least 2^k and each resulting cap-point loop has word length at most 5^{2k} in the original finite cap-point generators, their absolute labels are at most 5^{2k}B. It suffices that 2^(−floor((2^k−1)/2)) · 5^(2k) · B < 1.

Reserve one bottom surface stage; the other stages supply the indicated number of contraction pairs. The displayed quantity tends to zero because its base-two logarithm is at most −2^(k−1)+O(k)+log2 B. Thus a least successful k is obtained by a terminating integer search. No infinite choice or convergence theorem is hidden in this numerical step. Sections3–4 prove the required height and word-length estimates starting from the asymmetric height1.5 input.

Section3: The asymmetric bootstrap. Start with one symplectic pair above the bottom surface: on the positive side a framed capped surface R of positive genus, on the negative side a framed immersed cap W. The same construction is repeated for the finitely many pairs; their body blocks are disjoint, and interactions between cap systems are allowed. All operations fix a neighborhood of the original bottom surface and its attaching region.

Take two small framed parallel copies of R, at negative and positive normal coordinate, and join their boundary circles by the normal annulus crossing the attaching curve of W once. The result is an embedded closed surface of genus twice that of R. Its caps are the corresponding parallel copies of the caps of R. This is a sphere-like capped surface F: compressing its symplectic system gives a sphere. Its body meets W in one point and is disjoint from R; its cap intersections with all the old caps are unrestricted. The annulus and the single crossing are read in the supplied symplectic handle block, not in a hypothetical neighborhood of an arbitrary locally flat surface.

Construct a small parallel copy of F and contract its top stage to an immersed framed sphere T. During contraction, push every other cap sheet off T by the ribbons of Section1, excluding the one prescribed body intersection T with W. This includes the original caps of F, the caps of R and the remaining parts of W. All intersections newly produced by those pushes are between those cap sheets. The sphere T is therefore disjoint from both R and the body/caps of F, and from the caps of R, and meets W at its one prescribed point only. Self-intersections of T are allowed. The ribbons have support near cap arcs and their central squares, away from the lower body, so they do not alter any of the displayed body intersections.

Use T as a Norman sphere to remove every intersection of W with a cap of R or F: for each such point, choose a path in W from that point to the unique point W meets T. Remove a small disk from the intersecting cap and connect it by an untwisted tube, along a parallel copy of that path, to T with a small disk removed. The second endpoint consumes the one intersection of that parallel T with W. The two disks removed and the tube are the boundary of a product neighborhood of the path, so the resulting cap misses W at the chosen point and has the same boundary and framing. Choose the finitely many paths as branches of a tree in the abstract disk W, with separate parallel approach paths at their common endpoint; their interiors avoid all other double-point preimages. Normal widths are chosen successively small. The added copies of T are framed immersed spheres; their self- and mutual intersections belong entirely to the modified CAPS. Thus they may create cap intersections but no body intersections. Their original intersections with R or the body of F are empty, and sufficiently small parallel copies retain that property. After this finite operation W is disjoint from every other part except its own self-intersections and the single point on the body of F. In particular F remains a proper capped surface transverse to W.

For each self-intersection of W, tube one of its sheets to a separate small parallel copy of the BODY of F, along the other sheet and then along a chosen arc in W to the transverse point. This is the same local Norman cancellation, now using a capped surface instead of the contracted sphere. The new body consists of W with that sheet disk deleted, the punctured closed body of F, and their connecting annulus. It is a one-boundary surface with additional genus, with the old W boundary/framing. The caps of the added F provide the symplectic caps for its new handles. Its body is embedded: before the tubing, distinct body copies were disjoint and met W only at their distinct transverse endpoints; the tube arcs avoid every other double-point preimage. Choose disjoint parallel branches of a finite tree in W and nested normal widths. The tube neighborhoods consequently miss all other body sheets. Every old self-intersection of W is removed once, and none is introduced into the new body. Its new caps may intersect each other or the old caps, but they miss every body: the prior operation made the old caps disjoint from W, and properness of F makes its caps disjoint from all the parallel body sheets. At the attaching collars this assertion is the defining product chart. Consequently W has become an actual proper framed capped surface, not merely a surface with unaccounted body intersections.

This changes the side heights (1,0) to (1,1), preserving the original bottom stage. All choices and moves were finite. The scorecard is: R/body(F) embedded and mutually disjoint throughout; T immersed auxiliary, intersecting W once; old caps arbitrary among themselves; after the Norman cancellations all old caps miss W; after the final tubing every new intersection lies among caps. The auxiliary T is discarded after its copies have been inserted into caps. No disjointness of its parallel copies is needed, because their intersections are allowed cap intersections.

Section4: Height raising and word length. Here the two side heights count stages ABOVE the fixed bottom surface. Section3 gives (1,0) to (1,1). The following side extension, used only finitely often in this bootstrap, supplies (1,1) to (2,1) to (2,3). Let P be the side to be raised and Q its other side, of heights p,q at least one. Construct F transverse to the bottom surface of P by taking two normal parallels of Q and closing their rims with the normal annulus. It is sphere-like of height q. Its body is embedded, meets P only at the specified first-stage point, and misses all other body stages; intersections of its caps with the old caps are allowed.

First make a contracted parallel sphere T transverse to the bottom of P, using Section1 and pushing off all original cap sheets and the caps of F. Its only body intersection is that prescribed transverse point. To make the caps of F disjoint from the old P caps, push an F-cap sheet at each intersection down the old P cap to its attaching stage and then down successive P stages to its bottom. A finger across one attaching collar replaces one intersection with two on the immediately lower stage; this is two parallel paths around the two ends of that collar. After p stages there are finitely many points on the bottom of P. Tube them off over parallel copies of T. These changes affect only caps of F. They cannot reintroduce intersections with old P caps because T misses those caps and the finger paths on the old caps avoid the other finitely many double-point preimages. The new intersections are among caps of F and possibly the old Q caps, which are still caps. All old body stages are fixed.

