Face-relative Morse decomposition into the ten sewing generators

If $M=\mathrm{\varnothing }$, use the empty function and empty generator word; all assertions are vacuous. For nonempty $M$, the finite bump and partition constructions used below also work in its corner charts: extend a chart to an ambient Euclidean open set, use [F8] on nested balls there, restrict to the half-space or quadrant, and extend by zero outside the chart. A strict support margin makes that extension smooth, including on boundary faces. Smaller restricted balls cover each compact set in question; choose a finite subcover. The corresponding nonnegative bumps $b_j$ have positive sum near that compact set, so $b_j\mathrm{/}{\sum }_i{b}_i$ form a smooth finite subordinate partition there. Each selected bump is one on a smaller chart neighborhood. For a cutoff equal to one near an entire compact set and supported in a prescribed open neighborhood use $1-{\prod }_j(1-b_j)$; its support lies in the finite union of the strict chart supports. This proves the corner-chart partition and cutoff assertions directly from [F8], rather than applying a boundaryless partition theorem to corners. There are finitely many corners: the depth-two stratum of a compact surface with faces is a closed discrete zero-manifold, so its singleton neighborhoods have a finite subcover. Choose disjoint quadrant charts at these corners. The two coordinate directions in each quadrant give product collars of the two incident faces. Delete smaller quadrant neighborhoods, round their new junctions inside those neighborhoods, and apply [F2] to the remaining ambient smooth surface with boundary. Restrict its collars to the remaining original face arcs. Interpolate the associated inward transverse fields with the coordinate normal fields on the finite overlaps, keeping the latter fields fixed in smaller quadrant charts. Positivity of the inward component is preserved by the partition-of-unity interpolation. The two fields commute and are tangent to the opposite face in those smaller charts, so their local ambient flows from [F10], restricted to inward times and a sufficiently small common collar width, give exactly the product corner collars; elsewhere the faces are disjoint. Thus the black and free faces have compatible collars. On a black collar prescribe $f=t$ at an incoming face and $f=1-t$ at an outgoing face.

Each free boundary component is either a circle or an interval between corners. On a circle use $g(s)=\frac{1}{2}+\delta \cos (2\pi s)$ with $0<\delta <1\mathrm{/}4$. For an interval, its endpoint values are $0$ or $1$. If they differ, use the monotone profile $g(s)=s$ or $1-s$. If both are $0$, take a smooth nonnegative bump ${\rho }_0$ symmetric about $1\mathrm{/}2$, supported in $(1\mathrm{/}4,3\mathrm{/}4)$ and positive at $1\mathrm{/}2$; such a bump is obtained from [F8] using a ball centered at $1\mathrm{/}2$. Normalize it by its positive finite integral to obtain $\rho$ with ${\int }_0^1\rho =1$. Define $\theta (s)=1-2{\int }_0^s\rho (u)du$ and $g(s)={\int }_0^s\theta (u)du$. Then $\theta$ is nonincreasing, equals $1$ and $-1$ near the respective endpoints, satisfies $\theta (1-s)=-\theta (s)$, and has its unique zero at $1\mathrm{/}2$ with ${\theta }^{\mathrm{\prime }}(1\mathrm{/}2)=-2\rho (1\mathrm{/}2)<0$. Thus $g^{\mathrm{\prime }}(s)=\theta (s)$ has one transverse zero, while $g(0)=g(1)=0$, $g(s)>0$ for $0<s<1$, and $g(s)\le 1\mathrm{/}2$; its endpoint derivative germs are $+1$ and $-1$. If both endpoints are $1$, use one minus that profile. The endpoint germs match the black-corner collar coordinates. A smooth inward extension $g(s)+th(s,t)$ is chosen with $h=0$ near the endpoints, so it agrees with the black-corner collars, and $h>0$ at each free-face critical point, so the full differential is nonzero there. Every critical value lies strictly between $0$ and $1$. Do not scale a profile after its endpoint germs have been fixed. Instead choose pairwise disjoint free-face coordinate neighborhoods of its finitely many critical points, with closures away from corners. By [F8] choose a cutoff ${\beta }_j$ supported in each neighborhood and identically one on a smaller neighborhood of the critical point. Replace $g$ there by $g+c_j{\beta }_j$. On the transition annuli $\mathrm{\mid }{g}^{\mathrm{\prime }}\mathrm{\mid }$ has a positive minimum; choose $\mathrm{\mid }{c}_j\mathrm{\mid }\mathrm{\parallel }{\beta }_j^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}$ smaller than that minimum. Hence no new free-face critical point occurs, while the derivative and Hessian near the old point are unchanged and its value shifts by exactly $c_j$. Choose the finitely many constants successively, avoiding the finitely many previously selected critical values. The supports are away from corners, so every endpoint germ remains fixed. Extend each perturbation constantly in the inward variable on a smaller collar and cut it off farther in using [F8]; its inward derivative on the face is zero, so the prescribed nonzero normal derivative at the critical point is unchanged. The compact perturbation supports have positive distance in value from $0,1$, so sufficiently small constants retain the range and the prescribed extreme levels.

