Parametric index grouping preserves a relative descending path

Exact input. Let V be a compact smooth d-dimensional bordism, with fixed horizontal incoming/outgoing faces and a fixed vertical product collar, on which the slicing function is t. Suppose a smooth one-parameter family (F_s,X_s), constant near the parameter endpoints, is supplied with the following proved properties. Each X_s is descending for F_s, is the fixed vertical product field near the protected collar, has the supplied smooth hyperbolic descending models at regular critical points, and is the exact negative cubic gradient on protected fold charts. No normalization of every regular eigenvalue is assumed. There are finitely many function events and finitely many transverse isolated same-index connecting-orbit events. No orbit joins a critical point of smaller index to one of larger index. Away from births the stable/unstable sections are smooth in the parameter. The actual compactness theorem for trajectories, including the fold endpoint sections, is part of the input. Near every birth the newborn critical points have no same-index connecting orbit to any other critical point. Their small birth block and the retained critical charts may be chosen uniformly. No slice-by-slice transversality of positive-index-drop trajectories is required. The proposed output is a smooth family g_s with the same critical points and indices, the same cubic births, X_s descending for g_s, and all critical values ordered by index. All boundary heights and protected collars are unchanged. Same-index heights may coincide at isolated crossings. Its full relative handle presentation is index grouped. Its actual equal-index trajectory events are unchanged because X_s is unchanged. The field theorem is used precisely through [F1].

A value change with smooth orbit cutoffs. First take a compact regular-parameter neighborhood J with fixed Morse critical charts. At s=s₀ consider two adjacent critical levels, belonging to P and Q, with no connecting orbit from the upper point to the lower. Choose outer regular levels a<b containing just these two critical points. Their stable and unstable trajectory pieces in this slab, closed by their intersections with the outer levels, are compact and disjoint. Indeed the supplied broken-trajectory compactness permits a limit outside the indicated piece only by a broken orbit through the other critical point, which would give the excluded connection. The pieces stay away from the vertical collar, since the product trajectories there cannot approach an interior critical point. These separation properties persist on a smaller parameter neighborhood: a contrary sequence has a broken limit giving the same forbidden connection. Smooth dependence of the field and the Morse sections then applies uniformly. There is a smooth orbit-constant β_s on this slab, equal to one near the P trajectory piece and zero near the Q trajectory piece and the vertical collar. Here are the extension details. On the lower regular face choose a bump equal to one on a neighborhood of the compact P unstable exit section and zero near the Q unstable exit section and the boundary. The sections are disjoint compact sets, so the bump can be chosen uniformly for s near s₀; let it be one on one fixed open neighborhood containing every nearby P exit section. Extend by flow to every trajectory meeting the lower face. Set the extension one on trajectories ending at P and zero on those ending at Q. Near either regular critical trajectory piece the resulting function is constant, without requiring an exact exponential chart. Indeed, if regular points approaching that piece had lower exits outside the chosen constant neighborhood, take convergent parameters and such points. The supplied uniform broken-trajectory compactness gives a limiting broken orbit through the corresponding critical point. The absence of a connection between P and Q, and strict descent excluding a return to the same critical height, force its last segment to leave that same critical point through its unstable exit section. This contradicts the choice of exits outside an open neighborhood of the entire compact exit section. The same argument applies to points on the stable sheet and uniformly for nearby parameters, since the exit neighborhoods were chosen to contain every nearby compact section. Thus the extension is identically one, respectively zero, on a full neighborhood of each critical trajectory piece. There is therefore no nonsmooth quotient at the limiting sheet. On its remaining domain the exit time and exit point are smooth by the implicit function theorem for a regular level. This proves smoothness jointly in s and space. A minimum with no unstable exit has an open attracting set on which the value is set constantly one or zero. Its possible boundary inside the slab would limit to the other critical point; the excluded connection rules that out near the chosen trajectory piece. The same argument applied to the reversed field handles a maximum and an upper-face construction. The extension is zero near the vertical product trajectories because the lower bump is zero there, and the product flow has no critical endpoint there. Thus X_sβ_s=0 everywhere. A cutoff in the level coordinate may be used outside this outer slab; take it only where the height change below is already identity, so it contributes no derivative term. Choose a smooth increasing interval diffeomorphism φ_s of [a,b], identity near its endpoints and affine with positive slope near the P critical value, carrying that value to a desired position still inside (a,b). Define $G_s(u,z)=(1-z)u+z{\varphi }_s(u),{\tilde{F}}_s=G_s(F_s,{\beta }_s)\mathrm{.}$ Then ∂_uG_s>0, and $X_s{\tilde{F}}_s=({\mathrm{\partial }}_u{G}_s)(F_s,{\beta }_s)X_s{F}_s<0$ off the original critical points. Near P the function is φ_s(F_s), near Q it is F_s, and near the vertical collar it is exactly F_s. This changes the P level across Q without changing either critical point or introducing a new one. The full Hessian near P is multiplied by a positive scalar. All ingredients can be chosen smoothly on the smaller parameter neighborhood. The same construction works for a compact critical birth block in place of P when its complete entry/exit pieces are disjoint from Q: use a bump constant one on a neighborhood of all its exit sections, so the extension is constant on a full uniform birth block. The fold entry/exit compactness in the input supplies that neighborhood. Inside the block the change is chosen positive affine in the function value, so the exact cubic model is retained.