Now for an intersection of an old P cap with a Q cap, push the Q-cap sheet down the P stages and tube it off over T. The Q cap is still an immersed disk; its boundary and framing are fixed. It may acquire self/intersections with other caps, which are allowed. It remains disjoint from the old P caps and all body stages: the punctured T used in the tubing has had its unique bottom-P intersection removed. Crucially T is also disjoint from the body of F, so the Q caps miss every body piece to be added in the next step.

Finally treat every self/mutual intersection among the old P caps by pushing one of its P sheets down the P stages and tubing it off over copies of the full F, rather than T. This replaces each old P cap by a capped grope: its first added stage is the old disk with punctured bodies of F attached, followed by the higher stages of those copies of F. Orient each added copy so its transverse intersection cancels the chosen point, and use the untwisted framed tubes. All old P-cap intersections have been removed; the added bodies are embedded and disjoint from each other because distinct body parallels of F are disjoint, and the tube paths avoid all other old preimages. The new caps are copies of F's caps. Those were made disjoint from ALL old P caps before this step, and properness makes them disjoint from F's parallel bodies. Consequently every new intersection is among caps, not on a new body. A cap with no intersections may be regarded as a genus-zero stage with no attached higher surface, as in the disk-like model. We have obtained side heights at least (p+q,q), with the previous body and its attaching framing preserved. The first and second steps are needed: merely asserting that tubing adds height would leave the old cap intersections on the new body.

All choices involve finite paths, parallels and normal widths in supplied framed blocks. Use the move first with p=q=1, then exchange the two sides and use p=1,q=2. Both sides then have height at least two, so the entire grope has height at least three. The induced integer labels on its finite plumbing generators are obtained by following their actual paths back to the original neighborhood; let B be the maximum of their absolute values. B exists without any growth estimate for this finite bootstrap.

Once both sides have height at least two, there is an efficient variant which contracts only the TOP stage of a transverse grope. For the side P to be extended, form the transverse grope from the opposite side Q. Its height is that of Q. Contract its top stage and push the old caps off that contraction by Section1. The transverse grope now has height one less and its caps are disjoint from all old caps. Push each intersection on a P cap down the P stages, and at the bottom tube off over this transverse grope. The same body/cap scorecard as Section4 applies: new bodies consist of separate parallel copies of the old embedded opposite-side bodies, and all cap intersections have been removed from the new body before it is declared a body. This replaces each old P cap by a capped grope with the height of the contracted transverse grope. In terms of full path heights a,b INCLUDING the shared bottom, the updated path height is a+b−2. Alternate the sides. Starting with a_{−1}=a_0=3, the successive updated path heights obey a_n=a_{n−1}+a_{n−2}−2. The uniform height is the smaller of the two current path heights. Writing a_n−2 as a Fibonacci recursion gives, after 2k steps, uniform height at least 2^k: consecutive pairs increase the smaller Fibonacci term by at least a factor two. Additional initial height only improves the bound.

For clarity, the required estimate concerns WORDS, not numbers of double points. Base every old cap-point loop by a path in the old grope. Body-stage paths contribute no new plumbing generator. Contraction copies an old point without changing its generator. A push off a contraction creates the difference loop of Section1, a word of length at most two in the OLD cap-point generators. A new intersection among caps of the parallel transverse grope has length one, since its sheets are old parallel sheets. The only remaining type is a loop using a newly pushed-down finger: follow its first sheet across at most two old cap-point loops to the parallel transverse copy, cross its new cap intersection (one old generator), and return across at most two old loops on the other finger. The two bottom-stage paths cancel, as do the parallel collar paths. This is a concatenation of at most 2+1+2 old generators or inverses. These exhaust the new intersections: they occur on unchanged parallels, on a contraction push, or between two inserted fingers. Thus each step sends every NEW generator to a word of length at most five in the PRECEDING generators. Induction gives length at most 5^(2k) after 2k steps, and absolute Z label at most 5^(2k)B. No count of the possibly much larger number of points is substituted for this word bound.

Choose the least k satisfying Section2's displayed stopping inequality (if B=0 no raising is needed). Height supplies the required number of contraction pairs, and the label estimate supplies their initial bound. Section2 then gives zero image on every cap-point loop, and final contraction produces a framed immersed disk with the original framed attaching circle. Everything occurs in the original supplied grope neighborhood. For an arbitrary homomorphism to Z this is the pi1-null disk property of the existing goodness Definition. The construction is finite for each input: no countable-stage choice, limiting disk, tower standardization or arbitrary TOP normal-bundle theorem has been used in this cyclic part.

The normal microbundle in [F1] is a neighborhood germ with its zero section and retraction, not merely an abstract stable vector bundle. On the embedded boundary of a proper locally flat sheet, its relative half-space chart is $({\mathbb{R}}_{+}^2\times {\mathbb{R}}^2,{\mathbb{R}}_{+}^2\times 0)$. Restricting the given normal trivialization to its boundary gives a TOP tube germ of that boundary circle in $\mathrm{\partial }X$; its normal fiber is the second ${\mathbb{R}}^2$. Compactness permits a sufficiently small representative around each of the finitely many circles, and disjointness permits these representatives to be disjoint. This uses the GIVEN normal germ, not a theorem constructing normal neighborhoods for arbitrary TOP surfaces. Apply [F4] to this boundary link in the given compact smooth three-manifold $\mathrm{\partial }X$. Denote its isotopy by $h_s$ and its smooth framing representatives by ${\eta }_i$.

