Relative one-parameter Morse paths with finitely many elementary events

A finite-dimensional family prescribing all required jets simultaneously. Choose finitely many coordinate neighborhoods $U_{\alpha }$ and smooth bumps ${\chi }_{\alpha }$ supported in them such that their positive sets cover $W$. Boundary charts are allowed here; extend their smooth coordinates to open sets across the boundary. The finite vector $E(x)=({\chi }_{\alpha }(x),{\chi }_{\alpha }(x)x_{\alpha }^1(x),\dots ,{\chi }_{\alpha }(x)x_{\alpha }^n(x){)}_{\alpha }\in {\mathbb{R}}^N$ is a smooth embedding. Indeed, if $E(x)=E(x^{\mathrm{\prime }})$, choose $\alpha$ with ${\chi }_{\alpha }(x)>0$. Equality of its first component implies ${\chi }_{\alpha }(x^{\mathrm{\prime }})>0$; equality of its remaining components then gives $x_{\alpha }(x)=x_{\alpha }(x^{\mathrm{\prime }})$, hence $x=x^{\mathrm{\prime }}$. Its differential is injective because on $\{{\chi }_{\alpha }>0\}$ one recovers $d{x}_{\alpha }^i=(d({\chi }_{\alpha }{x}_{\alpha }^i)-x_{\alpha }^{i}d{\chi }_{\alpha })\mathrm{/}{\chi }_{\alpha }$. Compactness makes the continuous injection a homeomorphism onto its image. This constructs the needed embedding without invoking a separate embedding theorem. Let $\mathcal{P}$ be the finite-dimensional vector space of real polynomials on ${\mathbb{R}}^N$ of total degree at most $15$. For any $1\le k\le 4$ and any distinct interior points $p_1,\dots ,p_k$, the map $\mathcal{P}⟶{⨁}_{i=1}^k{J}_{p_i}^3(W,\mathbb{R}),P⟼(j_{p_i}^3(P\circ E){)}_i$ is onto. Here a spatial 3-jet is the finite list of the value and all spatial derivatives of orders one through three; its coordinate changes are the usual smooth invertible chain-rule changes. To prove surjectivity, choose for every $i\mathrm{\ne }j$ an affine linear function ${\mathrm{\ell }}_{ij}$ with ${\mathrm{\ell }}_{ij}(E(p_j))=0$ and ${\mathrm{\ell }}_{ij}(E(p_i))\mathrm{\ne }0$. Put $Q_i={\prod }_{j\mathrm{\ne }i}{\mathrm{\ell }}_{ij}^4$. It has nonzero value at $E(p_i)$ and vanishes to order at least four at every $E(p_j)$, $j\mathrm{\ne }i$. At $p_i$, some $n$ components of $E$ have independent differentials; they are local coordinates by the inverse function theorem. Polynomials of degree at most three in those components realize arbitrary spatial 3-jets at $p_i$. Since multiplication by a jet with nonzero constant term is invertible in the finite Taylor algebra, choose such a polynomial $R_i$ whose 3-jet, multiplied by that of $Q_i\circ E$, is the desired jet at $p_i$. Then $P={\sum }_i{Q}_i{R}_i$ realizes all prescribed jets. Its degree is at most $4(k-1)+3\le 15$. The separating linear forms and the coefficients depend on the chosen points, but they are used only to prove surjectivity of this one fixed finite polynomial space; no varying family of perturbation spaces is needed. Begin with the affine path between $f_0,f_1$, reparametrized to be stationary on parameter intervals $[0,2\delta ]$ and $[1-2\delta ,1]$. Call it $H_s$. Choose a spatial bump $\gamma$, zero near the boundary, positive on an open set containing the possible critical points, and equal to one on the complement of a slightly larger product collar. Choose $\beta (s)$ zero for $s\le \delta$ and $s\ge 1-\delta$, and positive for $\delta <s<1-\delta$. Fix a basis $P_1,\dots ,P_m$ of $\mathcal{P}$ and consider $H_{s,z}(x)=H_s(x)+\beta (s)\gamma (x){\sum }_{\nu =1}^m{z}_{\nu }{P}_{\nu }(E(x))\mathrm{.