Compatible prescribed heights. Away from small disjoint parameter neighborhoods of the births, prescribe $h_p(s)=\frac{\mathrm{ind}(p)+1}{d+2}+ϵF_s(p),$ with one sufficiently small positive ε. The margin 1/(d+2) makes different index groups disjoint. For same-index points the prescription preserves the order of their F_s values, hence respects every actual connecting orbit. Every positive-index-drop orbit also has strictly ordered prescribed heights, and index-increasing orbits are absent by hypothesis. Near a birth of indices k,k+1 take the exact critical block and place its common critical height near (k+3/2)/(d+2). On a smaller parameter neighborhood prescribe a positive affine function of the original cubic there. The two values then differ by a positive constant times the actual cubic value gap, of order s^{3/2}, and fit together into a function smooth in the original (s,u) chart. They lie above all the old index-k values and below all the old index-(k+1) values. Interpolate to the ordinary prescription on the outer part of this birth neighborhood by a smooth nonnegative parameter cutoff, zero on the smaller neighborhood and one outside the larger one. Only where that cutoff is positive do the two new branches have separate uniform Morse charts; the interpolation is therefore smooth there, while near the fold it is exactly affine cubic. The birth neighborhoods are disjoint, so these choices do not conflict. There are no same-index orbits incident to these two branches on the neighborhood in question. Their reordered heights consequently violate no orbit inequality; positive index drops still go from an upper group to a lower group. The ordering within every old index group continues to be the original F_s ordering. These prescriptions are strictly compatible with the partial order generated by the actual connecting orbits. Broken chains impose no additional inequalities, because inequalities compose. Values at a same-index function crossing may agree, but there is no orbit between equal-height critical points by strict descent. Those pairs can be kept together as a two-point value cluster during the construction.

A finite transverse list of critical-value events. The interpolated heights need not initially have only finitely many crossings. Here is the additional finite-dimensional construction. Protect smaller closed parameter neighborhoods of the folds and of the stationary endpoints. On a sufficiently small fold neighborhood the two birth heights lie in their common middle gap, are unequal away from the fold, and are separated from every other critical height of their respective index; the old same-index heights are distinct there because the original function path has no second event at the fold. On endpoint neighborhoods the original excellent heights are stationary and distinct within each index group. Thus there is no unaccounted crossing on these protected neighborhoods. There are finitely many same-index trajectory parameters by the field input. Choose disjoint small regular neighborhoods of them, missing the protected fold and endpoint neighborhoods. At the center of each, separate any coincident prescribed heights by independent arbitrarily small shifts in the corresponding critical charts. Preserve the strict height gap of the one actual connecting pair by choosing the shifts smaller than one quarter of that gap; preserve the gaps between distinct index groups as well. There are no other same-index trajectories in a sufficiently small chosen neighborhood. The shifts can therefore be extended smoothly with a parameter bump there without violating any trajectory inequality. After shrinking to a smaller neighborhood of its center, all critical heights are pairwise distinct there. Protect that smaller closed neighborhood and the adjusted heights on it. These finitely many preliminary shifts change no birth model or endpoint. They retain every actual orbit and its direction, since X_s has not changed. The remaining possible crossing set is compact and contained in the ordinary parameter region with finitely many regular critical branches. Cover it by finitely many parameter intervals with compact closure in that region. For each such interval choose a parameter bump χ equal to one on a smaller interval, and independently for each critical branch on it allow additions $a_p\chi (s)+b_p(s-s_0)\chi (s)\mathrm{.}$ Extend them by zero off that interval and that branch's parameter chart. On the smaller interval their value and first derivative can be prescribed independently. The union of this finite list is one finite-dimensional height family. For a same-index pair p,q its difference evaluation is a submersion in these coefficient directions wherever its zero can occur. Its value/first-derivative jet also has the two independent coefficient directions. For two different pair equalities at one time the value evaluations have rank two, including the case of a shared branch (a triple height). Apply the actual finite-dimensional parametric-transversality theorem to the finitely many single-pair value maps and the finitely many two-pair maps. A generic sufficiently small coefficient vector makes every single-pair difference transverse to zero, and excludes the simultaneous vanishing of two different pair differences because its codimension is two on a one-dimensional parameter source. This is the finite critical-height jet/Sard step; no genericity of a newly chosen field is assumed. All perturbations have support outside the protected neighborhoods. Choose their coefficients small enough that no new zero occurs outside the chosen smaller intervals, whose complement had a positive value-gap margin; enlarge the finite cover before making this choice if necessary. Choose them also small enough to retain the positive index-group gaps. The only same-index connecting-orbit parameters lie in the protected neighborhoods, where their strict prescribed inequalities have been left unchanged. Hence every actual trajectory inequality is preserved. The birth height formulas and endpoint heights are unchanged. Every resulting same-index pair difference has a closed zero set in this compact ordinary parameter region, with no zero near its protected boundary. Its zeros are transverse and hence isolated, so compactness makes them finite. Two such zeros cannot occur at the same parameter by the two-pair condition. There are no triple values, no coincident two crossings, and no crossing at any same-index trajectory event. The only remaining critical-value coincidence outside this finite list is the intended birth itself. At any adjusted crossing there is no connecting orbit, precisely because all trajectory-event neighborhoods were protected. The adjusted prescribed heights satisfy the same trajectory inequalities. The local orbit-cutoff value changes already proved in step 2.1 can therefore be used in their realization, with the finite parameter patching below. This gives the asserted finite event list while retaining the same field and its terminal maps.

