Decorated ribbon graphs lift through a marked relative surgery calculus

For each filling torus choose oriented coordinates $V_i=D_z^2\times S_{\theta }^1$, with dual core $c_i=\{0\}\times S^1$ and filling meridian ${\mu }_i=\mathrm{\partial }{D}^2\times \{{\theta }_0\}$. Its boundary gluing sends ${\mu }_i$ to the specified framed parallel ${\lambda }_i$, with the sign induced by the fixed orientation convention. Extend its radial coordinate across the boundary into a disjoint exterior collar, so $0\le r\le 1+ϵ$, $V_i=\{r\le 1\}$. Those neighborhoods are disjoint from $C$. A finite ribbon graph has a finite fatgraph spine: each coupon retracts to its labeled vertex with ordered short incident arcs, each band to its core edge, and a coupon-free annulus to a circle edge. Shrinking coupon rectangles and band widths gives a genuine ambient isotopy to any sufficiently narrow thickening of that spine, not a collapse to zero width. In a band chart this is a smooth coordinate contraction of its width variable, extended by a cutoff to a surrounding box; in a coupon chart contract its disk and join the moved bases by short parallel strips, keeping their order. These finitely many charts glue on their product collars; on $C$ no width or base is changed. The ribbon's orientation and transverse width vector keep its full framing during this isotopy.

Perturb the compact interior spine slightly so all its vertices and edges avoid every $c_i$. A vertex is zero-dimensional and a core one-dimensional in dimension three, and an edge has dimension one, so their incidence has negative expected dimension. This can be obtained by [F4] with chart cutoffs away from the endpoint/coupon collars. Perturbations sufficiently small preserve the finite embedded graph, attachment order and framing; then narrow its thickening so it also misses the cores. If its smallest radial coordinate in a torus is $\delta >0$, choose a smooth increasing function $\varphi :[0,1+ϵ]\to [0,1+ϵ]$ that is linear near0, the identity near $1+ϵ$, and has $\varphi (\delta )>1+ϵ\mathrm{/}2$. Such a function is obtained by integrating a positive smooth derivative with most of its mass before $\delta$, then normalizing its total integral; the derivative can be constant near0 and1 near the outer endpoint. The interpolation ${\varphi }_s=(1-s)r+s\varphi (r)$ has positive derivative. The maps $(z,\theta )\mapsto ({\varphi }_s(\mathrm{\mid }z\mathrm{\mid })z\mathrm{/}\mathrm{\mid }z\mathrm{\mid },\theta )$ are therefore smooth ambient isotopies, including at the core by the linear behavior. They fix the outside of the extended torus and push every graph point in it to $r>1$. Applying these finitely many disjoint radial isotopies puts the whole graph in the exterior, proving part (1).

Now take a framed graph isotopy in the filled manifold between two exterior representatives. Work on the transported abstract spine, retaining its ordered labeled vertices and all fixed collar endpoints. On each edge, its track is a smooth map from an interval times the isotopy interval to $Y_L$; the inverse image of the codimension-two core set has dimension zero after a small relative perturbation. Vertex tracks have dimension one and can avoid the cores entirely. For precision, take finitely many normal-coordinate charts near the cores and finite source charts on the compact edge tracks, with cutoff perturbations of their two normal coordinates. At each proposed incidence these parameters span the normal directions; [F4] makes the tracks transverse. Independent perturbations at two distinct edge parameters exclude simultaneous double incidences: two core conditions impose four equations on the two edge parameters and their common time, a three-dimensional domain. A further jet condition excludes a nontransverse individual incidence. No incidence is allowed at the fixed time endpoints, graph vertices or boundary collars, all of which already have disjoint neighborhoods. The small perturbation remains a family of spatial embeddings: derivatives on each edge remain nonzero, and compact sets of separated source pairs remain separated; the fixed local vertex charts preserve attachment order. Frame vectors extend by the same nearby normal charts. Thus it is an isotopy, not a generic homotopy allowing graph crossings.

The core incidences are a closed zero-dimensional transverse subset of finitely many compact track strips, with none near their boundaries or vertices, hence finite. Separate their times. Between small disjoint event intervals the spine stays a positive distance from the cores, so choose the radial expansion of step 2.1 uniformly on each closed time interval and then a sufficiently narrow ribbon thickening. This gives a genuine exterior graph isotopy on that interval. Different expansions at an interval endpoint give exterior-isotopic graphs: interpolate their positive radial functions after first choosing a common lower clearance; their convex interpolation still puts every point at $r>1$. The initial and terminal representatives can be kept exactly as given by choosing the outer radial collars disjoint from their compact graphs and taking the time cutoff to vanish there. Therefore it remains only to compute the local change at each of the finitely many spine/core events.