Extend the actual isotopy over a smooth collar $c:\mathrm{\partial }X\times [0,a)\to X$ from [F5]. Choose a cutoff $\beta$ equal to one near zero and zero before $a$. Set $H_s(c(x,r))=c(h_{s\beta (r)}(x),r)$, and use the identity outside the collar. The inverse uses $h_{s\beta (r)}^{-1}$, so $H_s$ is an orientation-preserving ambient TOP isotopy with joint inverse. Transport all maps, whiskers and normal classes by $H_1$. Every old signed intersection and group label is transported by the same based group isomorphism; goodness is unchanged. For the positive circle homeomorphism ${\theta }_i$ of [F4], let ${\mathrm{\Theta }}_i:D^2\to D^2$ be its radial cone, and replace the disk temporarily by $H_1{f}_i{\mathrm{\Theta }}_i$. Its boundary MAP is now a smooth regular parametrization. At the end we undo both $H_1$ and ${\mathrm{\Theta }}_i$, so no change of the original boundary parameter is concealed.

Construct a NEW smooth source collar. In the product of a smooth boundary-circle tube and the ambient collar, write the transported map on a small source collar as $(b_i(\theta ,t),v_i(\theta ,t),r_i(\theta ,t))$, with boundary values $(\theta ,0,0)$ and $r_i>0$ for $t>0$. Uniform continuity gives a unique small real lift of $b_i-\theta$ after restricting the collar. For $\kappa (t)$ equal to one on a smaller collar and zero outside the chosen one, interpolate these coordinates to $(\theta ,0,\rho t)$ by $((1-s\kappa )b_i+s\kappa \theta ,(1-s\kappa )v_i,(1-s\kappa )r_i+s\kappa \rho t)$. Choose the finite positive $\rho$ and collar small enough to stay in the coordinate tube. The last coordinate is positive for every interior point throughout this homotopy, and the boundary is fixed. At its end the smaller collar is smooth. The homotopy may pass through nongeneric maps: the conclusion requires a homotopy, not a regular homotopy of the old TOP ribbons. It can be arbitrarily small, and its support misses all old double points.

Use [F3]'s relative smooth approximation and finite affine-jet perturbations simultaneously on the disks and spheres, fixed on these smaller smooth collars. In a target with boundary perform approximation in its smooth double, with error less than half the positive target-boundary distance on the remaining compact source interiors; the output therefore stays in $X$. The same retraction interpolation gives a homotopy fixed on the collars. Independent value and first-jet bumps avoid the rank-deficient jet strata, make the two-point evaluation transverse to the diagonal and avoid triple points; compactness then gives only finitely many double points. This is exactly the finite genericity operation already proved in [F3]. Denote the resulting smooth generic maps by $f_i^{\mathrm{\prime }},g_j^{\mathrm{\prime }}$.

Here is the full group-ring comparison. At each old transverse crossing choose its supplied oriented TOP sheet chart and two small parameter disks. On their four-dimensional product the chart difference of the old sheet maps has one zero, of local degree the old sign, and no zero on the product boundary. For a sufficiently close approximation its chart difference is joined to the old one by a straight homotopy nonzero on that compact boundary. The signed sum of its transverse zeros is therefore the same local degree. All their group labels are the old label, since their connectors lie in that contractible chart. This uses a degree of continuous maps in a TOP chart, not differentiation of that chart. Outside these finitely many products and a neighborhood of each source diagonal, the compact source-pair set has positive image separation. Thus there are no other intersections between DIFFERENT source surfaces. Near a source diagonal both preimages lie in one of finitely many embedded-sheet charts, so a new self-double loop is null in $X$. It follows that every off-diagonal $\lambda$, including every disk–sphere pairing, retains its full original value, and $\mu (f_i^{\mathrm{\prime }})=n_{i}1$, $\mu (g_j^{\mathrm{\prime }})=m_{j}1$ for integers $n_i,m_j$. These equalities are in the actual sheet-involution quotient, not just after augmentation.

The self-pairing $\lambda (g_j^{\mathrm{\prime }},g_j^{\mathrm{\prime }})$ is still zero. To compute it without assuming TOP parametric transversality, take a small parallel using the original framed normal germ, with the same whisker. Apply the preceding degree comparison to that pair; it computes the original self-pairing. Approximate the two homotopic representatives smoothly and interpolate their maps in the smooth double, relative any protected collar. The ordinary finite smooth two-value transversality argument computes homotopy invariance of the pairing: the one-dimensional intersection tracks have cancelling signed interior endpoints, with unchanged group labels along each track. Thus the new self-pairing is the old zero. The same argument for a disk and its boundary parallel computes the relative intersection integer determined by a specified boundary framing.

Track the actual relative Euler obstruction rather than declaring it zero. For a smooth oriented disk $u$ and a boundary frame $\eta$, a generic normal section extending its first boundary vector has isolated zeros; their signed sum is $e(u,\eta )$. A parallel sheet meets $u$ at those zeros, and at both ordered branches of every self-point. Hence its relative intersection integer is $e(u,\eta )+2{\sum }_pϵ(p)$. The analogous equivariant sphere identity is $\lambda (u,u)=e(u)1+\mu (u)+\overline{\mu (u)}$. These follow by the stated local zero count; the two self-point labels are inverses. Twisting a boundary frame changes its relative Euler integer by the corresponding signed winding, so every desired integer value is realized by a smooth frame, and the difference from $\eta$ is recorded. Vanishing relative Euler is precisely the obstruction to extending that frame over a disk: extend on a finite cell subdivision and move the integer zero indices into one interior two-cell, where index zero extends across the cell.