Local realization and parameter patching. At a fixed parameter, the original and prescribed orders are two linear extensions of this trajectory partial order, allowing the just-described clusters. A finite sequence of exchanges of adjacent incomparable elements transforms one extension to the other: move the first element of the target extension to the first position of the source by successive exchanges; every passed element is incomparable because both extensions respect the order; then induct on the remaining list. Each exchange has no trajectory between the two pieces and is realized by step 2.1. Choose the local target gaps slightly larger than needed, then adjust all ordered values to the exact prescribed values by increasing reparametrizations in disjoint value intervals, again with orbit cutoffs zero near the vertical collar. The same construction is smooth on a sufficiently small parameter neighborhood. If an original value crossing occurs at its center, first use a small smooth local value separation of the two critical charts, whose supports are disjoint; they have no connecting orbit near that crossing. If the prescribed heights cross within one index group, the final two local chart values are simply the prescribed smooth functions, and the same no-connection value adjustment realizes their equality. No exchange across a connecting orbit is performed. A birth block is treated as one piece until its value is in the prescribed gap between the old k and k+1 groups, using the block version of step 2.1. Other critical values can be moved past this block only when the target inequalities require it. The no-increasing-orbit condition and the stated absence of same-index birth incidences then exclude any connecting orbit between the two pieces. In a smaller parameter neighborhood the function is exactly positive affine on the whole birth block, as required in step 3.1. These operations give a local smooth family with all prescribed critical heights and no other critical points. Cover the compact parameter interval by finitely many such neighborhoods, taking a single birth neighborhood for each fold. Each local realization is positive affine in F_s near every ordinary critical chart, with its constant term determined by the common prescribed h_p(s). Shrink the chart radii uniformly. At a birth choose the affine function on the common smaller birth block once and require all local realizations meeting that parameter neighborhood to equal it there; this is possible because subsequent value changes have support away from that block. Take a smooth partition of unity ρ_α(s) subordinate to the finite parameter cover and set $g_s(x)={\sum }_{\alpha }{\rho }_{\alpha }(s)g_s^{(\alpha )}(x)\mathrm{.}$ Then X_sg_s is a convex sum of strictly negative numbers at every noncritical point. No new critical point occurs. On an ordinary critical chart it is a positive affine function of F_s, with exactly the prescribed critical value. On a smaller birth block it is the common positive-affine cubic. Near every protected face all summands equal the original fixed height, so g_s has that height. Smoothness follows from the finite smooth sum. Choose the cover and partition stationary on endpoint collars. This proves the proposed parametric grouping conclusion from the exact field input.

The retained framed data and maps. Use X_s to carry the local framed unstable disks to regular levels of g_s. Because X_s is unchanged and g_s is descending for it, its unparametrized trajectories and the actual terminal entry/exit maps are unchanged. Changing a regular section only inserts the explicit regular-flow comparison. Thus the index grouping retains the framed elementary attachments and outgoing parametrizations, including through positive-index-drop tangencies. The same field therefore retains each finite isolated trajectory passage and each disjoint value-crossing comparison. Their interpretation in a full based handle movie is a separate construction, and is not a premise of the present grouping lemma.