}$ For every $s$ with $\beta (s)>0$ and every configuration of at most four distinct points where $\gamma >0$, parameter differentiation in $z$ maps onto all their spatial 3-jets simultaneously: multiplication by the nonzero smooth scalar $\beta (s)\gamma$ is invertible on each finite Taylor algebra, so the preceding interpolation applies. Restrict $z$ to a sufficiently small open ball. On the fixed product collar $H_{s,z}$ is unchanged; on its transition region the original spatial differential has a uniform positive lower bound, so no critical points appear there. On the compact perturbation support the original $H_s$ has a positive margin from $0,1$, so the range and the endpoint fibers are retained. By openness of Morse nondegeneracy and of the finitely many distinct endpoint critical values, the stationary parameter ends remain excellent even on the portions where $\beta$ begins to be positive. Thus every relevant exceptional event occurs where $\beta \gamma >0$ and the jet evaluation is submersive.

The jet strata, their smoothness, and their codimensions. Write a jet as $(v,d,A,T)$: value, gradient covector, symmetric Hessian matrix, and symmetric third derivative. At $d=0$, $A$ is intrinsic. At $d=0$ and $A$ of corank one, $T$ restricted to the kernel line is also intrinsic: the extra third-derivative chain-rule terms contain either $d$ or $A$ paired with a kernel vector, and vanish. Consequently the following conditions define well-defined smooth strata in the appropriate spatial jet bundles. The locus of symmetric matrices of corank exactly $r$ is smooth of codimension $r(r+1)\mathrm{/}2$. For a direct proof choose an invertible $(n-r)\times (n-r)$ principal block $A_0$ and write the matrix in blocks $\left[\begin{array}{cc}{A}_0& B\\ B^t& C\end{array}\right]$. Its rank is $n-r$ precisely when the symmetric Schur complement $C-B^t{A}_0^{-1}B$ is zero. Its $r(r+1)\mathrm{/}2$ entries are independent equations because differentiating with respect to $C$ gives the identity. Such charts cover the rank stratum (after a fixed linear change of basis where necessary). Hence the stratum $B_r$ defined by $d=0$ and Hessian corank $r$ has codimension $n+r(r+1)\mathrm{/}2$. The corank-one sub-stratum $B_{1,0}$ adding $T{\mathrm{\mid }}_{\ker A}^{\otimes 3}=0$ has codimension $n+2$: choose a smooth local nonzero kernel vector and vary its cubic coefficient independently. The sign of that vector does not affect the zero condition. For simultaneous jets at distinct spatial points at one parameter, the conditions below have the stated codimensions. Gradient conditions at the different points are independent, and value equalities are independent of all gradient/Hessian/third-derivative coordinates. The codimension/source-dimension cases are: One $B_1$; $n+1$; $n+1$. One $B_r$, $r\ge 2$; $n+r(r+1)\mathrm{/}2\ge n+3$; $n+1$. One $B_{1,0}$; $n+2$; $n+1$. Two Morse critical points of equal value; $2n+1$; $2n+1$. One $B_1$ and another critical point, equal values; $2n+2$; $2n+1$. Two $B_1$ points at one parameter; $2n+2$; $2n+1$. Three critical points with one common value; $3n+2$; $3n+1$. One $B_1$ and an equal-value pair at two other points; $3n+2$; $3n+1$. Two equal-value pairs at four distinct points; $4n+2$; $4n+1$. In the last three rows no Hessian restrictions on the ordinary critical points are needed. One may partition into fixed corank strata if a condition is already expressed on a corank locus; there are finitely many possibilities. The first equal-value row is restricted to invertible Hessians, an open condition, so it is an embedded submanifold as written. The $B_1$ rows use the smooth corank-one stratum, not the singular closed determinantal variety.