A signed interior cusp changes $(\mu ,e)$ by $(\pm 1,\mp 2)$ and changes no relative map homotopy class or off-diagonal pairing. For concreteness the birth chart is $(x,y)\mapsto (x,y^2,y^3-ty,xy)$: for $t<0$ it is immersed without a double point, and for $t>0$ its sole double pair is $(0,\sqrt{t}),(0,-\sqrt{t})$, with independent tangent planes. Reflecting a normal coordinate changes its sign. Restrict to a small inner chart and join the unchanged outer sheet through the surrounding annulus; the modification is a homotopy in that coordinate ball fixed on its boundary. The change of $e$ is opposite twice the double-point sign by the parallel-sheet count of step 7.1, since the boundary parallel intersection is unchanged. The balls and joining sheets can miss every other source surface. Thus any finite prescribed sum of signed identity-labelled cusps is available without changing a nonidentity Wall coefficient.

Let $e_i=e(f_i^{\mathrm{\prime }},{\eta }_i)$. Choose an auxiliary smooth boundary frame ${\eta }_i^a$ with $e(f_i^{\mathrm{\prime }},{\eta }_i^a)=-2n_i$, and record its integer difference from the prescribed ${\eta }_i$. Insert cusps with signed sum $-n_i$. The resulting smooth disks have $\mu =0$ and extending auxiliary frames. For spheres, steps 5.1–7.1 give $0=\lambda (g_j^{\mathrm{\prime }},g_j^{\mathrm{\prime }})=(e(g_j^{\mathrm{\prime }})+2m_j)1$, so the cusps of sum $-m_j$ give both $\mu =0$ and $e=0$. Their oriented smooth normal bundles are therefore trivialized: after choosing frames on the one-skeleton, the only two-cell obstruction is this Euler integer. All disk–sphere and off-diagonal pairings remain exactly those of the hypothesis.

If the prescribed original disk framing extended, its original boundary-parallel intersection integer was zero, since its Euler obstruction and original $\mu$ were zero. Steps 6.1–7.1 show that $e_i+2n_i=0$, so in that case ${\eta }_i^a$ is the prescribed class ${\eta }_i$ itself. If only an external boundary framing was specified, keep its recorded integer difference from ${\eta }_i^a$ on every later boundary tube. This preserves the specified external framing class and does not assert that a nonextending external class has become extending. Both readings of the literal datum are thus respected; no assertion that merely prescribing a frame forces $e_i=0$ was used.

More explicitly, let ${\tau }_i$ be an auxiliary extending frame on an output product normal tube. Along its boundary write the recorded external frame as ${\eta }_i={\tau }_i{A}_i$, with $A_i:S^1\to G{L}^{+}(2,\mathbb{R})$; the smooth boundary representatives of [F4] allow this description. On a sufficiently thin boundary annulus use $(z,t,v)\mapsto (z,t,A_i(z)v)$, shrinking the fiber radii if necessary. Its inverse uses $A_i(z{)}^{-1}$, so this is a genuine change of collar coordinates carrying the entire prescribed boundary frame. It changes neither the zero-section disk nor its homotopy. An arbitrary winding of $A_i$ is allowed because these coordinates are required only on the annulus, not over the disk. Thus a nonzero relative Euler obstruction forbids an extending frame with that boundary value, but does not forbid retaining that value on the boundary of the embedded disk. The framing in the Statement is explicitly this boundary framing.

Apply [F3] in the supplied smooth neighborhood $V=X$ to these corrected smooth framed maps. Their group is the original good group under the transport of step 2.1, and all hypotheses hold over its actual group ring. It gives pairwise disjoint proper locally flat disks and fixed-boundary homotopies, with extending auxiliary frames. Their actual product normal tubes carry the original external classes by the recorded boundary winding of step 10.1; when the input frame extended, it is already the extending output frame. No TOP-generic smoothing theorem or controlled five-dimensional normal-bundle theorem is being imported in this application.

Concatenate the finite collar, approximation, cusp and [F3] homotopies, then apply $H_1^{-1}$ and precompose each disk homotopy with ${\mathrm{\Theta }}_i^{-1}$. At every time the boundary is $H_1^{-1}(H_1{f}_i{\mathrm{\Theta }}_i{\mathrm{\Theta }}_i^{-1}){\mathrm{\mid }}_{S^1}=f_i{\mathrm{\mid }}_{S^1}$ pointwise. Properness and local flatness survive the ambient and source homeomorphisms. The full normal-class comparisons of [F4] and step 10.1 pull back to the prescribed original boundary framings. This proves the stated TOP-input conclusion in the given smooth ambient manifold. Only the finite boundary operation and the accepted CC smooth construction were used; the seven accepted construction/Alexander/open-problem carriers require no alteration.

Give the reflected surface $r(F)$ its transported orientation. Its boundary is $r(K)$ with the transported knot orientation. Therefore the Seifert surface for the specified reverse mirror is $F^{†}=-r(F)$, with the opposite surface orientation. Let $(u,v)$ be a positive tangent frame of $F$ and let $n$ be its positive normal, so $(n,u,v)$ is ambient-positive by [F1]. A positive tangent frame for $F^{†}$ is $(-dr(u),dr(v))$: reversing exactly one tangent reverses the surface orientation. Since $r$ reverses ambient orientation, the frame $(dr(n),-dr(u),dr(v))$ is ambient-positive. Consequently the positive normal of $F^{†}$ is $dr(n)$, and the positive push of the image basis curve $r(a_i)$ is precisely $r(a_i^{+})$. This computation includes both orientation reversals; reversing both tangent vectors instead would preserve the surface orientation.

