Marked relative surgery presentations and complete moves for three-dimensional ribbon bordisms

Marked surgery existence in the fixed base. On one connected component apply [F1] to the actual $N$ and $M$ with their given boundary parametrizations. Its marked closed-cap filling, interior one/three-handle trades and full framed-link consolidation give a four-dimensional two-handle trace $W$ from $N$ and a finite link $L\subset \mathrm{int}N$ with the specified outgoing identification $N_L\to M$. This does not identify three-dimensional surface handle attachments with four-dimensional link handles: the dimension-changing bridge is the actual four-bordism and disk-factor construction inside [F1]. The standard-complement base in the Statement exists by placing the standard handlebodies in disjoint balls of $S^3$ and keeping their explicit boundary parametrizations; its complement is connected. For a closed component use $S^3$. Pull the actual ribbon graph back by the given output identification; the independent exterior-representative part of [F6] moves its finite framed spine off the filling cores and radially out of the filling tori, with its endpoint ribbons and coupons retained. This gives the stated decorated presentation and all its markings.

The common actual trace with the prescribed output map. First disregard the graph, and let $L_0,L_1$ be two link presentations with the specified comparison of their output three-manifolds fixed pointwise on the external collar. Form their four-dimensional two-handle traces $W_0,W_1$ from the same $N$. Their entire three-boundary is $Y=(-N)\cup (\mathrm{\partial }N\times I)\cup M$, with the given output comparison used on its $M$ face and the identity on the incoming $N$ face. Thus $Y$ is an actual marked common boundary, not merely a common diffeomorphism type. By [F2], first add split signed unit-framed unknots as needed for the signature correction and then finite $O3$ pairs from the two ends. The resulting links describe the same actual middle trace $W_{\ast }$ with this entire marked boundary. Every move has its boundary-relative local identification, so the original prescribed output map is retained in the two descriptions.

A finite based movie on that trace. Construct the endpoint data from each supplied two-handle presentation of $W_{\ast }$ as follows. Use [F3]'s common simultaneous attached-stage model, with its disjoint actual framed attaching tubes. On each tube use its quadratic two-handle band $Q(x,y)=-\mathrm{\mid }x{\mathrm{\mid }}^2+\mathrm{\mid }y{\mathrm{\mid }}^2$, normalized to have the same critical height $c$; glue their side product coordinates to the unchanged regular complement. This is the explicit parallel version of the local band construction in [F3]: its angular coordinates and $y$ disk factor give exactly the prescribed attaching tube and its full framing. All these local bands lie off the protected vertical collar, where the complement keeps the given height. The resulting Morse function $H$ has only index2 critical points, all at $c$. Choose its descending field $X$ standard near every quadratic origin, using the displayed quadratic flow and its regular side-height field, and patch them in the regular complement by positive descending combinations. Thus $dH(X)<0$ away from the critical points and the prescribed vertical product field is retained. There is no $2\to 2$ trajectory: a nonconstant descending orbit strictly lowers $H$, while both endpoint critical values would be $c$. Choose disjoint bumps ${\rho }_j$ supported in the critical blocks, equal to one near their origins, and put $F=H+{\sum }_j{\delta }_j{\rho }_j$ with distinct sufficiently small constants ${\delta }_j$. On each transition annulus the compact support has a positive minimum of $-dH(X)$ and a finite bound for $\mathrm{\mid }d{\rho }_j(X)\mathrm{\mid }$; choose the constants so ${\sum }_j\mathrm{\mid }{\delta }_{j}d{\rho }_j(X)\mathrm{\mid }<-dH(X)\mathrm{/}2$ there. Near each critical origin the bump is constant, so its Hessian and the field model are unchanged. Hence $X$ remains descending for the excellent endpoint function $F$, with the SAME trajectories as for $H$, and still has no same-index orbit. There is no index-increasing orbit because there are no critical points of another index. With no handles use the regular product data. The actual band gluing and unchanged flow carry the endpoint framed attaching data and outgoing map from the given presentation; changing the local critical values inserts only its recorded regular-level comparisons. Extend these endpoint pairs constantly on small parameter-end neighborhoods. Use the fixed corner/vertical face convention of [F3]. The finite generic function path and the actual weak descending-path supplier give finitely many function events and finitely many isolated equal-index trajectory events, with exact local birth models and the complete elementary outgoing maps. Apply [F4] to order index heights while keeping that field and the vertical height. Its finite height-jet construction ensures that all new same-index value crossings are finite, simple and separated from the trajectory events. Its based-movie bridge transports the full unstable thickenings to the common cuts in uniformly finite time, isolates the unique equal-index passage, uses the reversed index1 calculation for index3, and includes the necessary slides of older attachments through a cancelling birth block. All later attaching embeddings and terminal maps are carried by those actual comparisons. Hence the precise finite based mixed-handle movie required by [F5] has been proved, rather than inferred from function genericity.

Marked undecorated completeness. Apply [F5] to the movie in step 3.1. Rooted-tree collapse, dual three-handle trades, forward one-handle trades and every change of arc or flattening produce the stated finite $O2\mathrm{/}O3$ moves, retaining their actual cumulative output map. Its four-dimensional levels may change by interior framed circle surgeries; no assertion that arbitrary diffeomorphic two-handlebodies are two-equivalent is being used. Compose with the $O1\mathrm{/}O3$ sequences from step 2.1, reversing the sequence from the second end. The resulting finite $O1\mathrm{/}O2\mathrm{/}O3$ and isotopy sequence begins at $L_0$, ends at $L_1$, and realizes exactly the specified output identification up to isotopy fixed on the external collar. This proves the marked undecorated calculus required in [F6], for arbitrary compact connected oriented $N$ and arbitrary fundamental-group classes.

Decorations and actual coupon data. Invoke the decorated lifting conclusion of [F6] only after step 4.1 has discharged its hypothesis. At each canonical local $O1\mathrm{/}O2\mathrm{/}O3$ map it gives a finite exterior graph preparation and carries the resulting graph through that very map. The final transported graph is isotopic in the filled output to the prescribed final graph, since the cumulative output identification is the specified one. Its independent graph/core crossing argument converts that isotopy to finite exterior colored framed graph isotopies and slides over the actual surgery longitudes. All original coupons remain the same morphisms, with their attachment order, orientations, colors and complete framings; all collar ribbons and endpoints are fixed. This proves the full decorated statement. No scalar evaluation or variable-coupon naturality is substituted for graph-move completeness.

Components and external weighted data. Compactness gives finitely many connected components. Perform step 1.1, step 2.1, step 3.1, step 4.1, step 5.1 separately on the fixed matching component partition and take their finite disjoint union. Surgery and all the local moves preserve this partition. An empty bordism uses the empty link and empty move sequence; an empty incoming or outgoing surface merely removes that boundary face, while a closed component still has its nonempty fixed base $S^3$. The collars used throughout are noncritical product heights, so no boundary critical-point event or reparametrization of a fixed cap has been introduced. Every external boundary handlebody, its parametrization and its marked cap ribbons remain the chosen ones. Any integer weight and object Lagrangians are carried unchanged because the theorem compares geometric presentations of the same weighted morphism; the anomaly and evaluation identities are separate consumer results. This proves every stated case.