Use the finite corner-chart cutoff construction of step 1.1 to choose $\chi =1$ near the compact boundary with support in the neighborhood carrying the prescribed smooth collar function $f_{\mathrm{c}\mathrm{o}\mathrm{l}}$. Shrink that neighborhood so $f_{\mathrm{c}\mathrm{o}\mathrm{l}}$ lies in $[0,1]$, with equality only on the prescribed black faces. Set $f_0=\chi f_{\mathrm{c}\mathrm{o}\mathrm{l}}+(1-\chi )\mathrm{/}2$, interpreting the first term as zero off that neighborhood. The strict support margin makes it smooth, it agrees with the prescribed function near every face, and its range and extreme levels are correct. On the resulting boundary neighborhood the full differential is nonzero: on black faces the collar derivative is nonzero; on free faces either the tangential derivative or the chosen normal derivative is nonzero; at corners the tangential germ is the nonzero linear endpoint germ. Compactness therefore leaves a compact set $K$ in the interior outside which no perturbation is needed. Cover $K$ by finitely many coordinate charts $U_i$ and smaller open sets $V_i$ whose union contains $K$. Choose the $V_i$ with compact closures in $U_i$ and apply [F9] on the boundaryless interior to obtain bumps ${\chi }_i$ supported in $U_i$ and equal to one near ${\bar{V}}_i$. The differentials of the finitely many functions ${\chi }_i{x}_i^j$ span every cotangent fiber on a neighborhood of $K$.

If $K=\mathrm{\varnothing }$, the constructed differential is already nonzero everywhere in the interior, so there are no interior critical points and no parameter choice or density argument is needed. Retain $f_0$ in this case. If $K\mathrm{\ne }\mathrm{\varnothing }$, at least one two-dimensional chart is present and the parameter space below has positive dimension. For parameters $a_{ij}$ in an open ball about $0$, set $f_a=f_0+{\sum }_{i,j}{a}_{ij}{\chi }_i{x}_i^j$. Put $W={\bigcup }_i{V}_i$ and $Q={\bigcup }_{i,j}\mathrm{supp}({\chi }_i{x}_i^j)$, extending each chart-supported function by zero outside its chart. Then $W$ is an open neighborhood of $K$, $Q$ is compact in the interior, and $W\subseteq Q$: on each $V_i$, the differentials $d({\chi }_i{x}_i^j)=d{x}_i^j$ span the cotangent fiber, so at least one of those functions has nonzero differential at each point and that point lies in its support. The parameter derivatives therefore span every cotangent fiber on $W$. On $Q\setminus W$, $d{f}_0$ is nonzero because $K\subseteq W$ and $d{f}_0$ has no zeros outside $K$. For each point of this compact set, continuity of $(p,a)\mapsto d(f_a{)}_p$ gives a product neighborhood on which the differential remains nonzero; a finite subcover yields a common parameter neighborhood of $0$ preserving nonvanishing throughout $Q\setminus W$. Outside $Q$ the perturbation vanishes, so the differential remains $d{f}_0\mathrm{\ne }0$. Restrict the parameter ball to that neighborhood and small enough that the range stays in $[0,1]$. On $W$ the evaluation map $(p,a)\mapsto d(f_a{)}_p$ is transverse to the zero section because its parameter derivatives span each cotangent fiber. Apply [F3] to this family on the positive-dimensional manifold $W$: its bad parameter set is null. The parameter space is positive-dimensional, so [F4] makes the good parameters dense; in particular the restricted open ball contains one. For this choice, all critical points lie in $W$ and are nondegenerate. By [F6] the interior restriction is Morse. Its critical set is closed and contained in the compact set $Q$, so it is compact; because each nondegenerate critical point is isolated, it is finite.

