Finite relative mixed-handle movies give marked Kirby moves

Movie input and the assignment to a surgery link. Fix connected compact oriented three-manifolds N,M, a specified identification of their boundaries, and the protected vertical collar. A connected four-dimensional bordism W between them is given with its complete boundary markings and its stated orientation. Use that orientation on every forward and reversed handle stage. All one-handle attachments are orientation-compatible, so their cores closed by base arcs have orientable rank-three normal bundles. The framed circle surgeries retain an orientation by the product gluing in [F5]; hence this remains true after earlier trades. Thus every application of the arc-choice comparison [F3], including the reversed trades, satisfies its normal-bundle hypothesis. Suppose two all-two-handle presentations are joined by a finite index-grouped mixed-handle movie on W: elementary framed attaching maps vary smoothly between events; events are cancelling births/deaths 0/1,1/2,2/3,3/4, same-index handle slides and critical-value interchanges of disjoint attachments. All terminal maps are retained. Positive-index-drop trajectory tangencies are permitted, since the elementary-band attaching maps remain smooth through them. This exact input is stronger than a generic function path. Its explicit supplier is Index-grouped descending paths give based framed handle movies: common index cuts and bounded-time full framed attachments; compactness isolation at the single same-index passage; reversed index1 treatment of index3; birth/value-crossing comparisons; and actual transport of every higher attaching map and endpoint marking. That construction, together with the weak field and grouping proofs, supplies this input rather than assuming it as a genericity consequence. Assign a two-handle presentation at every ordinary time as follows. Collapse a rooted spanning tree in the initial zero/one-handle graph, eliminating every new zero-handle. Dually collapse a rooted spanning tree in the outgoing dual zero/one-handle graph, eliminating every four-handle. The two roots are the fixed nonempty connected N and M. These collapses leave a presentation with indices1,2,3. In either direction the horizontal face at the one-handle stage is connected: it starts at the connected root, absorbed balls do not change it, and orientation-compatible one-handles on a connected three-face retain connectedness. The joining arcs for the trades lie in this horizontal attaching face, not merely in the connected four-dimensional lower stage. On the reversed presentation trade every one-handle (the original index3 handles) by the explicit interior circle surgery, obtaining only indices2,3 in reverse and hence indices1,2 forward. Flatten the successive two-handle attachments into the boundary of the forward one-handle stage. Trade every remaining forward one-handle by its explicit core-plus-base-arc interior circle surgery and cancellation. The result is a finite link in N. Every trade preserves the full external boundary of the four-manifold and changes its interior. Every flattening and cancellation retains its outgoing comparison. The actual local trade, framed-circle-to-level isotopy and two-disk-factor surgery bridge are Interior one-handle trades and the fully framed circle-surgery pair and Framed circles in a four-dimensional two-handle trace move to its fixed base. The arbitrary replacement arc comparison is Changing the base arc in a one-handle replacement gives marked Kirby moves. The one-handle slide after trading is One-handle slides become Kirby slides after interior trading. No change of replacement arc is declared invisible: it is that comparison's actual O3/O2 sequence with its meridian correction.

