Isotopy Extension and Embedding Theory Beyond Whitney

The track $\bar{F}$ is smooth by [L4], its two components being $F$ and the projection $M\times I\to I$, which are smooth. It is injective: if $\bar{F}(x,t)=\bar{F}(x^{\mathrm{\prime }},t^{\mathrm{\prime }})$ then reading the second coordinate gives $t=t^{\mathrm{\prime }}$ and the first gives $F_t(x)=F_t(x^{\mathrm{\prime }})$, so $x=x^{\mathrm{\prime }}$ because $F_t$ is injective by [F2].

The track is an immersion. Let $(v,a)\in T_{x}M\oplus T_{t}I$ satisfy $d{\bar{F}}_{(x,t)}(v,a)=0$. Applying $d{\pi }_I$ and [L5] to ${\pi }_I\circ \bar{F}={\mathrm{p}\mathrm{r}}_I$ gives $a=0$; then applying $d{\pi }_N$ to ${\pi }_N\circ \bar{F}=F$ gives $d(F_t{)}_x(v)=0$, so $v=0$ because $F_t$ is an immersion. Hence $d{\bar{F}}_{(x,t)}$ is injective for every $(x,t)$.

The track is proper: $M\times I$ is compact by [L6] and $\bar{F}$ is continuous by [F1] and [L5]. For a compact $K\subseteq N\times I$, the target being Hausdorff by [L6], $K$ is closed, so ${\bar{F}}^{-1}(K)$ is closed in the compact space $M\times I$ and hence compact by [L6].

A continuous injective map from the compact space $M\times I$ into the Hausdorff space $N\times I$ is a homeomorphism onto its image: it sends closed sets to compact, hence closed, sets by [L6]. With step 1.2 this makes the track an embedding. For its smooth inverse, choose source coordinates $u$ in a Euclidean open set or half-space of dimension $\dim M$ and target coordinates $y$ near one track point. Select $\dim M$ target components $p(y)$ for which $D_u(p\circ F)$ is invertible. The map $(u,t)\mapsto (p(F(u,t)),t)$ has invertible block differential. Locally extend its coordinate functions across any source boundary face and time endpoint and apply [L2]'s inverse function theorem in Euclidean open sets. Its smooth inverse recovers $(u,t)$ from $(p(y),t)$ along the track. The other target components are smooth functions of these coordinates, giving the local graph model and the smooth inverse on $S$, including its source boundary and endpoint faces and their intersections. If $N$ has boundary, extend its coordinate functions in Euclidean space for this calculation and then restrict back; no neatness or boundaryless slice theorem is invoked. Finally $S$ is compact and therefore closed by [L6].

The velocity $Y$ is well defined: a point of $S$ has the form $\bar{F}(x,t)$ for a unique $(x,t)$, because $\bar{F}$ is injective by step 1.1, so the prescription $Y(\bar{F}(x,t)):=d{F}_{(x,t)}(0,{\mathrm{\partial }}_t)$ is independent of choices; this value lies in $T_{F(x,t)}N$ seen inside $T_{(F(x,t),t)}(N\times I)$ by [F2].

On $S$ one has $Y=V\circ {\bar{F}}^{-1}$, where $V(x,t):=d{F}_{(x,t)}(0,{\mathrm{\partial }}_t)$. The inverse is smooth in the local graph coordinates of step 2.2. The target components of $V$ are time partial derivatives of the coordinate functions of $F$, hence are smooth, including at product corners by differentiating their local extensions. Thus $Y$ is smooth along $S$ in the stated local-extension sense.

The tangent identity $d{\bar{F}}_{(x,t)}({\mathrm{\partial }}_t)=(Y(\bar{F}(x,t)),{\mathrm{\partial }}_t)$ holds: by [L5] applied to the two components ${\pi }_N\circ \bar{F}=F$ and ${\pi }_I\circ \bar{F}={\mathrm{p}\mathrm{r}}_I$, the first component of $d{\bar{F}}_{(x,t)}({\mathrm{\partial }}_t)$ is $d{F}_{(x,t)}(0,{\mathrm{\partial }}_t)=Y(\bar{F}(x,t))$ and the second is $d{\mathrm{p}\mathrm{r}}_I({\mathrm{\partial }}_t)={\mathrm{\partial }}_t$.

Consequently $Y$ is a smooth field along the closed track, takes values in $TN\oplus 0$ by construction, and satisfies the tangent identity of step 4.2; this is clause 2.

Assume $F$ takes values in $\mathrm{\partial }N$. Fix $(x,t)\in M\times I$ and a boundary chart of $N$ at $F(x,t)$ with last coordinate $u_n$; by [L7] the last coordinate of the curve $s\mapsto F(x,s)$ is identically $0$ near $s=t$, so its derivative, which is the $TN$-component $Y(\bar{F}(x,t))$ of the velocity, has last coordinate $0$ and therefore lies in $T(\mathrm{\partial }N)$. If instead $F$ takes values in the interior of $N$, then the base point $F(x,t)$ of $Y(\bar{F}(x,t))$ lies in the interior by hypothesis. This is clause 3.