If there are no interior critical points, skip these interior perturbations. Otherwise choose pairwise disjoint small interior chart neighborhoods of the finitely many critical points and use [F9] to obtain bumps supported there and equal to one on smaller closed neighborhoods. Choose a smooth Riemannian metric on $M$ by covering the compact surface with finitely many corner charts and patching their Euclidean metrics with the finite smooth partition construction of step 1.1. For its induced cotangent norm, $\mathrm{\mid }df\mathrm{\mid }$ is continuous and positive on the compact complement of those smaller neighborhoods, so it has a positive minimum. Choose the bump supports and then the constants small enough that the norm of the perturbation differential is smaller than this minimum on the transition annuli; no new critical point is created there. Near each critical point the bump is constant, so its differential and Hessian are unchanged while its critical value shifts by that constant. Choose the constants successively to avoid the finite set of free-face and already fixed interior values. The boundary collars stay fixed, and the constants can be small enough that the range remains in $[0,1]$. Thus all critical values are distinct. On a component with no incoming or outgoing black face, first place the free/interior function in a compact subinterval of $(0,1)$; its absent endpoint level is then empty.

An interior critical point of index zero or two has a disk birth or death neighborhood by the Morse lemma, giving ${\eta }_C$ or ${ϵ}_C$. An interior saddle is a rectangular band attached to two disjoint subintervals of the preceding regular level. This level is a compact one-manifold, so each affected component is a circle or an interval. The band is untwisted, because its orientation extends that of the surface. If both affected level components are circular, or both feet lie on one circular component, the usual two circle configurations give ${\mu }_C$ or ${\mathrm{\Delta }}_C$. If an interval occurs, the only possibilities are two distinct intervals, two feet on one interval, or one interval and one circle. Indeed these exhaust whether the two feet belong to the same or different level components and whether those components have endpoints. Two feet on a single interval split it into an interval and a circle; reversing the slab gives the interval/circle merge. Two different intervals are reconnected into two intervals. A twisted reconnection would give a nonorientable band and is excluded.

The six mixed representatives and their types are the following. A permutation of the listed input or output slots is implemented by the symmetry cylinders; it never changes an interval's own endpoint labels. In rows 1–2 the canonical type is $I_{ab}\otimes I_{cd}\to I_{ad}\otimes I_{cb}$. These are the two choices of which incoming interval is split in a two-interval reconnection. Rows 3–6 have the types shown. The letter at a newly exposed free edge is determined by the free edge already meeting that end of the slab, so no arbitrary color is introduced.

At a free critical point the tangential restriction has a nondegenerate extremum and the inward normal derivative is nonzero by construction. In coordinates $t\ge 0$ inward and $s$ tangential its chart can be taken as $f=f(p)\pm s^2\pm t$: apply [F7] to the map $(s,t)\mapsto (s,f(s,t)-f(s,0))$, whose Jacobian determinant is the nonzero inward normal derivative, followed by the one-dimensional Morse coordinate on the boundary. A free minimum with positive inward normal derivative births an interval and gives ${\eta }_a$; a free maximum with negative inward derivative kills one and gives ${ϵ}_a$. A free maximum with positive inward derivative joins the two local level ends: if they close the one interval into a circle the slab is $i_a^{\ast }$ (two corners), and otherwise they join two intervals into one, giving ${\mu }_{abc}$ (six corners). A free minimum with negative inward derivative is the reverse: $i_a$ has two corners and ${\mathrm{\Delta }}_{abc}$ has six. The label at the extremum is the label of the single free face through that point; all other endpoint labels are carried by product collars. There is no zero-normal-derivative case in the function constructed in the preceding boundary-relative construction, and no assertion about a full boundary Hessian is needed.

