Vanishing Cycles, Novikov and Taut Foliations

A nonempty manifold with a regular codimension-one foliation has dimension at least one. If $\dim M=1$, its leaves are points. The coorientation and finitely many smooth chart bumps give a smooth nowhere-zero positive field $X$, whose flow is complete by [F4]. Each orbit is open because its orbit map has nonzero derivative, so connectedness makes $M$ one orbit. The orbit map $\mathbb{R}\to M$ is a surjective local diffeomorphism. If it were injective, it would be a homeomorphism, contradicting compactness of $M$. Thus its period subgroup is nontrivial; it is closed and misses a neighbourhood of zero by local injectivity, so it has a least positive element $T$. The induced map $\mathbb{R}\mathrm{/}T\mathbb{Z}\to M$ is an embedded positive circle onto $M$, meeting every point leaf. This proves the claim in dimension one. In the remaining steps assume $\dim M\ge 2$.

For a closed transversal $\gamma$ let $N_{\gamma }$ be the union of the leaves it meets; transversality is open and in a product chart a transversal meets all nearby plaques, so $N_{\gamma }$ is open and saturated, and the tautness hypothesis [F1] says that the family of all $N_{\gamma }$ covers $M$.

By compactness choose finitely many positive closed transversals ${\gamma }_1,\dots ,{\gamma }_k$ whose saturations cover $M$; reverse their orientations if necessary. This covering property survives sufficiently small $C^1$ perturbations of the finite curves. Indeed, for each $x\in M$ a finite plaque chain joins $x$ to a point of one curve. Restrict a positive parameter arc of that curve to a foliation box, and choose a smaller closed transverse-height interval strictly inside its height range. Transport this smaller interval along the chain using [F5]. Its saturation contains a neighbourhood of $x$. A sufficiently close perturbed arc still crosses every height of that smaller interval, by its positive derivative and the strict endpoint inequalities. Choose finitely many such neighbourhoods covering compact $M$ and take the minimum of their finitely many perturbation margins. Every perturbed family within that margin therefore still meets every leaf. The same argument applies to a finite family of positive immersed curves, because only regular parameter arcs were used.

We use the following finite approximation construction. A compact piecewise $C^2$ positive curve with positive one-sided derivatives can be rounded at its finitely many seams in foliation boxes: mollify the continuous chart curve; positivity of the transverse derivative persists because the mollifier is nonnegative, and a cutoff restores the old curve off the seam with derivative error tending to zero. A $C^2$ curve can then be approximated in $C^1$ by a smooth ambient curve: cover its parameter circle by finitely many smaller intervals mapping into smooth ambient charts, mollify the coordinate functions on each interval, blend back by a fixed smooth cutoff, and carry out these finitely many replacements. On the smaller intervals the replacement is smooth; subsequent replacements preserve smoothness where already obtained, and errors can be made smaller than any prescribed total $C^1$ margin. For a positive piecewise curve, the same covering margin is available even at a seam: its continuous transverse height is strictly increasing through the seam, so choose a smaller closed height interval strictly between the heights at the ends of an arc crossing it. Rounding and approximation preserve these strict endpoint inequalities. Thus positivity and the leaf-covering property of step 2.1 persist. For an initially embedded finite disjoint family, small $C^1$ errors also preserve that property: local injectivity follows from a nonzero coordinate derivative on finitely many parameter intervals, and pairs of parameters outside these intervals have images a positive distance apart in finitely many compact coordinate neighbourhoods.

Assume first $\dim M\ge 3$. The finite intersection graph of the open saturated sets $N_{{\gamma }_i}$ is connected: otherwise the unions belonging to two graph components would separate $M$. Two positive curves whose saturations meet have points in a common leaf, joined by a finite leafwise path. A finite plaque chain along that path supplies a foliated strip $H:[0,1]\times (-a,a)\to M$: for each $t$ the path $s\mapsto H(s,t)$ stays in one leaf, and the transverse derivative in $t$ is positive; its end transversals are short arcs of the two curves. To construct the strip, subdivide the path into finitely many convex plaque charts, interpolate its leafwise coordinate in each chart, transport the transverse coordinate by [F5], and smooth the leafwise seams inside their common plaques. This uses only finitely many compatible chart pieces; the strip need not be an embedding.

The general-position perturbations needed here follow from [F6] with finite parameters. For a smooth immersed circle, a fixed neighbourhood of the parameter diagonal contains no distinct coincident image pair, by the local injectivity and finite compact cover just used. On the remaining compact set of possible coincident pairs, finitely many smooth bumps supported in disjoint parameter intervals move either image independently in all ambient coordinate directions. They can be realized by local ambient coordinate translations on the curve with cutoff in its parameter. The resulting two-point evaluation is a submersion near its inverse image of the target diagonal; shrink the parameter ball so this remains true and no other coincidence enters. Apply [F6] there. The diagonal has codimension $\dim M$, so when $\dim M\ge 3$ a good slice has empty two-point coincidence set and is an embedding. For a finite family of embedded circles in a surface, the same independent parameters for distinct circles make all pair evaluations transverse to the diagonal and all three-point evaluations transverse to the small diagonal. Their respective source dimensions and target codimensions are $2,2$ and $3,4$. Thus pair intersections are isolated and, by compactness, finite, while triple intersections are absent. Each individual circle stays embedded. A finite union of null bad-parameter sets cannot fill any parameter ball, so these perturbations can be arbitrarily small and preserve the margins of step 2.1.

Delete from the two curves the short arcs parametrized at the ends of that strip by $-ϵ\le t\le ϵ$, where $0<ϵ<a$. Join the first lower endpoint to the second upper endpoint by $H(s,t(s))$, and the second lower endpoint to the first upper endpoint by $H(1-s,t(s))$, with $t$ strictly increasing from $-ϵ$ to $ϵ$. These joins are positive. Following the undeleted part of each circle and these two joins gives one piecewise positive immersed circle. Every removed plaque is still met, because each join crosses the whole transverse interval and its height-$t$ point lies in the leaf of each end plaque of height $t$. Consequently its saturation contains the saturations of both old curves. Before rounding, the merged circle and the unchanged remaining curves still cover all leaves. Round with the finite covering margin of steps 2.1 and 3.1; the entire remaining family therefore still covers all leaves. Its intersection graph of open saturations is again connected by the argument of step 3.2. Choose an overlapping pair again and repeat. Each merge reduces the finite family size by one, so eventually a single positive immersed circle meets every leaf. Steps 2.1, 3.1 and 4.1 smooth it and perturb it to an embedded positive circle while retaining this covering property. This proves the result in dimensions at least three.

