The infinite cyclic group is good for disk embedding

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.