Rooted tree collapses and changes of tree. The zero/one-handle graph has one old root vertex N and one vertex for every new ball; edges are the one-handles. It is connected because higher-index attaching regions are connected and cannot merge components. Choose a spanning tree T. Starting at its leaves, the union of a leaf ball and its incident tree tube is a ball with one retained attaching disk. Absorb it into its neighbor's collar using the explicit ball-plus-tube product chart. Every other foot and every higher attaching embedding is carried by this very chart. Repetition removes all new zero-handles. The support stays off the protected vertical collar. Reversing gives the outgoing tree collapse. Two trees are compared by single-edge exchanges. If T′ differs from T, choose f∈T′−T. The unique cycle in T∪{f} contains an edge e∈T−T′; replacing e by f reduces the number of differing edges. Finite induction reaches T′. For this one exchange restrict to the cycle plus its attached tree branches; first absorb the branches by the same leaf-ball products. The cycle is now a cyclic sequence of balls and tubes attached to the root component by a tree path. Collapse all but the two edges e,f. The remaining model is two balls joined by two tubes, with its incoming root collar retained. Collapsing the first tube or the second tube gives the same ball-plus-one-handle model with different retained core paths. For the comparison, cut only the tube f at one transverse three-disk. The two balls joined by e form a single ball, with two named f-cut faces; cutting both tubes instead would give two separate balls and cannot give a four-face ball. Keep the root collar patch fixed and use the explicit e-tube product to absorb the other ball. In this description the surviving handle follows f and returns along e. If instead f is absorbed, the surviving handle follows e and returns along f. These are the same two tube pieces traversed in opposite order and direction. Identify their full products by reversing the interval coordinate and reflecting one normal coordinate, $(t,v_1,v_2,v_3)\mapsto (1-t,-v_1,v_2,v_3)$; the two sign reversals preserve the four-dimensional orientation. This merely reverses the parametrization of the surviving one-handle and exchanges its two parametrized feet. Exchange the foot balls in the root boundary patch by its product isotopy. No handle slide of a handle over itself is required. Every other foot on the absorbed ball is transported to the root along e in the first absorption and along f in the second. Compare these paths before removing the tubes: their difference is one passage through the surviving tube f followed by the reverse e-tube. Move that foot disk along this passage with its full normal D³ coordinates. By the defining core-disk push [F9], this is exactly a slide of that incident one-handle over the surviving one-handle, with the sign determined by the passage orientation. Move the finitely many foot disks one at a time in disjoint thin parallel strips; feet already on the root need no move. Carry higher attaching embeddings by the same supported product maps, rather than leaving them fixed. This also checks the marking: on the root patch the comparison is the chosen foot exchange, on the two tube products it is the displayed coordinate reversal and these disk pushes, and on each remaining incident product it is the specified foot transport. Choose the seam roundings by these same collar isotopies. Thus it is the actual comparison of the two absorption charts, up to the retained product isotopies, not just an equality of their core classes. Restoring the absorbed branches carries these finitely many slides and isotopies back to the original graph. Hence a tree change gives only one-handle slides and framed isotopies, with its actual lower-boundary identification. At a 0/1 birth the new vertex has its cancelling edge. Choose T to contain that edge; the leaf-ball product absorbs the born pair and leaves the assigned presentation unchanged. At a death first choose T to contain the dying edge. If other feet lie on its ball, the same product absorption carries them across the joining tube; in another tree convention this is precisely the finite one-handle-slide comparison just described. Thus the 0/1 row produces only the already controlled one-handle-slide/isotopy changes. Apply the reversed argument to a 3/4 birth or death. It also proves independence of the outgoing tree choice before the dual trades.

Reversal of a two-handle slide. We need the local duality calculation, rather than an assumption that a reverse movie has the same diagram. A four-dimensional two-handle is D²_x×D²_y. In the forward presentation x is its core disk and y its cocore; in the reversed presentation their roles are interchanged. For a slide of the first core disk over the second along a band, its new core is the boundary band sum of the first core with a parallel of the second. Choose the normal disk coordinates on that band. The dual cocore disks are then the unchanged first cocore and the second cocore band-summed with the negative first cocore along the transverse rectangle. This is an actual disk construction: cut the two handle products along their cocores, follow the two opposite faces of the core band, and join their transverse disk patches by the intervening normal rectangle. On the first handle product the two parallel patches cancel across a product rectangle; on the second product the remaining patch meets its core once. No other handle is crossed after the band and normal radii are decreased. The boundary of the transverse rectangle is a genuine embedded band in the reversed attaching three-level, and its normal strip is the exchanged disk factor. Thus a forward two-handle slide becomes the opposite dual two-handle slide in reverse, together with the displayed product isotopies. Framings are transported by these disk-factor coordinates, including the sign from the reversed orientation. This comparison is the restriction of the same core-disk-push diffeomorphism, read in the other direction. It therefore retains both external face markings when those markings are transported by the original elementary move. It does not assert that an arbitrary diffeomorphism of a two-handlebody is generated by such slides. It is only the reverse of one already supplied local slide.