Application of the actual finite-dimensional transversality theorem. For each row and each relevant $k$, the source is the smooth manifold $\{s:\beta (s)>0\}\times {\mathrm{Conf}}_k(\{x:\gamma (x)>0\}),$ of dimension $kn+1$. The target is the product spatial 3-jet bundle over that source. The evaluation map also depending on $z$ is a submersion in every vertical jet direction by step 1.1, and covers the identity on the spatial base; hence it is transverse to each displayed stratum. Apply [F1] to each of these finitely many maps/strata. The union of their exceptional parameter subsets is null, so its complement meets every parameter ball. Pick one $z$ in that complement and set $F_s=H_{s,z}$. If a target stratum has codimension greater than its configuration source dimension, its transverse inverse image is empty: at a point of the inverse image the differential would have to surject onto a normal space of strictly larger dimension. Thus all the high-codimension rows are avoided. The $B_1$ inverse image and the Morse equal-value-pair inverse image are zero-dimensional smooth submanifolds. This is an application of a supplied finite-dimensional theorem to explicit submersive families, not an application of an unproved infinite-dimensional or multijet version. At a $B_1$ point choose coordinates with Hessian kernel spanned by $u$ and invertible Hessian block in $y$. Let $b={\mathrm{\partial }}_s{d}_{x}F$ and $c={\mathrm{\partial }}_u^{3}F$. To see exactly what transversality says, project the linearization of the gradient onto its kernel covector: its spatial columns vanish there and its parameter column is $b(u)$. In the remaining $n-1$ gradient directions the Hessian is invertible. The differential of the corank equation in the remaining kernel spatial direction is $c$ (terms from differentiating the kernel vector vanish because $Au=0$). Therefore the determinant of the normal derivative to $B_1$ is, up to a nonzero factor, $b(u)c$. Its invertibility implies $b(u)\mathrm{\ne }0$ and $c\mathrm{\ne }0$. The separate avoidance of $B_{1,0}$ is compatible with, and corroborates, this calculation. At an equal-value pair with invertible Hessians, eliminate the spatial normal directions using the two Hessian blocks. The remaining derivative is the derivative of the difference of the two critical values along the parameter. Thus transversality is exactly nonzero crossing speed. The three- and four-point exclusions ensure that there is only one pair at a crossing parameter. The two-$B_1$ and $B_1$-plus-pair exclusions ensure that no birth coincides with another birth or crossing. The $B_1$-plus-equal-value exclusion ensures that a birth value is distinct from every other critical value.

Local splitting and the birth geometry without smooth preparation. Near a $B_1$ event put its parameter and critical point at zero. Apply the actual square-completion lemma successively in the $n-1$ nonkernel directions, treating $(s,u)$ and all remaining coordinates as parameters. The invertible transverse Hessian block remains invertible after each Schur-complement step. This gives smooth parameter-dependent spatial coordinates in which $F_s=g(s,u)+Q(y)$ with $Q$ a fixed signed diagonal quadratic form. At the origin $g_u=g_{uu}=0$, $g_{uuu}\mathrm{\ne }0$, and $g_{su}\mathrm{\ne }0$. These equalities are preserved by the splitting: the transverse first derivatives vanish on the chosen critical transverse section, and the kernel third derivative and the mixed kernel-gradient derivative agree with the original intrinsic quantities up to nonzero coordinate factors. Reverse $u$ if necessary so $g_{uuu}(0,0)>0$. By the implicit function theorem, $g_{uu}(s,u)=0$ has a unique solution $u=m(s)$ near zero. Translate by $m(s)$. Integral Taylor expansion in $u$ gives the exact expression $g(s,u)=a(s)+b(s)u+u^{3}C(s,u),C(s,u)=\frac{1}{2}{\int }_0^1(1-t{)}^2{g}_{uuu}(s,tu)dt\mathrm{.