Now let $\dim M=2$. Apply the surface perturbation of step 4.1 to the finite family from step 2.1. Around each of its finitely many crossings choose pairwise disjoint foliation rectangles containing exactly the two crossing arcs. With plaques $y=\text{constant}$, these arcs are graphs $x_1(y),x_2(y)$, since both are positive. They cross once transversely. Replace them inside the rectangle by the ordered graphs $x_{\pm }(y)=\frac{x_1(y)+x_2(y)}{2}\pm \frac{1}{2}\sqrt{(x_1(y)-x_2(y){)}^2+ϵ(y{)}^2},$ where $ϵ$ is positive near the crossing, zero on endpoint collars, and sufficiently small to keep both graphs in the rectangle. Away from the crossing its zero set lies where $x_1-x_2\mathrm{\ne }0$, so both replacements are $C^2$ and agree with the old pair on the endpoint collars. They are disjoint positive arcs, and every plaque meeting a removed arc still meets the replacement pair. Performing all these resolutions yields finitely many disjoint embedded $C^2$ positive circles whose union still meets every leaf: the remaining finite arc graph has degree two everywhere and has no crossings. Smooth this disjoint family by step 3.1, using step 2.1 for the covering margin. Denote its union by $C$.

Choose a connected component $Q$ of the orientation cover from [F7]. It is compact, oriented, and surjects onto $M$. Pull back $F$ and $C$; write $\tilde{C}$ for the finite disjoint union of embedded circles over $C$. The ambient orientation and normal coorientation orient the tangent line field of the lifted foliation. Its positive local $C^1$ sections, combined with finitely many smooth chart bumps, give a nowhere-zero $C^1$ tangent field $X$ on $Q$. Its flow is complete by [F8] and compactness. Each orbit is open in its one-dimensional leaf by a flow box, so a connected leaf is exactly one complete orbit. Every such lifted leaf meets $\tilde{C}$: its projection is a whole base leaf, because any leafwise path lifts through the covering, and the base leaf meets $C$. In particular no complete $X$-orbit avoids $\tilde{C}$.

Cut $Q$ along $\tilde{C}$. This construction needs no surface classification. Each oriented embedded circle has a two-sided smooth collar: choose along it a smooth transverse field in the side cone selected by $X$, patch with finitely many bumps, and use its short smooth flow and the local inverse theorem. Compactness makes the collar injective after a common shrink, by local injectivity and separation of distant circle parameters. Replace each collar coordinate $r\in (-a,a)$ by its two labelled halves, retaining two copies of $r=0$. Glue to the unchanged complement. The resulting $P$ is a compact surface with boundary, and its natural map to $Q$ is a local diffeomorphism on each half-chart. Its boundary circles are the two copies of each component of $\tilde{C}$, and the pulled-back $X$ is everywhere transverse to them. There are finitely many connected components of $P$ by a finite cover by connected disk and half-disk charts. Each component has boundary: a boundaryless component would map to a nonempty open-and-closed subset of connected $Q$ disjoint from $\tilde{C}$.

In each component $P_0$ of $P$, every maximal interior trajectory of $X$ reaches its boundary in finite positive and negative time. For otherwise a forward trajectory remaining in compact $P_0$ for all $t\ge 0$ has a nonempty compact limit set $K$, the intersection of the closures of its tails. Local flow continuity shows that $K$ is invariant in both time directions: translate a sequence of times tending to infinity by any fixed sufficiently small positive or negative time, and then iterate. It cannot meet $\mathrm{\partial }{P}_0$, since a field transverse to that boundary has, in one time direction, points outside the half-chart; invariance would put such points back in $P_0$. Thus $K\subset \mathrm{Int}{P}_0$, and any point of $K$ has a complete orbit in $K$. Its image in $Q$ is a complete orbit avoiding $\tilde{C}$, contradicting step 6.1. The negative-time argument is identical. Nonemptiness of the limit set uses the finite-intersection property of compact sets, not a choice of successive times. Sequence arguments, when used in the local metrizable charts, require at most the standing ${\mathrm{A}\mathrm{C}}_{\omega }$.

Let $B_{-}$ be the incoming boundary of $P_0$. The sign of the boundary crossing is constant on each boundary circle. Step 8.1 gives, for each $p\in B_{-}$, a finite first exit time $T(p)>0$ at an outgoing boundary. This is a $C^1$ function: the exit boundary is a regular scalar equation along the flow, so [F8] gives the local hitting time; the compact segment between its incoming and outgoing collars stays in the interior, excluding any earlier exit for nearby initial points. The map $(p,s)\mapsto {\phi }_{sT(p)}(p)$ is a $C^1$ product identification $B_{-}\times [0,1]\to P_0$. It is onto by tracing each point backwards to its first boundary hit, injective by uniqueness and absence of intermediate boundary hits, and has a local $C^1$ inverse by flow boxes and transversality at the two ends. Connectedness of $P_0$ implies that $B_{-}$ is one circle; its outgoing boundary is also one circle. Thus the two boundary circles of each cut component lie on exactly the same lifted leaves.

Form the finite graph with vertices the components of $\tilde{C}$ and one edge for each cut component, incident to the circles which are the images of its two boundary circles; loops are allowed. This graph is connected, since otherwise the unions of the cut components and circle collars belonging to its distinct graph components would separate $Q$. Step 9.1 says that the saturations of the vertices at the two ends of every edge are equal. Therefore all these circle saturations are equal; their union is $Q$ by step 6.1, so any one component of $\tilde{C}$ meets every lifted leaf. Its image in $M$ is one of the embedded positive circles of $C$. Every base leaf has a lifted leaf in $Q$, since $Q\to M$ is onto and leafwise paths lift. The selected base circle consequently meets every base leaf. Together with step 1.1 and step 5.1 this proves the statement in every dimension, using only finite constructions and the standing countable choice.

