Isotopy extension for a compact source with boundary

The product $V\times I$ is compact: for an open cover and each time, compactness of $V$ supplies finitely many product neighbourhoods covering that time slice; intersect their time intervals to obtain a neighbourhood of that time, and compactness of $I$ supplies finitely many such neighbourhoods. Thus a finite subcover exists. The image $K=F(V\times I)$ is compact, being the continuous image of the compact space $V\times I$, and $K\subseteq W$; every point of $K$ has by [F6] an open neighbourhood with compact closure contained in $W$, finitely many of these cover $K$, and their union $V^{\mathrm{\prime }}$ is an open neighbourhood of $K$ with $\overline{V^{\mathrm{\prime }}}$ compact and $\overline{V^{\mathrm{\prime }}}\subseteq W$.

Extend $F$ in the time direction by $\tilde{F}(t,x)=F(x,0)$ for $t\le 0$, $\tilde{F}(t,x)=F(x,t)$ for $0\le t\le 1$ and $\tilde{F}(t,x)=F(x,1)$ for $t\ge 1$: the prescriptions agree on the overlaps because $F$ is constant for $t\le \epsilon$ and for $t\ge 1-\epsilon$, so $\tilde{F}:\mathbb{R}\times V\to N$ is smooth and each ${\tilde{F}}_t$ is a smooth embedding; let $\mathrm{\Theta }:\mathbb{R}\times V\to \mathbb{R}\times N$, $\mathrm{\Theta }(t,x)=(t,\tilde{F}(t,x))$, be the graph map.

The graph map is an injective immersion with invertible differential at every point: injectivity is immediate from the first coordinate, and at $(t_0,x_0)$ a boundary chart of $V$ at $x_0$ and a chart of $N$ at $\tilde{F}(t_0,x_0)$ present the coordinate representative of $\mathrm{\Theta }$ as a map smooth on a relatively open subset of a half-space in the sense of [F2], hence as the restriction of a smooth map $\mathrm{\Phi }$ defined near $(t_0,u(x_0))$ in an open set; the differential of ${\tilde{F}}_{t_0}$ at $x_0$ is invertible by [F1] because ${\tilde{F}}_{t_0}$ is an embedding between $n$-manifolds, so the differential of $\mathrm{\Phi }$ there is invertible and [F3] restricts $\mathrm{\Phi }$ to a local diffeomorphism, exhibiting $\mathrm{\Theta }$ locally as the restriction of an ambient diffeomorphism to the source half-space. At a boundary point its image is a half-space neighbourhood, not an ambient open set; in the interior it is open.

The graph map is proper: for a compact $L\subseteq \mathbb{R}\times N$ the time projection ${\pi }_1(L)$ is compact, ${\mathrm{\Theta }}^{-1}(L)$ is closed in the compact set ${\pi }_1(L)\times V$ by continuity and closedness of $L$, hence ${\mathrm{\Theta }}^{-1}(L)$ is compact; a proper continuous map into this locally compact Hausdorff target is closed: for a closed source subset $A$ and $y$ outside its image, choose a compact target neighbourhood $L$ of $y$; the image of $A\cap {\mathrm{\Theta }}^{-1}(L)$ is compact and hence closed in the Hausdorff target, and deleting it from the interior of $L$ gives a neighbourhood of $y$ missing the image of $A$. Therefore $\mathrm{\Theta }$ is closed, its image $S:=\mathrm{\Theta }(\mathbb{R}\times V)$ is closed in $\mathbb{R}\times N$ and $\mathrm{\Theta }$ is a homeomorphism onto $S$ whose inverse is smooth by step 2.1, so $S$ is an embedded smooth submanifold with boundary of $\mathbb{R}\times N$ in the sense of [F11] with $\mathrm{\partial }S=\mathrm{\Theta }(\mathbb{R}\times \mathrm{\partial }V)$.

