Gradient-like perturbation separates adjacent critical levels

If $n=0$, regularity gives $f^{-1}(v)=\mathrm{\varnothing }$ and [F2] gives empty crossing sets. Take $X^{\mathrm{\prime }}=X$: every trajectory is constant, so there is no connection between the distinct levels, all required disjointness holds, and the field change is zero. Henceforth assume $n\ge 1$. Since $f^{-1}(v)$ is compact and $U$ is an open neighbourhood of it, choose $a\in (c,v)$ such that the closed band $K:=f^{-1}[a,v]$ is contained in $U$; every value in the open interval $(c,c^{\mathrm{\prime }})$ is regular because $P$ and $Q$ are consecutive critical levels, so $a$ and $v$ are regular. The band $K$ is compact and contains no critical point, and $f^{-1}(v)$ is a closed embedded $(n-1)$-manifold.

For $q\in Q$ and $p\in P$ let $A_q,B_p\subseteq f^{-1}(v)$ be the crossing spheres of [F2]. They are compact embedded spheres, possibly empty, and by [F2] each carries a product neighbourhood in $f^{-1}(v)$. Moreover the spheres $A_q$ with $q\in Q$ are pairwise disjoint, and so are the spheres $B_p$ with $p\in P$: a point of $f^{-1}(v)$ lies on a unique trajectory, and a trajectory has at most one limit in each of the two time directions, so the critical point whose local disk the trajectory passes through is determined by the point.

Compute the dimensions: $\dim A_q=\mathrm{ind}(q)-1$ when $\mathrm{ind}(q)\ge 1$ and $A_q=\mathrm{\varnothing }$ when $\mathrm{ind}(q)=0$, while $\dim B_p=n-\mathrm{ind}(p)-1$ when $\mathrm{ind}(p)\le n-1$ and $B_p=\mathrm{\varnothing }$ when $\mathrm{ind}(p)=n$. For a nonempty pair, $\dim A_q+\dim B_p=(\mathrm{ind}(q)-1)+(n-\mathrm{ind}(p)-1)=n-2-(\mathrm{ind}(p)-\mathrm{ind}(q))\le n-2<n-1=\dim f^{-1}(v)$, because $\mathrm{ind}(p)\ge \mathrm{ind}(q)$ by hypothesis.

For a finite pairwise disjoint compact family $A_i$ and finite family $B_j$ satisfying the dimension inequalities, choose disjoint product tubes of the $A_i$. In the projection construction of [F3], avoid the union of the finitely many projected $B_j$ inside each tube: every projected image is null, their finite union is null, and its complement is dense. With a fixed product-tube cutoff, a single arbitrarily small nonzero translation therefore avoids every $B_j$ at once, and its compactly supported flow stays inside the tube. Composing the flows on the disjoint tubes gives an isotopy $l$ with $l(A_i)\cap B_j=\mathrm{\varnothing }$ for every pair. Set $h=l^{-1}$; then $A_i\cap h(B_j)=\mathrm{\varnothing }$. By the fixed-cutoff parameter-flow argument in [F3], the whole isotopy and its inverse may be chosen arbitrarily $C^{\mathrm{\infty }}$-close to the identity. Unlike successive moves against different $B_j$ in the same tube, this argument preserves every avoidance condition.

Apply step 1.4 on $V=f^{-1}(v)$ with $A_i$ the nonempty spheres $A_q$ and $B_j$ the nonempty spheres $B_p$, and prescribed neighbourhood $N$ a tubular neighbourhood of ${\bigcup }_q{A}_q$ in $f^{-1}(v)$. The dimension inequality is step 1.3, and the product neighbourhoods and the pairwise disjointness are step 1.2; hence there is a diffeomorphism $h$ of $f^{-1}(v)$, isotopic to the identity and compactly supported, with $A_q\cap h(B_p)=\mathrm{\varnothing }$ for all $q,p$.