All local selection in the compact constructions below is finite. In a $C^2$ chart choose nested coordinate balls and compose a Euclidean smooth bump with the chart; extension by zero gives a $C^2$ bump with support in that chart. Finite compact-core covers and normalization by the positive finite sum give $C^2$ partitions. In a smooth ambient manifold the same construction gives smooth bumps and partitions near a compact set. For a compact abstract $C^2$ surface $S$, choose finitely many such charts $x_i$ and bumps $b_i$ whose positive sets cover $S$. The map $J=(b_i,b_i{x}_i{)}_i$, extending each block by zero, is a $C^2$ embedding into a finite-dimensional Euclidean space: equality of the $b_i$ blocks and any positive $b_i$ recovers equality of $x_i$, hence of the points. If $dJ(v)=0$, then $d{b}_i(v)=0$ and $b_{i}d{x}_i(v)=0$ for a positive block, hence $v=0$. A continuous injective map from compact $S$ to a Hausdorff space has continuous inverse on its image; the local inverse in chart projections is $C^2$ by [F1]. Half-space charts give the same argument at the boundary.

First treat a closed cooriented embedded hypersurface $S$ in a smooth three-manifold. In finitely many ambient neighborhoods take $C^2$ defining functions $F_i$, zero precisely on $S$ there, with their differentials positive on the chosen common normal side. Multiply by an ambient smooth partition ${\rho }_i$ whose sum is one near $S$ and put $F={\sum }_i{\rho }_i{F}_i$. On $S$, $dF={\sum }_i{\rho }_{i}d{F}_i$, because $F_i=0$ kills every term $F_{i}d{\rho }_i$; this differential is nonzero and positively normal. Local implicit uniqueness and a finite shrinking therefore give a neighborhood of $S$ on which $F^{-1}(0)=S$. Positive local smooth transverse vectors, shrunk so they remain positive, glue by smooth nonnegative weights to a smooth field $V$ with $dF(V)>0$ near $S$. Finite coordinate convolutions and an ambient smooth partition, using [F2], give a smooth $G$ arbitrarily close to $F$ in $C^2$ on a compact smaller neighborhood.

Write ${\mathrm{\Phi }}_t$ for the smooth flow of $V$. Compactness gives a uniform short flow strip on which $dF(V)>c>0$; the map $(x,t)\mapsto {\mathrm{\Phi }}_t(x)$ is a $C^2$ diffeomorphism on a smaller strip about $S$. Its differential is invertible on the zero section; if no uniform injective strip existed, coincident points in successively thinner strips would have convergent base points, their limits would coincide, and local injectivity at that point would contradict the coincidences. For $G$ sufficiently close, $dG(V)>c\mathrm{/}2$ and $G$ has opposite signs at the strip ends. Thus $G({\mathrm{\Phi }}_t(x))=0$ has a unique root $t=\tau (x)$, $C^2$ by [F1]. The map $x\mapsto {\mathrm{\Phi }}_{\tau (x)}(x)$ identifies $S$ with the smooth level in that strip. Its $C^2$ inverse is obtained by the unique root of $F=0$ on the same flow orbit. A time cutoff extends the displacement to a $C^2$ ambient diffeomorphism supported in the strip: in these coordinates use $(x,t)\mapsto (x,t+\eta (t)\tau (x))$; choosing the approximation small makes $1+{\eta }^{\mathrm{\prime }}(t)\tau (x)>0$.

The abstract embedding in step 1.1 can have higher codimension; a scalar defining function is not asserted for it. Here is the needed finite replacement. A $C^2$ embedded surface is locally a graph $v=h(z)$ over a fixed two-plane by [F1]. On a compact graph patch replace $h$ by a smooth $h^{\mathrm{\prime }}$ arbitrarily close in $C^2$, and use the ambient map $(z,v)\mapsto (z,v+\eta (z,v)(h^{\mathrm{\prime }}(z)-h(z)))$, with a smooth compactly supported $\eta$ equal to one near the compact graph core. If the difference is sufficiently small in $C^1$, this is a $C^2$ diffeomorphism: its difference from the identity has derivative norm less than one, which proves injectivity by the mean-value estimate, and the local inverse is $C^2$; compact support gives surjectivity. The moved surface is a smooth graph near that core. To preserve already smoothed compact cores $K$, take $h^{\mathrm{\prime }}=h$ near their projections wherever the patch meets $K$. Such projections have an open neighborhood where $h$ is smooth. A smooth cutoff $\mu$, supported in that neighborhood and one near the relevant compact projections, gives $h^{\mathrm{\prime }}=\mu h+(1-\mu )h_{ϵ}$, which is smooth and $C^2$ close to $h$. Cut the ambient support away from any other protected cores. The displacement is then the identity near $K$.

Choose compact chart cores covering the original surface. Carry those cores and the remaining graph neighborhoods along each displacement. Apply step 4.1 successively to the finitely many cores, with the union of the earlier moved cores protected. Compactness allows finite refinements into graph patches and arbitrarily small displacements so that all needed graph projections remain regular. After the last patch the image is smooth near every core, hence everywhere. For boundary charts, extend the local graph $C^2$ across its half-plane boundary before convolution; the definition of $C^2$ regularity in a boundary chart supplies precisely such local extensions. This initially smooths the underlying surface graph, leaving a regular $C^2$ boundary curve in those smooth graph coordinates. Apply the same finite graph displacement argument, now to one-dimensional boundary graphs within these smooth surface coordinates, relative to previously smoothed boundary cores. Its coordinate displacements are $C^2$, have support in these patches, and smooth the boundary; every boundary core then has a smooth half-space chart. This proves the abstract assertion without an atlas-smoothing theorem.