Reflection negates linking. Indeed an oriented normal disk to an oriented curve $D$ is positive when its disk orientation follows the tangent of $D$ in the ambient orientation. The reflected disk has the opposite normal-disk orientation relative to $r(D)$, since $r$ reverses the ambient orientation. Thus $r$ takes the positive meridian class to minus the positive meridian class, and taking meridian coefficients in the complement gives $\mathrm{lk}(r(C),r(D))=-\mathrm{lk}(C,D)$. Using the actual positive pushes from step 1.1, the image-basis entries are therefore $\mathrm{lk}(r(a_i^{+}),r(a_j))=-\mathrm{lk}(a_i^{+},a_j)=-V_{ij}$. This proves the matrix is $-V$, including the empty matrix in genus zero.

By the actual determinant supplier, the supplied Seifert matrix has Laurent-unit determinant. Its skew part is the integral unimodular surface intersection form. The following finite matrix reduction applies to this very matrix and preserves the positive-push convention.

Let V be an integral 2g-by-2g matrix and J=V−V^T unimodular alternating. Suppose P(t)=det(V−tV^T) is a Laurent unit. Since P is a polynomial and P(t)=t^(2g)P(t^−1), it is t^g: a unit polynomial is ±t^m, reciprocity gives m=g, and P(1)=det J=1 fixes the sign. (The determinant of an integral unimodular alternating matrix is +1: choose any primitive vector, use the primitive row of J and Bezout to split off a symplectic pair, and repeat on its integral orthogonal direct summand. This determinant argument does not require a kernel vector of V, so it precedes the following deduction det V=0 without circularity.) For g>0 its constant coefficient det V is zero. Choose a nonzero rational kernel vector of V, clear denominators and divide by its coordinate gcd. This gives a primitive integral x with Vx=0.

The row x^T V equals x^T J, since x^T V^T=(Vx)^T=0. It is primitive because multiplication by the unimodular J preserves primitive vectors. Bezout therefore gives an integral y with x^T V y=1. Then omega(x,y)=x^T J y=1. Put L={z:omega(x,z)=omega(y,z)=0}. Every z decomposes uniquely as [F1, F2, F3, algebra] z = omega(z,y)x + omega(x,z)y + z_0, z_0 in L.

Thus Z^(2g)=Zx direct-sum Zy direct-sum L, and omega restricted to L is unimodular alternating. L is a free integral direct summand; its basis is obtained by applying the displayed integral projection to the coordinate generators and elementary Euclidean column operations. Repeating the primitive-vector/Bezout splitting in L gives an integral symplectic basis there. In that basis, followed by x,y, the Seifert matrix is [F1, F2, F3, algebra] [[B,0,u],[0,0,1],[u^T,0,w]].

Indeed Vx=0 gives its zero x-column, and x^T V=x^T J gives the x-row. Orthogonality to y gives equality of the two old-to-y coupling vectors. Replace y by y−wx. This is a symplectic integral basis change and makes w zero, without changing B,u. We have exhibited exactly the algebraic tube enlargement in the ACTUAL S-equivalence Definition, not a rational enlargement.

Expanding det(tV−V^T) along the x-row and then x-column gives [F1, F2, F3, algebra] det(tV−V^T)=t det(tB−B^T).

For even matrix size det(tV−V^T)=P(t) by transposition and multiplication by −1. Hence the reduced matrix B again has unit polynomial t^(g−1) and unimodular alternating skew part. Induct on g. This supplies a finite sequence of integral congruences and exact inverse tube additions down to the empty matrix. No stabilization theorem is assumed. It proves the full matrix assertion needed by the original S-equivalence carrier once its actual Seifert determinant/skew-pairing interfaces are applied.

Full AC implies Countable Choice, so choose the actual connected smooth Seifert surface supplied by [F2]. The proved reduction [F3] splits its given lattice by symplectic pairs and the exact inverse tube enlargements of [F4].

Reversing these reductions also produces the claimed trivial Alexander block. Keep all short vectors first and all long vectors second. A reverse tube addition introduces a new short x with zero column and only the entry 1 in its own long column, and a new long y with zero diagonal. Its couplings to earlier short vectors occupy precisely the new LAST column of U, so U stays strictly upper triangular. Its couplings to earlier long vectors give a symmetric new row/column of W, with zero new diagonal. Starting from the empty matrix therefore gives [[0,I+U],[U^T,W]] with exactly the advertised conditions. This is an integral symplectic basis of the original lattice; the finite construction below realizes it geometrically rather than assuming that arbitrary integral basis changes are surface homeomorphisms.

Fix a standard geometric basis a_i,b_i on the actual compact oriented genus-g surface with one boundary circle. Its existence follows by cutting a finite disk-and-handle model of the surface, not from an arbitrary homology basis. Put omega(a_i,b_j)=delta_ij. An annular Dehn twist about an embedded curve v acts by the transvection T_v(z)=z+omega(z,v)v, choosing the twist sign accordingly. This formula is proved by cutting each transverse arc at the annulus: the twisted arc acquires one oriented copy of its core. Curves representing a_i+a_j, b_i+b_j and a_i+b_j for i≠j are embedded band sums on distinct handles. The joining arc lies in the complement of the other standard handles; the band gives the indicated homology class. Thus these transvections and their inverses are actual boundary-fixed surface homeomorphisms (smooth diffeomorphisms in the smooth model).

For orthogonal u,v, direct expansion gives T_(u+v) T_u^−1 T_v^−1(z) =z+omega(z,u)v+omega(z,v)u.