At one event write the original time offset as ${\tau }_0=t-t_0$. Transversality says the derivative of its two normal coordinates with respect to $(u_0,{\tau }_0)$ is invertible. Rotate the target normal plane so the edge derivative is $(a,0)$, $a>0$, and write the time derivative $(b,c)$ with $c\mathrm{\ne }0$. Make only the fiber-preserving linear changes $u=a{u}_0+b{\tau }_0$ and $\tau =c{\tau }_0$. Thus each new time slice is one original time slice, possibly with a constant reversal of its order, and the normal map is $z=(u,\tau )+R(u,\tau )$ with $R=O((\mathrm{\mid }u\mathrm{\mid }+\mathrm{\mid }\tau \mathrm{\mid }{)}^2)$ and $DR=O(\mathrm{\mid }u\mathrm{\mid }+\mathrm{\mid }\tau \mathrm{\mid })$. In a sufficiently small source disk choose a cutoff $\chi$ equal to1 on its central half and zero near its boundary, and interpolate to $z_s=(u,\tau )+(1-s\chi )R$ while leaving the original time parameter fixed. On a disk of radius $\rho$, $\mathrm{\mid }{z}_s\mathrm{\mid }\ge r-C{r}^2\ge r\mathrm{/}2$ for $r=\sqrt{u^2+{\tau }^2}>0$ after choosing $\rho$ small; its derivative is $I+O(\rho )$, including the cutoff derivative because $D\chi =O({\rho }^{-1})$ and $R=O({\rho }^2)$. Hence no extra core zero or degenerate zero is introduced, and every fixed-time arc remains embedded. The perturbation is supported away from the local arc's endpoints, all other graph pieces and coupon/boundary collars; its smallness preserves the spatial embedding family and framing. The resulting normal map is exactly $z=u+i\tau$ on the central event region, with unchanged nonzero connecting arcs outside it. Its longitudinal coordinate is a smooth function of those normal coordinates; subtract that function in a static small target core chart to obtain $\theta ={\theta }_0$. This target coordinate change mixes no source time with the edge parameter. Thus the before/after local arcs are the upper/lower detours $z=u\pm i\eta$ in the punctured meridional disk. Multiply the displacement by a source cutoff equal to1 near the event and zero near the ends of the local arc; equivalently round the two short end connectors in the core-free outer collar. This makes their endpoints exactly the same, joined to the identical unchanged exterior arcs. After radial expansion, the two detours differ by one circuit of $\mathrm{\partial }{D}^2\times \{{\theta }_0\}$, with sign determined by the transverse incidence. To verify this without a drawing, choose a continuous argument on each detour: its change from negative to positive $u$ is approximately $-\pi$ on the upper arc and $+\pi$ on the lower, so the difference of the two relative homotopy classes is one signed $2\pi$ meridian. The region between either detour and the corresponding boundary arcs is an embedded disk after a small collar rounding, giving an actual relative arc isotopy, not only a homotopy classification. Gluing by the given torus map converts that based meridian circuit to a band sum with $\pm {\lambda }_i$, precisely the framed open-strand slide over the original surgery component.

This local calculation lifts to the whole framed ribbon. Choose a trivialization of its oriented normal plane over the event track disk, which is contractible; the original framing is an actual nonzero vector field there and extends in that trivialization. Take the small product strip thickening of the before/after detours and of the connecting meridional disk. The filling boundary gluing carries that strip along the specified surgery framing on ${\lambda }_i$. Radial expansion transports the same vectors by its differential. Hence the resulting band slide has exactly the transported normal framing; no Reidemeister-I curl is dropped. Any preexisting rotation of the strip is retained on both sides. Both signed incidences and both strand orientations are handled directly; if one chooses instead to reverse a label its dual and its actual bend maps are used, rather than identifying the two oriented band moves without their twists. Coupon neighborhoods have core-free vertex tracks, so the finite event balls can be chosen disjoint from all coupons and boundary collars. After each event choose a positive small width for the actual ribbons at its two exterior slices; a zero-width ribbon at the event time is neither required nor used. Outside the event ball the original product strips and coupons are unchanged. Combining all finite events and regular intervals proves part (2).

Every colored operation in part (2) keeps the original coupon morphisms as actual fixed maps. For a simple local strand $j$, its band slide is [F2]'s identity on the full $j^{\ast }\otimes j$ factor after bending, with its fixed exterior pre/post-composed. For a general color $X$, insert the actual finite resolution $1_X={\sum }_{j,a}{u}_{j,a}{v}^{j,a}$ on a short segment disjoint from coupons; absorb those two actual morphisms into the adjacent fixed exterior portions, apply the fixed-$j$ identity, then sum. The inverse dual-mate snakes unbend the equality to the original open graph morphism before closure. Only the surgery-component variable is dinatural; no arbitrary family of graph coupons depending on variable $X$ is introduced. Exterior colored isotopies preserve evaluation by [F1]. Thus the extra moves used to compare two exterior representatives have the full arbitrary-coupon algebraic invariance required in the Statement.