For an actual cooriented surface region with boundary, one can retain the normal-root construction. Extend its local surface graphs across the regular boundary and apply step 2.1 on a finite neighborhood of the region; at the boundary the extensions agree on the interior side, which is all that is needed for the root projection of the region itself. The moved region lies in smooth local surface graphs and has $C^2$ boundary. Alternatively step 5.1 provides an unambiguous finite ambient construction there. On the smooth underlying surface near the boundary, take a $C^2$ signed boundary defining function $H$ using finitely many boundary graph charts and smooth weights; its inward differential is nonzero. Smooth it to $H^{\mathrm{\prime }}$ and take a smooth inward-transverse surface field $W$. The equations $H^{\mathrm{\prime }}({\mathrm{\Psi }}_t(y))=0$, with $y$ on the old boundary, give a $C^2$ boundary displacement with inverse obtained from $H=0$. Extend it with a cutoff in its flow collar exactly as in step 3.1. This maps the region to a smooth-boundary region. A compact intrinsic leaf subsurface has embedded ambient inclusion: the leaf inclusion is an injective immersion, and its restriction to a compact intrinsic subsurface is a homeomorphism onto its image by compactness and Hausdorffness. The above finite construction therefore applies; other, possibly dense, parts of that leaf are not part of this carrier.

Let $\psi :S^1\to S^1$ be an orientation-preserving $C^2$ diffeomorphism. Its increasing $C^2$ lift $a:\mathbb{R}\to \mathbb{R}$ satisfies $a(\theta +2\pi )=a(\theta )+2\pi$ and $a^{\mathrm{\prime }}>0$. Choose a smooth radial cutoff $\beta$ zero near $r=0$ and one near $r=1$. In polar coordinates put $E(r,\theta )=(r,\theta +\beta (r)(a(\theta )-\theta ))$. Periodicity makes this well-defined, and its angular derivative $1-\beta (r)+\beta (r)a^{\mathrm{\prime }}(\theta )$ is positive. Thus each circle map is bijective; [F1] gives a $C^2$ inverse on the annulus, and $E$ is the identity near the origin. This is a $C^2$ disk diffeomorphism restricting to $\psi$, with product collar expression near the boundary. For an orientation-reversing map first compose with the fixed reflection of the disk. The same formula is smooth for smooth data.

We give the smooth normal-form argument explicitly. Apply [F4] to the smoothed surface with empty incoming face and boundary as outgoing face; use both faces empty in the closed case. Obtain finitely many disks (zero-handles), rectangles (one-handles) and capping disks (two-handles), with all zeros before ones before twos. Before forming the graph, transport any attaching windows on earlier rectangles back to zero-disk boundary arcs: along a rectangle side use its product collar, move the finite windows in order to its endpoint collar, and continue onto the zero-disk boundary, shrinking windows to leave disjoint gaps. For an increasing interval map $f$ fixed at the endpoints, the collar map $(s,t)\mapsto ((1-\alpha (t))s+\alpha (t)f(s),t)$ has positive first derivative; successive such maps give each transport and carry the affected band with it. Induction over the finitely many bands gives a presentation with every band attached to disjoint intervals of zero-disks. Its core is then an edge between the zero-disks. The resulting graph is connected when the surface is: each capping disk attaches along a connected circle and cannot join two components. Choose a finite spanning tree. The union of its disks and bands is diffeomorphic to a disk, by successive leaf disk-and-band absorptions described next. Every other attaching interval is transported in these absorptions, rather than discarded.

For the absorption move, two disks joined along a single band are a disk after rounding their four corners: straighten the two attaching intervals in boundary collar coordinates, rescale the band's product coordinates to a rectangle, and identify the resulting union with a planar rounded disk-and-rectangle model. Its boundary is parametrized by the successive arcs and the two sides of the rectangle; a collar of that boundary and radial coordinates on a smaller interior disk give the usual disk parametrization. More explicitly one may straighten the disks into end caps of the rectangle and choose the rounded union to be a convex stadium, hence star-shaped about its midpoint; its radial boundary function is positive and smooth, and radial rescaling, cut off to a constant linear rescaling near the center, identifies it with a round disk. Boundary arc parametrizations are transported by increasing interval maps. If attaching windows of other bands lie on the absorbed disk, they remain disjoint ordered intervals on the new disk boundary. To put them in prescribed positions choose an increasing boundary map with the prescribed maps on those finitely many disjoint intervals; on the complementary intervals interpolate positive derivatives with the required integral. On a boundary rectangle $(s,t)$ extend its isotopy by $(s,t)\mapsto ((1-\alpha (t))s+\alpha (t)f(s),t)$, with the interval endpoints fixed and $\alpha$ zero at the inner edge. Its Jacobian in the $s$ direction is positive. Composing finitely many such collar maps transports all attaching windows and their band coordinates. This is the explicit surface handle slide/straightening needed for each tree contraction; it invokes no higher-dimensional handle-slide theorem.

After primal contractions there is one zero-disk. View the same presentation upside down: a two-disk is a dual zero-disk, a rectangle is a dual rectangle with its factors interchanged, and the zero-disk is a dual capping disk. The dual graph is connected by the same connectedness argument. A dual spanning tree and the absorptions of step 9.1 leave one dual zero-disk, hence one original two-disk. Transport later attachments at every contraction by the same rectangle collar maps. These operations are actual diffeomorphisms of the entire surface: the disk-plus-band replacements agree on the boundary collars used to attach the rest. Each primal contraction decreases $h_0,h_1$ by one, and each dual contraction decreases $h_1,h_2$ by one. If $\chi =0$, [F4] now gives $0=1-h_1+1$, hence precisely two remaining bands. This calculation concerns the actual Morse handles, regardless of how many cells were present in any original topological cellulation.

Orientability forbids a twisted band. The first untwisted band on the remaining disk gives an annulus: straighten its two windows by the collar maps in step 9.1; the standard disk with that rectangle is the planar annulus. The second band must join its two boundary circles. Indeed, attachment to intervals on the same boundary circle of an oriented annulus increases the number of boundary components to three, whereas joining the two circles leaves one, and the single final two-disk can cap only one circle. Straighten the two windows on the different annulus circles and extend their boundary maps over disjoint collars. An interval attaching map has no additional twist once the surface orientation is fixed; positive changes of its longitudinal parameter extend across the rectangle by the positive-derivative interpolation in step 9.1. Thus this is the standard annulus with one joining band, a once-punctured torus. Its remaining boundary is a circle; a circle attaching diffeomorphism extends over the final disk by step 7.1. Capping therefore gives the standard torus. Composing with the carrier diffeomorphism proves the $C^2$ torus assertion.