Taking u=a_i,v=b_j, i≠j, realizes the elementary block-GL operation a_j to a_j+a_i and b_i to b_i−b_j. Its powers realize arbitrary integral elementary additions. Handle permutations and simultaneous sign reversal of a_i,b_i are also realized; the latter is the square of the two-twist rotation on that handle. Euclidean elimination consequently realizes every block action (x,y) to (A x,A^−T y) for A in GL_g(Z). Likewise the a_i and a_i+a_j twists realize every symmetric upper shear, because the diagonal rank-one matrices and the identity (e_i+e_j)(e_i+e_j)^T−E_ii−E_jj=E_ij+E_ji generate all symmetric integral matrices; the b_i analogues realize every symmetric lower shear.

Here is a complete generation argument, so this is not an appeal to mapping-class surjectivity. Given a primitive vector v=(x,y), first interchange the a and b directions on a handle if necessary to make x nonzero. A block-GL operation makes x=(d,0,...,0), d>0. If some y_j is not divisible by d, reduce d strictly: for j=1 use the ordinary Euclidean two-twist SL_2 algorithm on (d,y_1); for j≠1 an upper diagonal shear inserts y_j in x_j, then block-GL Euclidean reduction replaces d by gcd(d,y_j). Repeat. The positive integer d decreases until every y_j is divisible by d; primitivity then forces d=1. A lower symmetric shear with first column −y clears y while keeping x=e_1. Thus an actual finite product of the specified twists takes every primitive vector to a_1.

Let A be an arbitrary symplectic integral matrix. Apply this procedure to its first column, so the resulting matrix fixes a_1. Its image of b_1 has the form b_1+c a_1+z, with z in the span W of the remaining handles. The displayed orthogonal-transvection combination for u=a_1 and each basis vector v of W fixes a_1. Its k-th power subtracts k v from that image of b_1 and only changes its a_1 coefficient in addition. Use these powers to clear all coordinates of z. Then a power of T_(a_1) clears c. The resulting matrix fixes both a_1,b_1, and hence restricts symplectically to W. Induct on g. We have reduced A to the identity by actual twists; reversing the finite list realizes A. In particular the symplectic lattice basis from the preceding primitive-vector/tube reduction is represented by an ACTUAL geometric basis, with disjoint short curves, disjoint long curves and exactly one transverse short–long intersection on each matching pair.

Use [F1] to fix the actual geometric trivial Alexander basis. The following integral shear preserves its short–long intersection convention and makes its long–long block zero.

The current general disk carrier demands vanishing mutual/self Wall numbers for its dual spheres, not merely the triangular disk–sphere matrix in the short GT application. The following integral operation removes a potential extra obstruction rather than assuming those two interfaces coincide.

Suppose a supplied geometric trivial Alexander basis has matrix V=[[0,I+U],[U^T,W]], with U strictly upper triangular and W symmetric. Put N(C)=CU+U^T C on symmetric integral g-by-g matrices. Left multiplication by U^T and right multiplication by U commute and each has gth power zero. The binomial expansion therefore gives N^(2g−1)=0: in every summand one of the two exponents is at least g. Hence I+N has the integral inverse sum_{r=0}^{2g−2}(−N)^r (the g=0 case is empty). Set C=−(I+N)^(−1)W. This is an explicit symmetric INTEGRAL matrix, since N preserves symmetry. The block change S=[[I,C],[0,I]] is symplectic: its new long basis vectors are ell'_j=ell_j+sum_i s_i C_ij, and their intersections are −C_ij+C_ji=0; short–long intersections remain the identity. Direct multiplication gives S^T V S = [[0,I+U],[U^T,W+C(I+U)+U^T C]] = [[0,I+U],[U^T,0]].

Thus the same triangular block admits an entirely zero long–long Seifert block. This does NOT assert minimal Seifert rank: U may be nonzero. The initial basis is supplied by [F1]. This particular shear has an explicit geometric realization, so no general surjectivity assertion about the mapping-class group is needed for THIS step. For any vector v among e_i and e_i+e_j, realize its short-class sum by an embedded circle: use s_i for e_i; for e_i+e_j use a band sum of parallel copies of s_i,s_j along a simple arc avoiding all short curves. Such an arc exists because cutting along all the short curves leaves a connected surface. If that cut were disconnected, the oriented boundary of one complementary component would give a nontrivial integral relation among the independent short classes, a contradiction. The band sum and its small annular neighborhood can be chosen disjoint from every original short curve. A Dehn twist in that annulus, written in annular coordinates as (theta,r) to (theta+2pi rho(r),r), is a homeomorphism fixed on the annulus boundary; choose rho constant zero/one near its two ends. Tracking an oriented transverse arc gives on homology the transvection x to x+(x intersection v)v, with sign reversed by the inverse twist. It fixes every short class and sends the long classes by the symmetric rank-one shear vv^T, with the sign chosen according to the intersection convention. Now E_ij+E_ji = (e_i+e_j)(e_i+e_j)^T−E_ii−E_jj.

Every symmetric integral C is a finite integer sum of these rank-one matrices. Compose the corresponding positive/negative twist powers. Each action fixes the short Lagrangian, so their shear matrices ADD even when their supporting annuli are not disjoint. The composition is an actual orientation-preserving boundary-fixed surface homeomorphism. Applying it to the entire original geometric basis preserves all its geometric intersection counts and realizes S. In the smooth category choose smooth annular cutoffs to obtain a diffeomorphism. Thus the zero-W basis is genuinely geometric and has exactly the same triangular U. The basis supplier [F1] supplies that initial input.

Apply the preceding shear to [F1]. We obtain a smooth geometric basis s_i,ell_i with V=[[0,I+U],[U^T,0]], not only the existential lattice statement. The following construction retains the ambient ball and all attaching data. It is stated before invoking any disk-embedding conclusion.

