Continuous one-handle surgery gives relative Kirby moves

Exact movie input and the retained marking. Let N be a compact oriented smooth three-manifold with a protected boundary collar. Consider a four-dimensional relative handle presentation based on N×I. A distinguished one-handle h=I×D³ attaches at two disjoint parametrized balls in int(N). Other one-handles, if present, have disjoint feet. All two-handle attaching embeddings and their framings are part of the presentation. A finite movie is given on a compact parameter interval, with these conditions: Away from finitely many events the feet and framed attaching curves move by smooth isotopy. No one-handle slide involving h occurs during the interval under consideration; slides among other one-handles are allowed. The distinguished h persists from an isolated cancelling 1/2 birth to an isolated cancelling 1/2 death. A small birth/death column is disjoint from every other handle attachment. Other births, deaths and slides have already supplied local models, and their supports miss the feet of h. Between those endpoints every event involving two-handles is a framed two-handle slide with its actual band, or a birth/death supported away from h. If further one-handles are present, they remain part of the background handle presentation; the replacement arc in N is chosen off their feet. This is possible because finitely many disjoint interior foot balls do not separate a connected three-dimensional component between the two distinguished feet. The full outgoing boundary comparison with a fixed target is recorded at every isotopy, attachment and cancellation. All motions and all supports avoid the protected collar. Choose an embedded framed arc B in N joining the feet. It misses every other attaching curve at the initial parameter. Let A′ be a parallel of the core of h. The union A′∪B, rounded at its two ends, is a framed interior circle γ in the four-dimensional stage. Carry B by a compactly supported ambient isotopy of N extending the motion of the feet. Such an extension follows directly by extending the velocities on the two moving closed balls and integrating a supported field. It fixes the boundary collar. Its values on B supply a smooth family B_t; it need not keep B_t disjoint from the two-handle curves for every t. We prove that replacing h by its framed circle surgery along γ_t gives a finite relative surgery movie whose new elementary changes are isotopy, two-handle slides O2, and O3 at the distinguished birth/death; other one-handles and their moves remain in the background. It is a framed-link movie once no such handles remain. Every outgoing boundary comparison is transported in this conversion. The replacement changes the four-manifold in its interior and is not asserted to preserve that four-manifold.

The actual link after the replacement. Take disjoint small foot balls and a thin product tube about B. Together they form a three-ball: explicitly straighten the arc in its own product chart I×D², join the two end half-balls, and round the two seams. The union of this ball's collar with h is S¹×D³, with the core circle γ. Perform interior circle surgery there. The disk-factor construction in Interior one-handle trades and the fully framed circle-surgery pair gives two two-handles C_B and U_B before the subsequent cancellation. The first C_B goes once through h along A′ and returns along B, with its supplied framing. The second U_B is its small disk-framed meridian. This is the two-hemisphere decomposition of D²×S², with the second disk factors interchanged. The attaching circle C_B meets the belt sphere of h in one transverse point. The product normal coordinates in the construction identify its attaching strip with one foot-to-foot traversal of h. Therefore the actual one/two-handle cancellation supplier applies, including its outgoing boundary diffeomorphism. Cancel h with C_B. In this cancellation every arc of another attaching curve that went through h is carried to a parallel of B in the chosen tube, with its normal strip carried by the product cancellation map. To verify that description, cut h at its cocore D³: its boundary becomes the two foot spheres with a common equatorial seam. The cancellation identifies the two sides by the core disk of C_B; along its remaining boundary that disk is exactly B. Thus the new boundary chart joins the two foot patches along B. Each disjoint parallel strip is joined by this same chart; its ordering and framing are those of the chosen tube. No new twist may be inserted independently. The remnant U_B is the boundary of a small cocore disk transverse to this routed bundle. It is an unknot with its displayed disk framing zero. It encircles the routed strands once, with their actual signs and multiplicities. Call the resulting link R_B(L)∪U_B, where R_B means this explicit routing of every old two-handle strip through the B-tube. In particular its integer framings are obtained by carrying entire strips, rather than by retaining arbitrary numerical labels after a drawing change. The complement of the ball/tube and the protected boundary collar are unchanged. The cancellation map and the original circle-surgery boundary map determine its outgoing identification.