We next show that the canonical undecorated local moves admit the needed graph lifts. For an $O2$ band slide, put the graph spine transverse to the attaching rectangle and move its finitely many coupon vertices off it. There are finitely many interior intersections, since its short attaching edges lie on the surgery tori and the exterior graph misses those tori. Each of the rectangle's two long edges is free: it is not part of the surgery link. For one graph intersection choose a simple path in the rectangle to the interior of a free long edge, avoiding the other intersection points and both attaching edges. A thin strip about this path is a disk whose neighborhood is a ball disjoint from the surgery link, all coupon vertices, and all graph portions except the intersecting short arc. Push that arc's transverse crossing along the strip and around the free edge. In disk-product coordinates the move replaces a short normal interval through the strip by a U-shaped detour outside its boundary; the swept disk has interior disjoint from the rest of the graph and the link. The cutoff flow in this ball realizes an actual framed graph isotopy, with the normal framing transported. The new arc misses the rectangle, and the strip's choice prevents new intersections. Repeat finitely many times. This preparation is entirely an exterior graph isotopy; no unjustified naturality or disk bounding an arbitrary longitude is needed. After narrowing the ribbons, the actual graph misses a neighborhood of the band and both surgery tori. The standard canonical $O2$ filling identification is the identity outside that neighborhood, so the undecorated surgery-component slide now carries this graph unchanged in exterior coordinates. Both directions use the same construction with their respective rectangles.

For an $O1$ insertion, its ball and a slightly larger ball miss the old surgery link. The graph can be moved out of the support ball by an exterior ambient isotopy: choose a point there off its thin spine, shrink the support ball into a tiny ball about that point using a smooth radial contraction extended to the slightly larger ball, and apply the inverse contraction to the graph. The tiny ball is disjoint from the graph, so its inverse image is disjoint from the original support ball. This fixes collars and the old link. Now introduce the separated $\pm 1$ unknot with graph unchanged; the canonical filling map fixes its exterior. For deletion, use the canonical filled-ball identification to perform this graph-avoidance isotopy in that ball first, then translate it to the current surgery exterior by parts (1)–(2). The graph is then disjoint from the deletion support. If one uses the equivalent local signed twist-surgery description with strands passing through its disk, its complete colored lift is also [F3]'s arbitrary-coupon identity; no additional graph move is required.

For an $O3$ insertion, perturb its introduced framed knot $K$ slightly off the finite graph spine, keeping it disjoint from the old surgery link and all boundary collars. This is an isotopy of the prescribed local move data, with its framing transported, so its canonical marked identification changes only by the corresponding rel-collar isotopy. Narrow the graph and choose the support tube $N(K)$ disjoint from it, then introduce its small zero-framed meridian there. The canonical circumcision filling identification is the identity outside that tube, so the graph is unchanged. For deletion, carry the current graph to the output of that canonical identification. A spine can first be made disjoint from $K$ and then pushed outside $N(K)$ by the explicit core-avoidance/radial construction of step 1.1, step 2.1, now applied to that ordinary tube. Pull that isotopy back through the specified identification and translate it to the current surgery exterior by parts (1)–(2). The deletion support is then graph-free. Its evaluation invariance is [F3]'s complete circumcision identity, with the preliminary graph changes covered by [F2]. No nullhomotopy of $K$ or assumption that the ambient manifold is simply connected is used.

Invoke the marked undecorated hypothesis only now. Given a marked bordism, take its supplied undecorated surgery presentation and pull the actual decoration back along its specified output identification. step 2.1 puts it in the exterior, preserving every terminal parametrization/collar ribbon. Given two marked decorated presentations of the same decoration, take the supplied finite undecorated sequence realizing their specified output identification up to rel-collar isotopy. Lift each exterior link isotopy by its ambient isotopy and each $O1\mathrm{/}O2\mathrm{/}O3$ by step 6.2, step 7.1, step 7.2, inserting their finite preparatory graph moves as necessary. Every lifted step represents the graph transported by that very canonical move map, up to a rel-collar graph isotopy. This is why the marked realization hypothesis, not bare existence of an unspecified diffeomorphism, is the exact global input needed here.

After that finite lifted sequence, the graph in the last exterior is isotopic in the filled output to the prescribed final graph: their output markings agree up to the given rel-collar isotopy, and they represent the same marked decoration. Part (2) converts precisely this last graph isotopy into the allowed finite exterior isotopies and open-strand slides. All coupon labels, endpoint intervals, attachment orders, band orientations and framings were carried throughout; no new boundary intersection was introduced because every perturbation/expansion was supported outside $C$. The argument is componentwise over finitely many manifold components and allows closed graph components, zero-valence coupons and empty boundary components. The total extra graph-move sequence preserves evaluation by step 6.1, and the local link-move algebra is [F2]–[F3]. This proves the full stated decorated lifting implication, conditional only on the actual marked undecorated calculus in part (3), without claiming that global calculus or any colored Cerf theorem has thereby been proved.