Realize $h$ by a perturbation of the field. The isotopy from the identity to $h$ constructed in step 1.4 has compact support, so the construction in the proof of [F4] applies inside the compact regular band $K=f^{-1}[a,v]$ with the isotopy of the level $f^{-1}(v)$ and produces a complete downward gradient-like field $X^{\mathrm{\prime }}$ for $f$, equal to $X$ outside $f^{-1}(a,v)\subseteq K\subseteq U$, whose trajectory transport ${\phi }^{\mathrm{\prime }}$ from level $a$ to level $v$ satisfies ${\phi }^{\mathrm{\prime }}=h\circ \phi$, where $\phi$ is the transport of $X$. In particular $X^{\mathrm{\prime }}$ is again an adapted field: it equals $X$ near $\mathrm{\partial }W$, where it still points outward along $M_0$ and inward along $M_1$, it has the same critical points as $X$ because $f$ is unchanged and no point of $f^{-1}(a,v)$ is critical, and it has a complete collar carrier. Extend its compact interior field change by zero to the carrier of $X$ and multiply the ambient result by a cutoff equal to one near $W$ and compactly supported in a relatively compact carrier neighborhood. It is complete by [F8] and restricts to $X^{\mathrm{\prime }}$. This replaces the global completeness step of [F4]'s construction; its level-flow and isotopy construction uses only the interior band. With this band and its normalized coordinates fixed, the field is $\lambda F_{\ast }(-{\mathrm{\partial }}_s)$ as in [F4]. As the level isotopy tends to the identity in $C^{\mathrm{\infty }}$, this formula tends to $X$ in $C^{\mathrm{\infty }}$ on the compact band, including its fixed endpoint neighbourhoods. Therefore the interior change can be made arbitrarily small.

Identify the new crossing spheres. Write $T:f^{-1}(v)\to f^{-1}(a)$ for the map that follows the $X$-trajectory downwards from level $v$ to level $a$, so that $B_p=T^{-1}(B_p^a)$, where $B_p^a$ is the crossing set of the $E_p$-trajectories at level $a$. The new crossing sphere of $p$ is the preimage of $B_p^a$ under the new downward transport from $v$ to $a$, which is the inverse of ${\phi }^{\mathrm{\prime }}=h\circ \phi$; hence $B_p^{\mathrm{\prime }}=({\phi }^{-1}\circ h^{-1}{)}^{-1}(B_p^a)={\phi }^{\mathrm{\prime }}(B_p^a)=h(\phi (B_p^a))=h(T^{-1}(B_p^a))=h(B_p)$. For $q\in Q$ the crossing sphere is unchanged, $A_q^{\mathrm{\prime }}=A_q$, because $X^{\mathrm{\prime }}=X$ above the level $v$: the trajectory through a point of $f^{-1}(v)$ agrees with the old one above that level, so its past limit is the same for $X^{\mathrm{\prime }}$ and for $X$.

A trajectory of $X^{\mathrm{\prime }}$ has a limit in $Q$ exactly when it passes through the local unstable disk of that limit and a limit in $P$ exactly when it passes through the local stable disk of that limit, by the same local model argument as in [F2]; the field $X^{\mathrm{\prime }}$ is complete and adapted by step 3.1, so the correspondence of [F2] applies to the pair $(f,X^{\mathrm{\prime }})$. Hence a trajectory of $X^{\mathrm{\prime }}$ whose limits lie in $Q$ and $P$ crosses $f^{-1}(v)$ exactly once, at a point of $A_q^{\mathrm{\prime }}\cap B_p^{\mathrm{\prime }}=A_q\cap h(B_p)$, and this set is empty by step 2.1. Therefore no trajectory of $X^{\mathrm{\prime }}$ has one limit in $P$ and the other in $Q$.

In the intermediate regular band the two trajectory sets are disjoint compact flow tubes over the disjoint crossing spheres. In a larger compact band containing just $P,Q$, their closures are still disjoint: any additional limiting critical trajectory would be a broken connection between $Q$ and $P$, and step 5.1 excludes such connections; the exact Morse-chart flow gives this compactness argument, as detailed in the interchange lemma's Proof 1.1–2.1. Thus the field change supplies the no-connection hypothesis needed for interchange.