A transverse crossing of an attaching curve with the replacement arc. The only new obstruction to a continuous routed-link isotopy is an intersection between an old attaching curve and B_t. Feet and ends are protected. A finite relative perturbation of B_t and the curves makes all such intersections isolated and transverse, with no intersections at an old event parameter. Here the map from (t,u,v), where u and v are the two arc parameters, to the two equal points in N has source dimension three and the diagonal has codimension three. To obtain the required submersive perturbation, choose a small product box at each possible intersection and add three independent translations of the interior of the B-arc with a bump equal to one in the smaller box. The evaluation difference has these three translations as a basis. Cover the compact parameter/arc track away from endpoints by finitely many boxes, apply the actual finite-dimensional parametric-transversality theorem to their summed perturbation, and additionally avoid simultaneous intersections using the two-point family with independently supported translations. Its two-diagonal codimension is six on a five-dimensional source (one time and four arc parameters). Near the parameter/arc diagonals, embeddedness and the nonzero tangents give one local intersection branch; distinct branches cannot accumulate there. Off these diagonal neighborhoods, the relevant source is compact. Thus transversality gives finitely many individual crossings. The other protected compact tracks remain disjoint after the perturbation is sufficiently small. Near one crossing choose oriented coordinates (x,y,z) in which the replacement arc is {(x,0,0)} and the old attaching strand, with its parameter, is {(0,y,t)}. This follows by the inverse function theorem applied to the three independent tangent vectors of the transverse intersection, preserving t by straightening the spatial arc at each t and then rescaling the nonzero normal velocity. Only the signs of the x/y/z coordinates affect the slide sign. Choose the B-tube x∈[−a,a], y²+z²≤ε². The two routings just before and after t=0 agree outside this box. Inside, join their corresponding endpoints by following the routed strip across the B-tube. The difference is the boundary of a small normal disk to B, connected to the old strand by a thin rectangular band. This boundary is precisely a framed parallel of U_B, transported along the B-tube from its original cross-section; the intervening annulus is the boundary of the product sub-tube between the two cross-sections. Its framing is constant in those product coordinates. Hence one routing is the band sum of the other attaching component with the appropriate signed framed parallel of U_B. The band interior misses all other strands after its width is decreased. This is the geometric definition of O2, with its full normal strip. It also proves the framing assertion: the two old strips and the parallel strip are joined by the displayed rectangle, so the resulting framing is the band framing, including any linking contribution. An unchanged numerical framing is not asserted. This verification is local even when the replacement arc represents a nontrivial relative homotopy class. Neither B nor the union B∪C below is assumed to bound a disk in N.

Isotopies, other slides and simultaneous strips. Away from the crossings in step 3.1 the ball/tube construction and cancellation vary smoothly. After shrinking their radii uniformly on the compact parameter interval, their strip routings give a framed-link isotopy, extended by the ordinary compact-source velocity construction. Near each old two-handle slide include its band as one more compact two-dimensional strip in the routing. Choose its width and the B-tube so that the band is in product position at every traversal of h. Routing commutes with joining the two strips along this band: both operations take their union and then use the identical product cancellation chart. Thus an old two-handle slide becomes a two-handle slide of the routed components. If an end of the band must be carried across U_B, the rectangle of step 3.1 gives the additional O2; these are finite because the band is a finite smooth compact product strip in the specified event model and can first be put in transverse position at its edges. A birth/death in a support disjoint from B and the feet is unchanged. Several strands may pass through h. Use distinct normal positions in D² of the B-tube, in the order determined by their attaching embeddings. Every operation above acts on their disjoint thickened strips. At a crossing only the intersecting strip is changed; all other strips are fixed after reducing the crossing box. This supplies the framed, many-strand version and does not replace an embedded strip by an unframed homology class.

Slides among other one-handles. Subdivide at each such supplied slide. Its moving foot and core strip avoid h, since h does not participate. The compact base portions of that strip's core path can be perturbed slightly off B and its two foot balls: two curves have negative expected intersection dimension in the three-dimensional base, using the relative chart translations of [F5]. Such a small perturbation is an isotopy of the supplied framed strip and retains its outgoing comparison by [F6]. Narrow its product neighborhood and the B-tube so they are disjoint. The slide comparison is then the identity on the entire ball/tube used in step 2.1. Its map and the distinguished surgery/cancellation therefore commute literally on their disjoint product supports. Transport every old two-handle strip by both maps; the two orders give the same framed embedding and outgoing map. Any preliminary change of B needed to compare these narrowed choices contributes only the arc-crossing O2 moves of step 3.1. Thus arbitrary finitely many slides among the other handles are covered, rather than forbidden. They remain background one-handle slides; this lemma does not assert their replacement by link moves before those handles themselves are traded.

The distinguished birth and death are O3. At a cancelling 1/2 birth, take the supplied cancelling chart. Just after the birth, the new two-handle attaching circle consists of a core arc A of h and a return arc C in N. It passes once through h. Every older two-handle is disjoint from this chart and hence has no traversal of the new h. The replacement operation in step 2.1 routes the cancelling circle to the closed curve K=C∪B, with corners rounded and its actual induced integer framing. The only other new component is U_B. It is a small meridian of K, because K uses B exactly once in the product tube. Its framing is the meridian-disk framing zero. Other attaching components avoid the chosen tube after B is initially chosen disjoint from them. Thus the result differs from the pre-birth link by the O3 pair K_f,U_0, supported in a small tubular solid torus of K. The integer f is whatever framing the cancellation and circle-surgery charts supply; the actual O3 filling theorem allows every integer. In particular K need not be nullhomotopic. This is why an O3 move, rather than an unsupported assertion that a split Hopf link was added, is the correct relative conclusion. At the death, before cancelling h with its supplied cancelling two-handle C, remove every other two-handle traversal of h by sliding that component over C. For each traversal take the band following its foot patch into the corresponding patch of C, then once through h with opposite signed traversal. The two parallel traversals bound a product rectangle in I×D³, so after the slide they isotope off h. The normal strip is the band strip and hence has exactly the slide framing. Choose parallel bands at different normal positions and perform the finitely many slides sequentially. No other one-handle is crossed by these local bands. Afterward only C goes through h, and the inverse of the birth construction removes the corresponding O3 pair. Routing the preparatory slides is permitted by step 4.1. This proves the distinguished death assertion without requiring the transported B to be isotopic to the short cancelling arc C.