The isotopy extension theorem

Clause 1, construction: by [L5] choose a relatively compact open neighbourhood $W^{\mathrm{\prime }}\subseteq W$ of the compact image $F(M\times I)$ with $\overline{W^{\mathrm{\prime }}}$ compact and $\overline{W^{\mathrm{\prime }}}\subseteq W$. Apply [L2] with $W^{\mathrm{\prime }}$ to obtain a smooth time-dependent field $G$ on $N$ whose supports lie in one compact subset of $W^{\mathrm{\prime }}$, with $G_t(F(x,t))={\mathrm{\partial }}_{t}F(x,t)$ and $G_t=0$ for $t\le \epsilon \mathrm{/}2$ and $t\ge 1-\epsilon \mathrm{/}2$, and apply [L3] to obtain its global evolution operator and the compactly supported ambient isotopy $H_t={\mathrm{\Psi }}_{t,0}$.

For clause 2 put $O:=\bar{F}(U\times I)$, the open track image. Each slice differential is an isomorphism, so in product coordinates the track has invertible block differential. The Euclidean inverse function theorem, applied to local extensions at the time endpoints, gives a smooth local inverse preserving time; injectivity makes these inverses agree on $O$. Thus $Z(\bar{F}(x,t)):={\mathrm{\partial }}_{t}F(x,t)$ is a smooth horizontal field on $O$. The compact set $C:=\bar{F}(A\times I)$ has a relatively compact open neighbourhood $O^{\mathrm{\prime }}\subseteq O$ with compact closure contained in $O$, by [L5]. Choose a smooth cutoff $\chi$ equal to one near $C$, with support in $O^{\mathrm{\prime }}$, by [L6]. Define $G=\chi Z$ on $O$ and zero outside $\mathrm{supp}\chi$. This zero extension is smooth; the projection of $\mathrm{supp}\chi$ to $N$ is compact and contains every slice support. By [L3] its evolution gives a compactly supported ambient isotopy.

Clause 4, field construction without endpoint stationarity: let $F:M\times I\to N$ be any smooth isotopy of the compact $M$, possibly with boundary. Its track $S$ and horizontal velocity $Y$ are still compact and smooth by [F2], which does not require stationarity. The local extension construction in [L1] applies on the finite interval itself: at an endpoint, smoothness in a product boundary chart means restriction of a smooth map across that endpoint, and injectivity of the track differential persists locally, so the same graph-coordinate extension of the velocity components is smooth up to $t=0,1$. Restricting each extension to $N\times I$ and patching by the partitions of [L6] gives a smooth horizontal field $Z$ on an open neighbourhood of $S$. By [L5] choose a relatively compact neighbourhood of $S$ inside that neighbourhood and $W\times I$, and by [L6] a cutoff $\rho$ equal to one near $S$ with compact support there. Define $G=\rho Z$ on its domain and zero outside its support. The zero extension is smooth, $G_t(F(x,t))={\mathrm{\partial }}_{t}F(x,t)$ including both endpoints, and its spatial support lies in a compact subset of $W$. No time reparametrization or vanishing end velocity is needed.

Clause 1, the identity $H_t\circ F_0=F_t$: fix $x\in M$. The curve $t\mapsto F(x,t)$ satisfies $\frac{d}{dt}F(x,t)={\mathrm{\partial }}_{t}F(x,t)=G_t(F(x,t))$ by the defining property of $G$, and the curve $t\mapsto H_t(F_0(x))$ satisfies the same equation with the same initial value $F_0(x)=F(x,0)$ by the defining ODE of the flow. By uniqueness of integral curves [L4] the two curves agree for every $t$.

Clause 1, support and stationarity: a point outside $\overline{W^{\mathrm{\prime }}}$ lies outside ${\bigcup }_t\mathrm{supp}{G}_t$, so its integral curve is constant and $H_t={\mathrm{i}\mathrm{d}}_N$ there; in particular $H_t={\mathrm{i}\mathrm{d}}_N$ outside $W$, as $\overline{W^{\mathrm{\prime }}}\subseteq W$. For $t\le \epsilon \mathrm{/}2$ one has $G\equiv 0$ on $[0,t]$, so ${\mathrm{\Psi }}_{t,0}={\mathrm{i}\mathrm{d}}_N$ and $H_t={\mathrm{i}\mathrm{d}}_N$; for $t\ge 1-\epsilon \mathrm{/}2$, $G$ vanishes on $[1-\epsilon \mathrm{/}2,t]$, so ${\mathrm{\Psi }}_{t,1-\epsilon \mathrm{/}2}={\mathrm{i}\mathrm{d}}_N$ and hence $H_t={\mathrm{\Psi }}_{t,1-\epsilon \mathrm{/}2}\circ H_{1-\epsilon \mathrm{/}2}=H_{1-\epsilon \mathrm{/}2}$, the maps being diffeomorphisms. This is clause 1.

Clause 2, the identity near $A$: by construction $\chi =1$ on a neighbourhood of $\bar{F}(A\times I)$ in $N\times I$, so compactness of $I$ gives an open neighbourhood $A^{\mathrm{\prime }}$ of $A$ in $U$ with $\bar{F}(A^{\mathrm{\prime }}\times I)\subseteq \{\chi =1\}$. Indeed the preimage of the open set where $\chi =1$ contains $A\times I$; finitely many product neighbourhoods covering each $\{a\}\times I$ supply one source neighbourhood of $a$, and their union over $a\in A$ gives $A^{\mathrm{\prime }}$. For $x\in A^{\mathrm{\prime }}$ the curve $t\mapsto F(x,t)$ solves the ODE of the field $Z$ and hence of $G$; the curve $t\mapsto H_t(F_0(x))$ solves the same equation with the same initial value, so [L4] gives $H_t(F_0(x))=F_t(x)$ for all $t$ and all $x\in A^{\mathrm{\prime }}$. This is clause 2.

Apply [L3] to this field on $I=[0,1]$ to obtain $H_t={\mathrm{\Psi }}_{t,0}$. Its inverse is ${\mathrm{\Psi }}_{0,t}$, it starts at the identity, and it fixes the complement of $W$. For each $x$, both $F(x,t)$ and $H_t(F_0(x))$ solve the same initial-value problem; [L4] therefore gives $H_t\circ F_0=F_t$ for every $t\in I$. This proves clause 4, including smoothness at the original endpoints. Stationarity of the extension is claimed only when the given isotopy is stationary, as proved for clause 1.

For the boundary-valued track use the tangent extension of [L1] and restricted product-chart cutoffs; multiplication and zero extension preserve boundary tangency. The boundary-tangent evolution argument in [L3] then supplies diffeomorphisms of $N$ preserving $\mathrm{\partial }N$ in both time directions. The ODE comparison of step 2.1 still gives $H_t\circ F_0=F_t$. If the track is interior-valued, perform the compact construction in $\mathrm{Int}N$, with support in a compact subset of $W\cap \mathrm{Int}N$, and extend the resulting diffeotopy by the identity near $\mathrm{\partial }N$. This proves clause 3 without applying a boundaryless flow theorem directly to a manifold with boundary.

The four clauses have been established, with countable choice used in the stated extension and cutoff constructions.