For a smooth region homeomorphic to a disk, cap its outgoing boundary with an auxiliary disk and mark that disk as an exterior dual vertex. A smooth circle parametrization needed for the cap is obtained by ordering finitely many regular curve charts around the circle and choosing a positive smooth speed on their overlaps. First contract the primal tree as above. The capped surface's dual graph is connected; choose a spanning tree rooted at the marked exterior vertex. Successively absorb the other dual disks towards that root using step 9.1. In original coordinates this deletes an internal two-disk together with a band; the exterior disk is not deleted, and its boundary collar is transported, so removing it at the end gives a diffeomorphism of the original region. There remain one zero-disk, no two-disks and $h_1$ bands. Its Euler characteristic is one, since it is homeomorphic to a disk, so [F4] gives $1=1-h_1$, hence $h_1=0$. The region is therefore a smooth disk, by an actual composition of disk/band collar moves. Pulling back by step 5.1 gives a $C^2$ parametrization of the original disk region; composing with the extension in step 7.1 realizes any prescribed boundary parametrization.

For completeness, an oriented compact annulus has a disk-and-band normal form by the same moves. Cap both boundary circles with marked exterior disks, contract the primal tree, and contract a dual forest rooted at those exterior disks (one root for each component of the forest). Every internal dual disk is absorbed into a root, so in the original annulus no internal two-disk remains. There is one zero-disk and, by $\chi =0$, one band. Orientability and the two boundary components make this the standard untwisted annulus. For two disjoint regular disks in a $C^2$ sphere, first smooth the sphere and then both marked boundary curves by the boundary moves in step 6.1, transporting the disks. Choose a common signed transverse-flow collar of each resulting smooth boundary curve. All disk-and-band moves can use these supplied collars: their interval maps extend as product maps on smaller collars and are cut off farther inside by the positive-derivative interpolation of step 9.1. For these disks their complement is connected with two boundary circles and Euler characteristic zero: paths can be rerouted around each removed disk along its boundary collar, and capping these two boundary circles recovers the sphere. The preceding normal form identifies this complement with an annulus. Given compatible boundary parametrizations, their increasing angle lifts $a_0,a_1$ extend across it by $a_t=(1-\lambda (t))a_0+\lambda (t)a_1$, with $\lambda$ constant near both ends; the angular derivative is positive. Parametrize the two disks by step 12.1 with those same collars, and use the constant-end interpolation on the annulus with the common signed collar coordinate. The maps on the two sides thus have identical product expressions on a whole seam neighborhood, so gluing is a $C^2$ diffeomorphism. This gives the simultaneous sphere diffeomorphism and prescribed windows. The scalar approximation assertion is [F2], after extending the function by a cutoff equal to one near its compact set and convolving.

Let $K=u(D^2)$, compact in the intrinsic topology. Choose finitely many precompact leaf charts with $C^2$ bumps $b_i$ positive on $K$, as in step 1.1. Their sum $b$ has compact support in $L$ and a positive minimum on $K$. Take a regular value $c$ strictly between zero and that minimum: cover the compact support by finitely many charts and apply [F3], so the finite union of bad value sets is null and cannot fill this interval. Then $P=\{b\ge c\}$ is a compact intrinsic $C^2$ subsurface with regular boundary and contains $K$ in its interior. Step 5.1 supplies a $C^2$ diffeomorphism $q:P\to P^{\mathrm{\prime }}$ with $P^{\mathrm{\prime }}$ smooth. Glue two copies of $P^{\mathrm{\prime }}$ along a smooth boundary collar to obtain its compact smooth double $Q$; collars can be read in the finite smooth boundary charts, gluing their inward fields by a finite smooth partition and taking their smooth flow. Thus $q(K)$ lies in the interior of the designated copy of $P^{\mathrm{\prime }}$ in the boundaryless smooth carrier $Q$.

By [F5] smoothly embed $Q$ into ${\mathbb{R}}^N$, writing $j$ for the embedding, and take a smooth retraction $R$ from a tubular neighborhood onto $j(Q)$. Let $v=jqu$. This is continuous on the disk and $C^2$ on the given open collar. Extend it continuously across the disk boundary by its boundary values constant on short radial rays, and multiply outside a larger disk by a continuous compact-support cutoff. Euclidean convolution gives smooth maps $v_{ϵ}$ uniformly approaching $v$ on $D^2$ by [F2]. Choose a smooth scalar cutoff $\gamma$ equal to zero near the specified smaller collar and equal to one off a slightly larger collar whose closure still lies in the original $C^2$ collar. Set $w_{ϵ}=v+\gamma (v_{ϵ}-v)$. Where $\gamma \mathrm{\ne }1$, the original $v$ is $C^2$; where $v$ is only continuous, $\gamma =1$ on a neighborhood and $w_{ϵ}=v_{ϵ}$. Thus $w_{ϵ}$ is $C^2$ everywhere and equals $v$ near the smaller collar.

The compact set $v(D^2)$ is inside the tubular domain and inside $j(\mathrm{int}{P}^{\mathrm{\prime }})$. By continuity of $R$ and compactness, sufficiently small uniform perturbations and their entire straight segments from $v$ remain in the tubular domain and retract into $j(\mathrm{int}{P}^{\mathrm{\prime }})$. Define $u_{ϵ}=q^{-1}{j}^{-1}R(w_{ϵ})$, using $j^{-1}$ on that copy. It is $C^2$, agrees with $u$ near the prescribed smaller collar, and converges uniformly to $u$. The formula $q^{-1}{j}^{-1}R((1-s)v+s{w}_{ϵ})$ gives a continuous homotopy fixed there. Uniform convergence in any compatible metric follows from uniform continuity of $q^{-1}{j}^{-1}R$ near this compact image; no regularity or injectivity of $u$ was used away from its collar.