Define the horizontal velocity along the graph by placing $Y(\mathrm{\Theta }(t,x)):=(0,{\mathrm{\partial }}_t\tilde{F}(t,x))$ in $\{0\}\oplus T_{\tilde{F}(t,x)}N\subseteq T_{(t,\tilde{F}(t,x))}(\mathbb{R}\times N)$: the assignment is well defined because $\mathrm{\Theta }$ is injective and smooth because ${\mathrm{\Theta }}^{-1}$ is smooth by step 3.1, it is a horizontal smooth field along $S$ in the sense of [F10], and $Y=0$ at every point of $S$ whose first coordinate lies outside $[0,1]$, because $\tilde{F}$ is constant in $t$ there.

At every point $q\in S$ the field $Y$ extends over an open neighbourhood in $\mathbb{R}\times N$ to a smooth horizontal field: choose $(t_0,x_0)={\mathrm{\Theta }}^{-1}(q)$ and, by step 2.1, an open neighbourhood $U$ of $(t_0,x_0)$ in $\mathbb{R}\times V$ mapped diffeomorphically onto ${\mathrm{\Omega }}_0:=\mathrm{\Theta }(U)$; shrink $U$ so that $\tilde{F}(t,x)\in V^{\mathrm{\prime }}$ for all $(t,x)\in U$, possible by continuity because $\tilde{F}(t_0,x_0)\in K\subseteq V^{\mathrm{\prime }}$. If $x_0\in \mathrm{int}V$ then ${\mathrm{\Omega }}_0$ is open in $\mathbb{R}\times N$ and ${\tilde{Y}}_q(\mathrm{\Theta }(t,x)):=(0,{\mathrm{\partial }}_t\tilde{F}(t,x))$ defines on it a smooth horizontal field restricting to $Y$ on $S\cap {\mathrm{\Omega }}_0$. If $x_0\in \mathrm{\partial }V$ then ${\mathrm{\Omega }}_0$ is only a half-space neighbourhood of $q$, but in boundary charts of $V$ the horizontal components of $Y$ are smooth functions on that half-space model, so by the local-extension convention of [F2] they extend smoothly to an open neighbourhood of $q$ in $\mathbb{R}\times N$ while the zero first component extends by zero, giving a smooth horizontal field on a neighbourhood $\mathrm{\Omega }$ that restricts to $Y$ on $S\cap \mathrm{\Omega }$; in both cases $\mathrm{\Omega }$ may be shrunk to lie in $\mathbb{R}\times V^{\mathrm{\prime }}$.

The compact set $S_0:=\mathrm{\Theta }([0,1]\times V)\subseteq S$ is covered by finitely many neighbourhoods ${\mathrm{\Omega }}_{q_1},\dots ,{\mathrm{\Omega }}_{q_m}$ from step 5.1, each contained in $\mathbb{R}\times V^{\mathrm{\prime }}$; let $({\psi }_0,{\psi }_1,\dots ,{\psi }_m)$ be a smooth partition of unity on $\mathbb{R}\times N$ subordinate to the open cover $\{\mathbb{R}\times N\setminus S_0,{\mathrm{\Omega }}_{q_1},\dots ,{\mathrm{\Omega }}_{q_m}\}$, which exists by [F4], and define $\tilde{Y}:={\sum }_{i=1}^m{\psi }_i{\tilde{Y}}_{q_i}$ with each term extended by zero outside ${\mathrm{\Omega }}_{q_i}$; the sum is smooth because $\mathrm{supp}{\psi }_i\subseteq {\mathrm{\Omega }}_{q_i}$, it takes values in the horizontal subbundle and so is a time-dependent vector field $\tilde{Y}(t,y)=X(t,y)\in T_{y}N$ on $N$ over $\mathbb{R}$ in the sense of [F10], and for $q\in S$ one has $\tilde{Y}(q)={\sum }_i{\psi }_i(q)Y(q)=(1-{\psi }_0(q))Y(q)=Y(q)$, because ${\psi }_0(q)\mathrm{\ne }0$ forces $q\mathrm{\notin }{S}_0$ and then $Y(q)=0$ by step 4.1; finally $\mathrm{supp}\tilde{Y}\subseteq \mathbb{R}\times \overline{V^{\mathrm{\prime }}}$ because each ${\mathrm{\Omega }}_{q_i}\subseteq \mathbb{R}\times V^{\mathrm{\prime }}$.