Use the literal ball of radius1.5 and its unit sphere. Place the original surface F in that unit sphere and add the exterior collar K times [1,1.5]. Round its junction with F, retaining the outer K product collar; call this proper smooth surface Sigma. Near F its normal coordinates are the radial coordinate x and the positive normal coordinate y of F in S3. Choose its open normal thickening there to be .9<x<1.1, |y|<epsilon, with the compatible product continuation near K. This is an ACTUAL smooth thickening, supplied by the radial product and the normal line of the oriented surface in S3. Put N=the complement of that open thickening, with corners rounded away from the marked circles. It is compact, connected, smooth, and the original ball/surface have not been replaced by an abstract four-manifold.

A disk-and-band handle decomposition of F has one disk and2g bands. Make its height increase from that disk across the bands to the boundary, and realize that height as a small radial displacement in the displayed collar. The complement has one0handle, one1handle from the surface minimum and2g2handles from its bands, with no higher handles. This local complement recipe follows by taking the product normal2disk to a surface0handle or1handle: its complement adds respectively the normal meridian1handle or the2handle whose core crosses the band. The product collars identify the attaching pieces; no arbitrary embedded4ball is identified with the standard ball. The resulting complement deformation retracts onto a2complex with one vertex, one1cell and2g2cells. Name the positive normal meridian of its one1handle a. The local2handle at an oriented surface band identifies its endpoint meridians by transport through the band: its attaching relation is m_1 w m_2^−1 w^−1. To see the signs, take the boundary of a normal2disk and transport it along the band core; the two boundary ends of the resulting normal-annulus strip have opposite induced boundary orientations. Both endpoint meridians come from the same original disk and represent a in the one-generator1skeleton. Any winding of a knotted band only changes the transport word w, which is a power of a there. Thus the attaching word is a a^k a^−1 a^−k=1, for every band. Consequently pi1(N)=Z, and the same2complex is homotopy equivalent to a circle wedge2g spheres. This is the actual group/attaching-word computation, not an inference from H1 alone.

Choose generic immersed disks Delta(s_i) in the inner unit ball with boundaries s_i-down. Run their boundary collars in the negative-y side of the marked normal strip, bending into the interior of N immediately off each boundary circle. Choose disks Delta(ell_i) bounding the long curves and trim their radial collars to x=.8; they then have boundaries ell_i at x=.8 and lie in the ball x<=.8. The short disks consist of disks in that inner ball and their mutually disjoint negative-y collars. Their union lies in a simply connected part of N: the inner ball together with those product collars retracts onto the inner ball. Thus all their equivariant intersection labels are1. Their ordinary intersections are the boundary linking numbers, which are V(s_i,s_j)=0. To justify this relative4-ball identity, compare one filling disk with a Seifert surface for the same boundary curve pushed into a thin boundary collar. Their difference is a two-cycle, null in the ball. Its intersection with the other proper disk is therefore zero, by the boundary-of-intersection identity for a bounding three-chain. The collar surface counts exactly the linking number on the boundary sphere, with the outward-normal-first orientation. This proves the identity for immersed fillings, rather than assuming disjoint embedded disk fillings. The boundary framings are the prescribed surface pushoffs. A generic filling disk with that zero longitude can be made framed: the parallel-intersection identity gives 0=2mu+e, so its relative normal Euler number e is even; an interior cusp changes e by2. Use finitely many cusps to make e0. The same identity then gives mu0. These changes are local in the inner ball and preserve the boundary circle/frame. Mutual intersection numbers are invariant under those rel-boundary homotopies. Accordingly lambda(Delta(s_i),Delta(s_j))=0 and mu(Delta(s_i))=0 in the precise integral group-ring convention, not merely after augmentation.

For each long curve construct the torus T_i=ell_i times the boundary of the normal rectangle [.8,1.2] times [−2epsilon,2epsilon]. The long curves are geometrically disjoint, so these tori are disjoint. They lie in N. A short disk meets T_i at the point of ell_i intersection s_j on its negative-y radial collar, hence with geometric signed count delta_ij. Remove an annular neighborhood of the long curve at x=.8,y=0 in T_i, and cap its two boundary circles by opposite oriented framed parallel copies of the trimmed Delta(ell_i). This is a smooth immersed sphere g_i in N. The long–long Seifert block is ZERO, so these long disks have zero boundary linking and can be chosen with the zero extending normal framing as above. The torus product framing matches both caps explicitly: along x=.8 the first common normal is the direction in F transverse to ell_i; the second rotates from the radial direction on the torus to the y-direction on each disk across the chosen rectangle corner. The opposite upper/lower cap orientations give opposite half-rotations, not an extra full twist. The long disk's extending zero frame supplies the remainder. Hence the untwisted attaching annuli give a trivial normal2plane bundle on g_i. Thus these are framed spheres, not tori called spheres without a framing check.

Fix orientations explicitly before the coefficient calculation. At a positive short–long intersection let u be the short tangent and v the long tangent, so (u,v) orients F. The ambient ball orientation is (x,y,u,v), with x outward radial and y the positive surface normal in S3. Orient the rectangle boundary counterclockwise in (x,y), and orient T_i by its rectangle tangent FIRST and its long tangent SECOND. At its x=.8 side the torus tangent basis is (−y,v), while the outward-radial collar of the short disk has basis(x,u); their ordered determinant relative to(x,y,u,v) is positive. Thus the old torus–short point contributes +delta_ij. Removing the annulus |y|<epsilon' at that side leaves its upper boundary oriented +ell_i and its lower boundary −ell_i. Consequently the surgery uses −Delta(ell_i) on the upper cap and +Delta(ell_i) on the lower cap.