The finite constructions prove the carrier, normal-form and relative approximation assertions. All selections particular to these compact carriers are finite. Countable choice is inherited only from the convolution, smooth-flow, Morse/handle, smooth-embedding and tubular suppliers; no arbitrary-index Axiom of Choice or general $C^2$ atlas-smoothing result is used.

By [F8] choose a $C^2$ diffeomorphism $\psi :S\to \mathrm{\Sigma }$ to a compact smooth oriented carrier and an excellent Morse function on $\mathrm{\Sigma }$, with finitely many critical points and a smooth gradient section. At an index-$\lambda$ point the gradient's derivative is the Hessian up to a positive metric isomorphism, so its index is $(-1{)}^{\lambda }$ by [F1]. Pull the section back by $D{\psi }^{-1}$ to a $C^1$ tangent section $s_0$ on $S$. At a zero its derivative is conjugate to the old derivative; differentiating the bundle map contributes no extra term because the section value is zero. Hence the same nondegenerate zeros and indices occur. The smooth handle CW model gives ${\sum }_p(-1{)}^{\lambda (p)}=\chi (\mathrm{\Sigma })$ by [F3], and homology functoriality under $\psi$ gives $\chi (\mathrm{\Sigma })=\chi (S)=V-E+F$ for the original topological cellulation. No differentiable handle per original topological cell is asserted.

The ambient and plane orientations coorient the normal line of $E$. Finite chart bumps give a C¹ positive transverse field $V$ near $W$, inward along its boundary, and a positive annihilator $\omega$ of $E$. Approximate their coefficients in finitely many ambient charts by smooth coefficients and patch with smooth bumps. Uniformly small errors preserve transversality and inwardness, giving smooth $V,{\omega }^{\mathrm{\prime }}$ with ${\omega }^{\mathrm{\prime }}(V)>0$. Put $E^{\mathrm{\prime }}=\ker {\omega }^{\mathrm{\prime }}$ and orient it so projection along $V$ gives an orientation-preserving bundle isomorphism $E\to E^{\mathrm{\prime }}$. This is invertible because both planes complement the same line.

Finite smooth bump multiples of local frames of $E^{\mathrm{\prime }}$ span each fiber over $W$. Their parameterized linear combination $s_a$, $a\in {\mathbb{R}}^P$, has a fiber-surjective parameter derivative. Solving for two parameters in each frame shows that its universal zero set over C² $W$ is a C² manifold of dimension $P+1$, with boundary the zero set over $\mathrm{\partial }W$, of dimension $P$. Apply F2 to the parameter projection in interior and boundary charts: C² suffices for dimension difference one, and C¹ suffices on the boundary. Countable chart covers and null unions use the stated countable choice. A common regular parameter gives a section transverse to zero on $W$ and on $\mathrm{\partial }W$. Its zero set $Z$ is a compact oriented C² one-manifold, with boundary $Z\cap \mathrm{\partial }W$. Subdivide a finite oriented interval-chart cover; its paired internal endpoints cancel, giving total signed boundary count zero.

Compare $s_0$ with a C¹ tangent section $s_1$ having isolated nondegenerate zeros. Both zero sets are finite, being closed and discrete in compact $S$. Refine a finite chart cellulation and move its graph slightly to miss both zero sets, with every face inside a tangent trivialization. Give $TS$ a C¹ metric by finite chart bumps, and use oriented orthonormal frames. On the graph the unit-section ratio $r=(s_1\mathrm{/}\mathrm{\mid }{s}_1\mathrm{\mid })(s_0\mathrm{/}\mathrm{\mid }{s}_0\mathrm{\mid }{)}^{-1}$ is a well-defined circle map independent of the frame, since frame changes are rotations. In each face remove small disks about its zeros and cut the remaining region into finitely many disks. Continuous argument increments cancel on paired cut edges, so the boundary winding of each section equals the sum of its local indices. Their difference is the winding of $r$. Summing over all faces cancels every edge increment of $r$ with its reversed occurrence, because $S$ has no boundary. The total index sums agree. This is the circle-degree and lifting calculus of The degree of a based circle loop and Degree defines a function $\mathrm{Deg}:{\pi }_1(S^1,[0])\to \mathbb{Z}$; it requires no C² Sard theorem for a C¹ homotopy.

The signs are uniform on each connected component of $W$: orienting $E$ by the rule that a positive frame of $E$ followed by the ${\omega }^{\mathrm{\prime }}$-positive direction $V$ is a positive frame of $TW$, the inwardness of $V$ gives $V$ a strictly negative outward-normal component at every boundary point, so comparison with the outward-normal-first rule [F7] shows that this induced orientation of $E{\mathrm{\mid }}_C$ is the negative of the boundary orientation of $C$ on every component; on a connected component of $W$ the given orientation of $E$ either agrees with the induced orientation at every point or disagrees at every point, so ${\epsilon }_C$ is one constant over the boundary components of each connected component of $W$.

Combining steps 1.1 and 2.1, every C¹ tangent section of a finitely cellulated closed oriented C² surface with isolated nondegenerate zeros has total index $V-E+F$, and [F3] identifies this count with the Euler characteristic of the cellulation.

On a boundary component $C$ pull $s_a$ back along $E\to E^{\mathrm{\prime }}$ to a C¹ section of $TC$ with nondegenerate zeros. At a zero, differentiating this bundle isomorphism adds no term from the zero section value, so corresponding fiber frames give the same signed determinant. F6 supplies a finite cellulation of closed $C$. Its ordinary tangent-index sum is $\chi (C)$ by step 3.1; with the given fiber orientation of $E$ rather than the boundary tangent orientation, the signed zero count is ${\epsilon }_C\chi (C)$.