By [F5] choose a smooth function $\beta :\mathbb{R}\to [0,1]$ with $\beta =1$ on $[\epsilon ,1-\epsilon ]$ and $\beta =0$ outside $(\epsilon \mathrm{/}2,1-\epsilon \mathrm{/}2)$, and put $X_t^{\mathrm{\prime }}:=\beta (t)X_t$ for $t\in [0,1]$; then ${\bigcup }_{t\in [0,1]}\mathrm{supp}{X}_t^{\mathrm{\prime }}$ is contained in the compact subset $\overline{V^{\mathrm{\prime }}}\subseteq N$, so [F9] provides a global evolution operator ${\mathrm{\Psi }}_{t,s}:N\to N$ for $s,t\in [0,1]$.

The isotopy identity: fix $x\in V$ and put $\gamma (t):=F_t(x)$ for $t\in [0,1]$; then $\gamma (0)=F_0(x)$ and ${\gamma }^{\mathrm{\prime }}(t)={\mathrm{\partial }}_{t}F(t,x)$ equals $X_t^{\mathrm{\prime }}(\gamma (t))$ for every $t$, because on $[\epsilon ,1-\epsilon ]$ one has $\beta =1$ and $X_t(\gamma (t))={\mathrm{\partial }}_t\tilde{F}(t,x)={\mathrm{\partial }}_{t}F(t,x)$ by step 6.1, while off $[\epsilon ,1-\epsilon ]$ the derivative ${\mathrm{\partial }}_{t}F(t,x)$ vanishes and is multiplied by $\beta (t)\in [0,1]$; the curve $t\mapsto {\mathrm{\Psi }}_{t,0}(F_0(x))$ solves the same equation with the same initial value by the defining property of the evolution operator in [F10], both curves are defined on all of $[0,1]$, and the local uniqueness in [F7] makes them agree near every point of the connected interval, so $H_t\circ F_0=F_t$ for $H_t:={\mathrm{\Psi }}_{t,0}$.

The remaining properties: $H_0={\mathrm{\Psi }}_{0,0}={\mathrm{id}}_N$ and each $H_t$ is a diffeomorphism with inverse ${\mathrm{\Psi }}_{0,t}$, since the cocycle law of [F8] gives ${\mathrm{\Psi }}_{0,t}\circ {\mathrm{\Psi }}_{t,0}={\mathrm{\Psi }}_{0,0}={\mathrm{id}}_N$ and ${\mathrm{\Psi }}_{t,0}\circ {\mathrm{\Psi }}_{0,t}={\mathrm{\Psi }}_{t,t}={\mathrm{id}}_N$; the map $(t,y)\mapsto H_t(y)$ is smooth because near every $(t_0,y_0)$ it agrees by [F7] with the local smooth evolution map of $X^{\mathrm{\prime }}$ through the point $H_{t_0}(y_0)$ at time $t_0$; if $y\mathrm{\notin }\overline{V^{\mathrm{\prime }}}$ then $X_t^{\mathrm{\prime }}(y)=0$ for all $t$ by step 6.1, so the constant curve at $y$ solves the equation of $X^{\mathrm{\prime }}$, [F7] gives $H_t(y)=y$, and hence $H_t={\mathrm{id}}_N$ outside $W$ because $\overline{V^{\mathrm{\prime }}}\subseteq W$; finally $X_t^{\mathrm{\prime }}=0$ for $t\le \epsilon \mathrm{/}2$ and for $t\ge 1-\epsilon \mathrm{/}2$, so the cocycle law gives $H_t={\mathrm{id}}_N$ for $t\le \epsilon \mathrm{/}2$ and $H_t={\mathrm{\Psi }}_{t,1-\epsilon \mathrm{/}2}\circ H_{1-\epsilon \mathrm{/}2}=H_{1-\epsilon \mathrm{/}2}=H_1$ for $t\ge 1-\epsilon \mathrm{/}2$.