Flattening choices and moves based at the opposite end. The finite geometric curve/core lemma in Decorated ribbon graphs lift through a marked relative surgery calculus, in its exterior-representative and framed core-crossing parts, is independent of its global completeness hypothesis. For an ordinary framed circle its statement is: in a three-manifold filled along an earlier link, avoid the finitely many filling cores and radially move a sufficiently small tubular thickening into the old exterior; two such choices connected by an isotopy are related by finite exterior framed isotopies and slides over the specified surgery parallels. A transverse crossing of one core gives exactly that slide, with its strip framing. Consequently successive two-handle attachments can be flattened into the original one-handle-stage boundary, and every change of flattening choice produces O2 moves over earlier two-handles. Their terminal maps are the original isotopy maps in the filled manifold. This is a local geometric input, not an appeal to global surgery completeness. An O3 operation obtained in the reverse presentation also has a forward description. Before any cancellation it is an interior framed circle surgery in the actual four-dimensional level. Choose the forward one-handle trade circles disjoint from this circle and its small tube, perturbing one-dimensional cores in dimension four and retaining the protected boundary. These surgeries commute because their displayed product neighborhoods are disjoint. After the forward trades the four-dimensional level is a two-handle trace relative to N. The already proved π₁-surjection and four-dimensional circle-isotopy construction put the remaining framed surgery circle into N, with its rank-three framing induced by a rank-two framing there. The two-hemisphere bridge records this actual surgery by K_f and its small zero-framed meridian, hence by O3 in N. The circle is chosen off the old surgery link and its tube small enough to miss that link. All further flattening choices are controlled by the preceding local curve/core lemma. Thus an opposite-end O3 is converted to a forward O3 with finite O2/isotopy comparisons; it is not treated as an unspecified diffeomorphism of the outgoing three-manifold. The same argument applies when other disjoint trade circles are present: choose pairwise disjoint cores and normal tubes, use their product commutation, then flatten one at a time. For a finite movie, these choices are made smoothly on each closed ordinary interval. At an isolated event compare its two endpoint choices by the explicit arc-choice or flattening lemma. No infinite choice limit or implicit naturality is used.

The remaining event rows. A same-index index1 slide becomes the O2 dual-meridian slide of One-handle slides become Kirby slides after interior trading; noncompatible replacement arcs are corrected by the finite arc-choice comparison. An index3 slide is that same calculation in the reversed presentation, followed by step 3.1 for its two-handle slides and step 4.1 for any arc-choice O3. An index2 slide is unchanged after the adjacent trades: choose the trade cores in the low forward one-handle stage and the high reversed one-handle stage, off its core-band support. When a base arc is moved in making this choice, use the arc-choice comparison. Carry all other attaching strips by the same local slide map. After flattening, it is O2 by step 4.1. Critical-value interchanges of disjoint equal-index attachments are their commuting product gluings and produce no additional move. For a 1/2 birth use the actual local cancelling chart, transporting all later attaching data by its cancellation map. An older two-handle may traverse the born one-handle through an allowed positive-index-drop trajectory; it is not declared disjoint merely by function genericity. Before the local conversion, slide each such traversal off the new cancelling two-handle using the very product bands used for the death calculation. There are finitely many traversals after transverse position. Each band removes one signed traversal and carries its full strip. This produces the birth-normalized attaching data, with the actual composition of slide maps followed by the cancellation map. Use the inverse slides on the other side if retaining the original convention. These finite O2 preparations are part of the birth row. Choose the other trade circles away from this small birth support. Such a choice exists: in the index-grouped presentation the other one-handle cores lie on their own lower or dual-upper stages, while an auxiliary arc can be moved off a ball by replacing its finitely many crossing subarcs with boundary-collar arcs; small distinct parallels preserve its embedding. Changing from any old choice is controlled by the arc-choice lemma. The distinguished forward one-handle replacement is then exactly the born K_f,U_0 pair of the continuous-surgery construction. At a death first slide every other traversing two-handle off the cancelling two-handle using the explicit product bands. Thus the row is O3, O2 and isotopy with the retained cancellation map. A 2/3 birth or death is the identical calculation in the reversed presentation, and step 4.1 converts its O3 to the forward base while step 3.1 converts its slides. A 0/1 or3/4 birth/death was handled by step 2.1. Positive-index-drop trajectory tangencies add no row. The supplied elementary-band theorem says their attaching embeddings and terminal maps remain smooth. After index grouping and the product trades they are therefore ordinary intervals, with auxiliary arc/core crossings already translated by the finite local geometric lemmas. This statement does not assume those tangencies are absent.

Assembly and the marked conclusion. The field movie has finitely many events and compact regular intervals. Choose the finite trees, arcs, thickenings and flattenings on each interval. Subdivide further only for the finite transverse auxiliary crossings. At its ends use the preceding comparisons to identify the choices. Every listed change becomes a finite O2/O3/isotopy sequence, with the actual product boundary maps. At the two endpoints the input presentations have only two-handles, so both tree collapses and every one/three-handle trade are empty; choose the flattening to be their given original links. The resulting sequence therefore connects the original links, rather than merely some unspecified links representing diffeomorphic manifolds. Its cumulative outgoing map is the given comparison induced by the marked input movie, because every event comparison was the restriction of its explicit product cancellation, slide or interior-surgery trace and the interval maps were the retained regular flows. All supports avoid the vertical boundary collar. This proves the stated marked comparison from the supplied finite movie. At no point has same-trace two-equivalence been assumed; every changed intermediate level was modified by an explicit interior surgery whose external boundary map was retained.