}$ Here the translated function is understood on the right-hand side. In particular $C(0,0)=g_{uuu}(0,0)\mathrm{/}6>0$, $b(0)=0$, and $b^{\mathrm{\prime }}(0)=g_{su}(0,0)\mathrm{\ne }0$, since the shift contributes $g_{uu}{m}^{\mathrm{\prime }}=0$ at the origin. Shrink the neighborhood so that $g_{uuu}>c>0$ and $b^{\mathrm{\prime }}$ has constant nonzero sign. For each fixed small $s$, $g_u$ has a unique strict local minimum at $u=0$: its derivative $g_{uu}$ is strictly increasing and vanishes at zero. If $b(s)>0$ there is no nearby root of $g_u$; if $b(s)<0$ there are exactly two nearby roots, one on either side of zero. To justify existence uniformly, at $u=\pm r$ the integral lower bound $g_u(s,u)\ge b(s)+(c\mathrm{/}2)r^2$ is positive for $\mathrm{\mid }b(s)\mathrm{\mid }<c{r}^2\mathrm{/}4$, and strict decrease/increase on the negative/positive half-interval gives uniqueness. At $b(s)=0$ they merge. Their nonkernel Hessian block is that of $Q$, while $g_{uu}$ is negative at the left root and positive at the right root. Their indices are therefore $\mathrm{ind}Q+1$ and $\mathrm{ind}Q$. Between these roots $g_u<0$, so integrating gives $g(s,u_{+})<g(s,u_{-})$. Consequently the two birth critical values never coincide while both critical points exist. This conclusion uses the actual function, without asserting that a formal cubic Taylor expansion is already an exact coordinate normal form. The spatial-gradient map on $[0,1]\times W$ is transverse to zero: where its Hessian is invertible this is immediate, and at a fold its parameter derivative supplies the missing kernel covector by step 3.1. Its zero set is therefore a smooth curve. At a fold the equation $g_u(s,u)=0$ solves $s=\psi (u)$, since $g_{su}\mathrm{\ne }0$. Differentiating gives ${\psi }^{\mathrm{\prime }}(0)=0$ and ${\psi }^{\mathrm{\prime }\mathrm{\prime }}(0)=-g_{uuu}\mathrm{/}{g}_{su}\mathrm{\ne }0$. Thus its parameter projection is a nondegenerate fold, with the two critical branches just described.

Finiteness, including exclusion of diagonal accumulation. The critical curve is closed in the compact parameter/manifold product and is disjoint from a fixed boundary collar. Thus it is compact, with the ordinary endpoint critical points as its only boundary. The degenerate subset is closed in it, since it is the zero set of the Hessian determinant. It is discrete by transverse $B_1$ intersection and has no higher-corank limit by step 3.1; compactness therefore makes it finite. The equal-value-pair set is discrete but initially lies in a noncompact configuration space, so discreteness alone does not establish finiteness. Suppose infinitely many crossings existed. By compactness of $[0,1]\times W^2$, take an accumulation point $(s,p,q)$. If $p\mathrm{\ne }q$, continuity makes both limiting points critical with equal values. Neither can be degenerate by the $B_1$-plus-equal-value exclusion, and then the equal-value transverse inverse image is locally discrete, a contradiction. If $p=q$ and this point is Morse, the parameter implicit function theorem gives a unique nearby critical branch, so two distinct critical points cannot converge to it. If $p=q$ and it is a fold, step 4.1 gives exactly the two nearby critical branches and proves that their values are unequal. This also contradicts crossing accumulation. Hence there are only finitely many crossings. The stationary endpoint neighborhoods exclude endpoint accumulation. Together with the simultaneous-event exclusions in step 3.1, this proves the stated genericity and finiteness conclusions.