Here is a boundary-relative verification of the cuts in that table. Write $X^{-}$ for an incoming black interval read in induced surface-boundary orientation, $U^{+}$ for an outgoing interval, and $F_s$ for an intervening free arc of color $s$. In rows 1–2, call the inputs $X=I_{ab},Y=I_{cd}$ and outputs $U=I_{ad},V=I_{cb}$. The one-saddle slab is two interval product rectangles joined by the untwisted rectangular band along one edge in each rectangle, hence is a disk: successively attaching each rectangle along a full band edge leaves a disk. Its boundary word is $U^{+}{F}_d{Y}^{-}{F}_c{V}^{+}{F}_b{X}^{-}{F}_a\mathrm{.}$ For row 1 cut the ${\mathrm{\Delta }}_{acb}$/${\mu }_{acd}$ seam $I_{ac}$; for row 2 cut the ${\mathrm{\Delta }}_{cbd}$/${\mu }_{abd}$ seam $I_{bd}$. Each composite consists of two disks joined along that seam. Reading the remaining edges gives precisely the displayed word. The apparent crossing in each formula is a permutation of black slots; it does not identify any free edges. In rows 3–4 the slab is an annulus with outer boundary word $U^{+}{F}_b{X}^{-}{F}_a$ and inner boundary the incoming closed circle $Z^{-}$. This can be seen directly from the untwisted band on a circle and an interval: slit the circle at a point away from the band foot, then cut through the intervening product strip to the indicated outer free arc; the band and the slit level collars unfold to a disk. For the composite, the zipper is an annulus with inner boundary $Z^{-}$; attaching the product disk along its interval seam replaces an interval of the outer boundary by the complementary boundary arc of that disk. This disk attachment preserves the annulus and gives exactly $U^{+}{F}_b{X}^{-}{F}_a$. Choose an auxiliary arc in the zipper from $Z$ to the surviving part of its free arc, of color $a$ in row 3 and $b$ in row 4. Its interior avoids the product seam. Cutting along this auxiliary root-to-hole arc gives one disk. It is not the interval gluing seam: cutting that seam instead separates the original disk and annulus. In either the slab or composite split the outer cyclic word at the auxiliary arc's endpoint, writing it $E_1{E}_2$; the cut word is $E_{1}s{Z}^{-}{s}^{-1}{E}_2$. Choose corresponding endpoints on the indicated free arc, so the two cut words and their paired seam sides agree, including all black parameterized subarcs and free labels. Rows 5–6 are the height reversals of these annular cut models, with the original interval parameterizations retained. Matching these disk words gives diffeomorphisms relative to black collars: after rounding their nonblack corners, an orientation-preserving circle map extends across the disk by interpolating its lifted angular map to the identity near the disk center; on a prescribed boundary collar use its product extension. Regluing each pair $s,s^{-1}$ recovers the annular diffeomorphism. Thus these are actual local factorizations, including the two interval reconnections and all four interval/circle orientations.

Choose regular values $a<b$ surrounding each critical value, and first consider a compact slab $Q=f^{-1}([a,b])$ containing no interior or free-face critical value. Extend it a little through its two regular black level faces using the product level charts supplied by [F7] from the nonzero tangential or interior differential; at an actual incoming or outgoing black face use the given collar to add a short exterior strip. This gives a neighborhood of $Q$ whose only boundary near $Q$ is the free boundary, with no remaining corner at the extended black levels. On a free face define the tangent field $X=({\mathrm{\partial }}_{s}f{)}^{-1}{\mathrm{\partial }}_s$ and extend it smoothly to a smaller collar; its denominator is nonzero throughout the regular slab. Near an interior point choose a coordinate direction with nonzero $df$ and divide its coordinate field by that derivative. The finite chart partitions from step 1.1 glue these fields while preserving $df(X)=1$ and tangency along the free faces. The extended neighborhood is a smooth manifold with boundary, and its entire boundary near $Q$ is the free face to which $X$ is tangent; thus [F11], not a theorem requiring tangency to the transverse level faces, gives its boundary-preserving local flows. In interior charts the same flow is supplied by [F10]; uniqueness makes the charts agree. At every point of compact $Q$ the open flow domain contains $(-{\epsilon }_j,{\epsilon }_j)\times U_j$ for some neighborhood. A finite subcover gives a common positive lower time bound. Since $df(X)=1$, a trajectory stays in $Q$ until its level reaches an endpoint; it cannot cross a free face by [F11]. Restarting in this finite cover with increments below the common bound constructs the trajectory through every time needed between $a$ and $b$. This proves finite-time existence on the slab without an unjustified completeness assumption. The map $f^{-1}(a)\times [a,b]\to Q$, $(p,t)\mapsto {\mathrm{\Phi }}_{t-a}(p)$, has inverse $q\mapsto ({\mathrm{\Phi }}_{a-f(q)}(q),f(q))$. These maps are smooth by [F10]–[F11] and preserve each free component, hence every color; they identify the slab with its product, with the level collars retained. Each critical slab is one of the interior, free-boundary or mixed local models already proved. Concatenating the finitely many slabs gives the generator word and identity cylinders. Apply the finite construction to each connected component and tensor the words; the empty component family gives the empty word.