Finite C2 surface carriers have smooth normal forms and relative cap approximations

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.