An explicit homotopy making the elementary cubic path. Use the split-and-shifted coordinates of step 4.1. Write $c_0=C(0,0)>0$ and $D(s,u)=u^3(C(s,u)-c_0)$. Let $\alpha$ be a smooth scalar bump equal to one on $[-1\mathrm{/}2,1\mathrm{/}2]$ and zero outside $(-1,1)$, and choose a radial spatial bump $\rho (y\mathrm{/}r)$ equal to one near zero and zero for $\mathrm{\mid }y\mathrm{\mid }\ge r$. Put $g_{\theta ,s}(u,y)=a(s)+b(s)u+u^{3}C(s,u)+Q(y)-\theta \alpha (s\mathrm{/}{r}^2)\alpha (u\mathrm{/}r)\rho (y\mathrm{/}r)D(s,u),0\le \theta \le 1.$ Extend this family by the original path outside the coordinate box. The bump factors make this a smooth compactly supported change; it is supported in $\mathrm{\mid }s\mathrm{\mid }<r^2$, $\mathrm{\mid }u\mathrm{\mid }<r$, $\mathrm{\mid }y\mathrm{\mid }<r$, and hence leaves the endpoint functions untouched. At $\theta =1$ it has constant cubic coefficient $c_0$ in a smaller box. Here are the derivative estimates ensuring that this is a homotopy through the same generic event, rather than a possibly singular interpolation. Smoothness gives $\mathrm{\mid }C(s,u)-c_0\mathrm{\mid }\le K(\mathrm{\mid }s\mathrm{\mid }+\mathrm{\mid }u\mathrm{\mid })$ on a fixed box, while the derivatives of $C$ of each fixed finite order are bounded there. On the shrinking box this gives $D=O(r^4)$, $D_u=O(r^3)$, $D_{uu}=O(r^2)$, $D_{uuu}=O(r)$, and $D_s=O(r^3)$. Differentiating the $s\mathrm{/}{r}^2$ cutoff once and the $u\mathrm{/}r$ cutoff once therefore gives a mixed $(s,u)$ derivative bounded by $O(r)$; three spatial derivatives likewise give $O(r)$. A first $y$ derivative of the correction is $O(r^3)$ and a second is $O(r^2)$, uniformly in $\theta$. All critical points in the support have $y=0$. Indeed, on the region where the $y$ cutoff varies, $\mathrm{\mid }y\mathrm{\mid }$ is bounded below by a positive constant times $r$, so $\mathrm{\mid }{d}_{y}Q\mathrm{\mid }$ is bounded below by a positive constant times $r$, whereas the correction's $y$ derivative is $O(r^3)$. These cannot cancel for small $r$. Where the cutoff equals one or zero its $y$ derivative vanishes and $d_{y}Q=0$ exactly when $y=0$. The transverse Hessian at $y=0$ remains the nondegenerate Hessian of $Q$. On $y=0$, the reduced function still has $g_{\theta ,uu}(s,0)=0$ and $g_{\theta ,u}(s,0)=b(s)$, because the correction has a factor $u^3$ and the $u$ cutoff is constant near zero. Its third $u$ derivative is positive by the uniform $O(r)$ estimate and the lower bound in step 4.1. Thus $g_{\theta ,uu}(s,u)$ has its only zero at $u=0$, and a degenerate critical point requires $b(s)=0$, hence $s=0$. The fold stays at the origin, with the same nonzero cubic and mixed derivatives; each nearby slice has exactly the root pattern of step 4.1. No new birth can occur anywhere in the support. Outside the support nothing changed. Choose the initial neighborhood to isolate this event from the finitely many other event parameters and from the other critical values at its parameter. The correction has size $O(r^4)$, so it retains a strict gap from all those other critical values after shrinking. The two local critical values remain distinct by the strict-minimum argument in step 4.1 for every $\theta$. Thus no crossing is created or destroyed. This proves the stated supported cubic normalization, with all boundary collars fixed.