Take t to be the counterclockwise meridian in the oriented (x,y) normal plane and base through the inner ball on the lower side. Traveling in the retained torus from the lower cap to the upper one goes around the rectangle's outer three sides counterclockwise; returning through the inner ball closes the positive meridian. Thus the upper cap path differs from the lower by t, with no arbitrary relabeling of the prescribed positive meridian. Use common basing paths through the inner ball. The old torus–short-disk point contributes delta_ij. Every intersection of the long disk with a short disk gives two sphere–short intersections of opposite signs. Their paths differ by precisely the short meridian, so lambda(g_i,Delta(s_j))=delta_ij+(1−t)lk(ell_i,s_j-down) =delta_ij+(1−t)(U^T)_ij.

For two DIFFERENT g spheres the tori and their annular surgery pieces contribute no points, because the long curves are geometrically disjoint. Each long-disk intersection contributes four points from the two cap parallels, with factors1,−t,−t^−1,1. Hence lambda(g_i,g_j)=(1−t)lk(ell_i,ell_j-down)(1−t^−1)=0.

For a self-double point the same parallel-sheet accounting gives two identity-label contributions and the negative t,t^−1 contributions, modulo the sheet-exchange relation. The long disk has framed relative Euler number0 and self-intersection sum0, so mu(g_i)=0. No extra intersections occur at the surgery seam: its two parallel attaching annuli are disjoint in the explicit product torus, with their normal framings already matched. This verifies the additional dual-sphere vanishings demanded by the ACTUAL local disk-data Definition, not just the weaker duality assertion quoted in AppendixA.

The displayed disk–sphere matrix is lower unitriangular. Set f_i=g_i−sum_(k<i) T_ik f_k, inductively, using whiskered parallel sphere copies and framed connected sums. For j<i the sum cancels T_ij; for j=i the earlier rows vanish and the diagonal stays1; for j>i both terms vanish. A whisker around t^r left-multiplies the first-variable coefficient by t^r in [F2], so each Laurent coefficient is realized with its stated sign. Its result has lambda(f_i,Delta(s_j))=delta_ij, and the zero lambda/mu conditions among the spheres persist by direct sesquilinearity and the self-sum formula (all cross terms vanish). Taking conjugate transpose gives lambda(Delta(s_i),f_j)=delta_ij in the original disk-data order. The output is therefore the ACTUAL good-group disk problem in this particular N, with exactly the short-circle boundaries and product normal framings needed to compress Sigma. Goodness of Z is [F3]; disjoint embedded replacement disks still require the separate tower/standardness proof. This construction makes no such replacement by definition.

The relative smooth filling step used above is justified without TOP smoothing. Approximate a filling map in the smooth ball, fix its supplied boundary collar by cutoff interpolation in the Euclidean embedding/retraction from [F5], and choose finite affine coordinate bumps which independently vary its first jets. Rank-one jets have codimension3>2, so parametric transversality gives an immersion. Independent bumps at distinct pairs make the double locus zero-dimensional; triple configurations have codimension8>6 and are absent. Compactness away from the uniform local-injectivity diagonal gives finitely many double points. Boundary collars are fixed throughout. The explicit cusp and sphere operations above are in those smooth charts.

Apply [F1] to obtain the actual surface $\mathrm{\Sigma }$, complement $N$, short curves $s_1,\dots ,s_g$, marked annuli and smooth disk/dual datum. By [A1] and [F2], [F3] applies with $V=N$. Its hypothesis concerns smooth maps and smooth collars; those are explicitly supplied here. Thus the compression disks have disjoint embedded standard product handle charts with the prescribed attaching framings. If $g=0$, the original $\mathrm{\Sigma }$ is already the required disk and the following finite surgeries are empty.

Join each produced disk to its short curve on $\mathrm{\Sigma }$ along the marked negative-normal annulus. These annuli are disjoint, and their interiors lie on the $\mathrm{\Sigma }$ side of $\mathrm{\partial }N$; the disks lie on its $N$ side. The supplied collars identify their product charts at the seam. The resulting joined disks therefore have product normal neighborhoods whose first boundary normal direction is the normal direction of $s_i$ within $\mathrm{\Sigma }$. Choose mutually disjoint sufficiently small product neighborhoods. Replace the annulus $s_i\times (-\eta ,\eta )$ in $\mathrm{\Sigma }$ by the two disk parallels at $\pm \eta$ in that first normal direction, with opposite orientations. The attaching collars give two coordinate half-planes meeting on their common boundary line. Rounding their quadrant in those product coordinates yields a locally flat surface. This uses the product charts actually supplied by [F3], rather than a normal-neighborhood theorem for an arbitrary TOP surface. All changes fix the outer knot collar.

Cutting along the geometric short curves leaves a connected surface. Indeed, if its complementary-component graph had more than one vertex, choose a proper nonempty vertex subset with an edge to its complement. The oriented boundary of the corresponding union gives a nonzero signed relation among the independent classes $[s_i]$: every crossing edge occurs once, while edges internal to the union cancel. This is impossible. Hence capping these cut curves leaves the result connected. Removing the annuli changes no Euler characteristic, while the $2g$ disk caps raise it from $1-2g$ to $1$. The connected oriented one-boundary surface is a disk: the geometric cut-system polygon is a sphere with $2g+1$ boundary circles, and the caps fill its $2g$ inner circles. Its only outer boundary is still $K$, and its original boundary product collar is unchanged. Finally radial rescaling of the literal radius-$3\mathrm{/}2$ ball to $B^4$ carries this locally flat proper disk to the desired one.