At a boundary zero take coordinates with $W=\{x_3\le 0\}$ and outward normal ${\mathrm{\partial }}_3$, and an oriented frame of $E^{\mathrm{\prime }}$. The boundary-coordinate derivative $A$ of the section is invertible. Solving for those two coordinates makes the zero curve a graph over $x_3$; its orientation compares with the positive $x_3$ direction by $\mathrm{sign}\det A$. Its endpoint sign is therefore the signed boundary-section zero count of step 4.1, up to one fixed orientation convention shared by all endpoints. By step 2.2, ${\epsilon }_C=\epsilon$ on every boundary component of one connected component of $W$. Zero signed boundary count gives $0=\epsilon {\sum }_C\chi (C)$ there. Add over the finitely many region components. Empty boundaries contribute zero, and steps 1.1–2.1 give the closed-surface assertion.

Let ${\gamma }_A$ be a $C^2$ leafwise path from $x$ to $c(0)$ inside $A$ and ${\gamma }_B$ a $C^2$ leafwise path from $c(1)$ to $y$ inside $B$. Such paths exist: cover each leaf by its foliated charts, whose plaques are convex and chartwise polygonally connected, use [F2] finitely many times to reach the target plaque, and smooth the finitely many corners inside plaque charts; the finite plaque transport preserves the $C^2$ regularity of the leafwise curves by [F3]. Only finitely many charts and plaques are selected.

Near the compact union of the three curves ${\gamma }_A,c,{\gamma }_B$ construct a smooth field $X$ and a $C^1$ form $\omega$ with $\ker \omega =TF$ and $X$ positive transverse: choose finitely many smooth ambient chart bumps over that union by [F4], take in each chart a constant vector that is positive for the continuous cooriented tangent planes on a smaller neighbourhood, and sum the bumped fields; patch the local positively cooriented annihilating forms of the $C^2$ atlas in the same finite way. On a compact flow domain the smoothness of $X$ and $\omega$ gives a uniform bound $\omega (X)\ge b>0$ and, by [F5], a uniform interval $\mathrm{\mid }a\mathrm{\mid }\le a_0$ of the flow ${\phi }_a$ on which the derivative $D{\phi }_a$ is bounded; [F6] reduces the neighbourhood data to finitely many compact charts.

For a $C^2$ leafwise path $\gamma$ near the union and a $C^2$ offset profile $a(s)$, the curve $s\mapsto {\phi }_{a(s)}(\gamma (s))$ has transverse derivative $v(s,a(s))+a^{\mathrm{\prime }}(s)\omega (X)$, where $v(s,0)=\omega ({\gamma }^{\mathrm{\prime }}(s))=0$ because ${\gamma }^{\mathrm{\prime }}$ is tangent to $F$; hence $\mathrm{\mid }v(s,a)\mathrm{\mid }\le A\mathrm{\mid }a\mathrm{\mid }$ on the compact parameter set for a constant $A$ by [F7] applied to $a\mapsto v(s,a)$. With the profiles $a(s)=\delta (e^{Bs}-1)\mathrm{/}B$ on the initial path and $a(s)=-\delta (e^{B(1-s)}-1)\mathrm{/}B$ on the terminal path one has $a^{\mathrm{\prime }}=B\mathrm{\mid }a\mathrm{\mid }+\delta$, so choosing $B>A\mathrm{/}b$ gives transverse derivative at least $b\delta +(bB-A)\mathrm{\mid }a\mathrm{\mid }>0$; both profiles vanish at the outer endpoints, so ${\phi }_a{\gamma }_A$ starts at $x$ and ${\phi }_a{\gamma }_B$ ends at $y$.

Along $c$ use a fixed $C^2$ profile scaled by $\delta$ that joins the positive offset at its start to the negative offset at its end; since $\min \omega (c^{\mathrm{\prime }})>0$, its perturbed transverse derivative stays positive for small $\delta$, and the three pieces join into a piecewise $C^2$ positive path from $x$ to $y$. At each of the two internal seams choose one $C^2$ foliated chart; the one-sided transverse-coordinate derivatives have a positive lower bound there, and mollifying the continuous piecewise $C^2$ chart curve with a nonnegative kernel keeps the transverse derivative positive, while the blending correction has derivative tending uniformly to zero and is made smaller than the lower bound; the resulting curve is $C^2$, positive, agrees near the outer endpoint collars and still joins $x$ to $y$.

Transitivity: if genuine segments realize $A{⪰}_{F}B$ and $B{⪰}_{F}C$, apply the endpoint adjustment of step 3.1 to the second segment so that it starts at the actual endpoint of the first, concatenate the two, and smooth the single internal seam by the same chartwise mollification; the equality cases are immediate from the definition, so $A{⪰}_{F}C$.

Reflexivity holds by the defining equality clause of [F1], and transitivity is step 4.1, so ${⪰}_F$ is a preorder in the sense of [F8]; the endpoint conclusion follows from step 3.1 applied to any genuine joining segment, and the formal equality clause alone is never used to produce an actual segment.

Mutual accessibility is reflexive and symmetric by definition and transitive by two applications of the transitivity in step 4.1, so it is an equivalence relation in the sense of [F8]; the construction used only finitely many charts, bumps, flow intervals and profiles, hence only the standing countable choice from [F9].

Cover compact $M$ by finitely many smaller product boxes with closures inside larger product boxes. The compact barriers are embedded by F2, so refine near them so that each barrier meets any box in one plaque or not at all; away from their union use boxes disjoint from all barriers. If $L$ had only finitely many plaques in each larger box, the compact plaque disks in those larger boxes containing its intersections with the smaller boxes would form a finite compact cover of $L$ in its intrinsic topology. This would make $L$ compact, a contradiction. Thus a box has infinitely many $L$-plaques, and finitely many barrier levels. This argument uses finite plaque disks and their intrinsic topology; the mere property of being a union of plaques does not imply that a leaf is embedded or closed.

Choose two of those infinitely many plaque levels in the same component of the transverse interval after removing the finitely many barrier levels. Let their heights be $a<b$, and join their central points from height $b$ to height $a$ by a $C^2$ leafwise path $\gamma$ using finitely many plaque charts. Its compact image is disjoint from the compact barriers. A sufficiently thin neighborhood of that image and the vertical segment between the endpoints therefore avoids all barriers.

Construct a genuine leafwise constant-label strip $P(r,s)$ along $\gamma$, with $P(r,0)=\gamma (r)$ and $\omega ({\mathrm{\partial }}_{r}P)=0$. Here is a finite construction: choose a smooth ambient transverse field near the compact path (patch finitely many positive local fields); its short flows give transverse curves over the path by F4. In each successive foliation box, solve for the flow coordinate whose transverse box coordinate equals the transported initial label. Its normal derivative is nonzero, so the $C^2$ scalar inverse gives a $C^2$ solution. On overlaps the transverse coordinate depends only on the previous transverse coordinate; equality of the transported labels and uniqueness of the scalar root identify these solutions. Finite shrinking and F3 give one $C^2$ strip on $[0,1]\times (-\delta ,\delta )$, with each fixed-$s$ path leafwise and $\omega ({\mathrm{\partial }}_{s}P)>0$. Choose the transverse fibers at the two endpoints along the original vertical segment. For small $\epsilon >0$ take $s(r)=\epsilon (2r-1)$. Then $\omega ((P(r,s(r)){)}^{\mathrm{\prime }})=2\epsilon \omega ({\mathrm{\partial }}_{s}P)>0$ exactly, rather than by a domination estimate. Its endpoints still have final height smaller than initial height, so the positive vertical interval closes it. The tilted part crosses $L$ at $s=0$, and the whole curve misses all barriers. Smooth its two corners by $C^2$ interpolation in boxes; the positive tangent half-space is convex, so sufficiently small interpolation preserves positivity and the interior crossing.

To make this $C^2$ positive closed immersion embedded, protect a small parameter arc containing the crossing of $L$. Its nonzero derivative gives local injectivity; compactness of the parameter circle gives a uniform $\eta >0$ such that sufficiently $C^1$-small perturbations remain injective on every pair with circular parameter distance less than $\eta$. The remaining pairs form a compact set. At a possible equality on this set, choose disjoint parameter neighborhoods of the two branches; at most one is in the protected arc. On the other branch use a bump supported away from the protected arc and three independent ambient chart fields, whose flows independently move its value in the three ambient coordinates. Compactness gives finitely many such bump/field triples covering all possible equality pairs; the complement of their neighborhoods has image separation bounded below and cannot acquire equalities under a small perturbation. Let $u\in {\mathbb{R}}^N$ be these finite flow parameters and write ${\gamma }_u$. For small $u$, on each shared target coordinate chart the equation ${\gamma }_u(r)={\gamma }_u(t)$ is a $C^2$ submersion in the full variables $(r,t,u)$: the chosen bump moves one value independently of the other. Its zero set has dimension $2+N-3=N-1$. Its countably many local Euclidean parametrizations project by $C^1$ maps into the $N$-dimensional parameter space; F5 makes their images null, and countable choice supplies their countable chart cover and countable null union. Therefore a small parameter outside these images exists. There are then no separated-pair equalities, and uniform local injectivity handles the near-diagonal pairs. Smallness preserves positivity, avoidance of the compact barriers, and the protected crossing. No finite-self-crossing assumption or incorrectly dimensioned Sard theorem is used.

The resulting curve is an embedded positive closed transversal through $L$, avoiding the specified finite compact barriers. Contraposition shows that absence of such a curve, for any specified finite family or for the empty family, forces intrinsic compactness. The geometric choices are finite; the null-image argument in step 4.1 uses the stated countable choice only. No full-AC smooth-manifold supplier is imported into the $C^2\mathrm{/}{\mathrm{A}\mathrm{C}}_{\omega }$ statement.

The source interior of the frontier occupies exactly one saddle quadrant of $q$: the two coordinate axes of the saddle normal form of [F2] give four quadrants, and if the source interior occupied three of them, the two unused characteristic separatrix half-rays would lie inside $\mathrm{\Omega }$, whereas every point of $\mathrm{\Omega }$ other than the center $p$ lies on a periodic orbit, and a separatrix half-ray is not a periodic orbit. Hence exactly one quadrant is occupied, and the maximality of the nested-circle basin and the absence of other zeros make $p$ the only interior center.

The rounded image of $\mathrm{\Gamma }$ being null in its leaf makes the holonomy germ of $\mathrm{\Gamma }$ the identity on both sides: the leafwise class of $\mathrm{\Gamma }$ is trivial, so by [F1] the holonomy of $\mathrm{\Gamma }$ is the holonomy of a null loop, namely the identity germ, and the same holds for the opposite side.

Choose one smooth positively transverse ambient field $V$ near the compact image of $\mathrm{\Gamma }$, by finitely many positive chart fields and smooth nonnegative weights. In each foliation box let ${\zeta }_i=0$ be the assigned plaque of the intrinsic leaf $L$ through the corresponding arc of $f(\mathrm{\Gamma })$. The equation ${\zeta }_i({\mathrm{\Phi }}_{-{\sigma }_i(x)}f(x))=0$ has a unique short $C^2$ root because its derivative in ${\sigma }_i$ is nonzero. On overlaps the same $V$ orbit and assigned intrinsic plaque give the same root and projected point. The two-sided identity holonomy of step 1.2 returns the same plaque branch after one circuit, so these expressions give a $C^2$ projection $B_C$ on a sufficiently thin neighborhood $N$ of $\mathrm{\Gamma }$, including its saddle, with $B_C=f$ on $\mathrm{\Gamma }$. No embeddedness of the whole leaf or exclusion of distant branches is used. Choose a regular inner circle $C$ of the basin inside $N$ and an enlarged source disk $\widehat{W}$ with exterior boundary inside $N$. The projection annulus, prescribed leafwise rounding collar and given null filling supply a continuous leafwise filling of $B_C{\mathrm{\mid }}_C$. Attach the prescribed $B_C$ germ on a collar of $C$, apply [F6] to the remaining filling disk, and extend by $B_C$ on $\widehat{W}$ outside $C$. This gives one $C^2$ leafwise cap $B$ agreeing with the entire projection germ near $\mathrm{\Gamma }$ and the exterior boundary; no flattening of the saddle jets is performed.