In a product chart $(y,t)$ near a track point, the graph coordinates of [F1] give a smooth local inverse $(p(y),t)\mapsto (u(y,t),t)$, where $p$ selects target coordinates. Extend the coordinate functions of $F$ locally across time endpoints and, if necessary, the target boundary chart. Define the $j$th target component of a local field by $({\mathrm{\partial }}_t{F}^j)(u(y,t),t)$. On the track it equals the $j$th velocity component. Interpret these coefficients in the coordinate basis of $TN$ and assign zero time component; horizontality follows directly, without assuming slice charts preserve the horizontal subbundle. Take finitely many such chart domains covering compact $S$. In their Euclidean extensions choose finitely many nonnegative smooth bumps with compact supports inside these domains by A Euclidean bump for a compact set inside an open set and positive sum near $S$, and restrict them to $N\times I$. Dividing each by their sum gives smooth weights summing to one on a neighbourhood $\mathrm{\Omega }$ of $S$. The weighted sum of the local horizontal fields is smooth, horizontal and equals $Y$ on $S$. These restricted coordinate bumps also handle the corners of $N\times I$ when $N$ has boundary. This proves clause 1.

If $F$ takes values in $\mathrm{\partial }N$, perform the graph construction of step 1.1 first in the boundary coordinates $y^{\mathrm{\prime }}$, choosing $p$ from those coordinates, since the slice differential is injective into $T\mathrm{\partial }N$. Extend the tangential coefficients independently of the inward coordinate $y_n$ and set the $y_n$ component identically zero. Restricted product-chart bumps patch these fields as in step 1.1. Each is tangent to $\mathrm{\partial }N$ there, so their sum is tangent too; boundary coordinate changes preserve this condition. If the track lies in the interior, restrict $\mathrm{\Omega }$ to $\mathrm{Int}N\times I$. These are the two alternatives of clause 2.

For a prescribed open neighbourhood $W$ of $S$, simply restrict the extension to ${\mathrm{\Omega }}^{\mathrm{\prime }}:=\mathrm{\Omega }\cap W$. This is an open neighbourhood of $S$ contained in $W$; no tubular theorem for a boundary or cornered track is required.

Clause 3, relative form: let $A\subseteq S$ be compact and let ${\tilde{Y}}_0$ be a smooth extension of $Y$ defined on an open neighbourhood ${\mathrm{\Omega }}_0$ of $A$ (it agrees with $Y$ at every point of $S$ in its domain). The set $A$ is closed in the manifold $N\times I$ by [F2], and both ${\mathrm{\Omega }}^{\mathrm{\prime }}$ from step 2.2 and ${\mathrm{\Omega }}_0$ are open neighbourhoods of $A$; by [L3] choose a smooth cutoff $\chi$ equal to $1$ on a neighbourhood $A^{\mathrm{\prime }}\subseteq {\mathrm{\Omega }}^{\mathrm{\prime }}\cap {\mathrm{\Omega }}_0$ of $A$ with support in ${\mathrm{\Omega }}^{\mathrm{\prime }}\cap {\mathrm{\Omega }}_0$. Define ${\tilde{Y}}_1:=\tilde{Y}+\chi ({\tilde{Y}}_0-\tilde{Y})$ on ${\mathrm{\Omega }}^{\mathrm{\prime }}\cap {\mathrm{\Omega }}_0$, extended by $\tilde{Y}$ outside $\mathrm{supp}\chi$. This is a smooth map into $TN$ because the fibrewise vector-space operations of a smooth vector bundle are smooth in local trivialisations ([L4]), it agrees with ${\tilde{Y}}_0$ on $A^{\mathrm{\prime }}$ and with $\tilde{Y}$ outside $\mathrm{supp}\chi$, and it restricts to $Y$ on $S\cap {\mathrm{\Omega }}^{\mathrm{\prime }}$; hence it is an extension of $Y$ agreeing with ${\tilde{Y}}_0$ near $A$. When both fields are boundary-tangent, their blend is boundary-tangent too.

The constructions prove all three clauses. They modify only the horizontal component and preserve the stated relative and boundary conditions.

A relatively compact neighbourhood of the track inside $W\times I$: $S$ is compact by [F1] and $S\subseteq F(M\times I)\times I\subseteq W\times I$, which is open. Since $N\times I$ is locally compact Hausdorff and $S$ is compact, [L2] applied at each point of $S$ yields finitely many open sets with compact closure covering $S$ and contained in $W\times I$; their union ${\mathrm{\Omega }}_0$ is an open neighbourhood of $S$ with $\overline{{\mathrm{\Omega }}_0}$ compact and $\overline{{\mathrm{\Omega }}_0}\subseteq W\times I$.

Choose a smooth cutoff $\psi :N\to [0,1]$ with $\psi =1$ on the compact image $F(M\times I)$ and $\mathrm{supp}\psi \subseteq W$ by [L3], applied to the closed set $F(M\times I)$ inside the open set $W$; and choose a smooth function $\beta :I\to [0,1]$ with $\beta =1$ on $[\epsilon ,1-\epsilon ]$ and $\beta =0$ on $[0,\epsilon \mathrm{/}2]\cup [1-\epsilon \mathrm{/}2,1]$, which exists by the smooth Urysohn lemma on the interval.

Apply [L1] with the prescribed neighbourhood ${\mathrm{\Omega }}_0$: there is an open neighbourhood $\mathrm{\Omega }\subseteq {\mathrm{\Omega }}_0$ of $S$ and a smooth horizontal $\tilde{Y}:\mathrm{\Omega }\to TN$ extending $Y$. By [L3] applied to the closed set $S$ inside the open set $\mathrm{\Omega }$, choose a smooth cutoff $\rho :N\times I\to [0,1]$ with $\rho =1$ on a neighbourhood of $S$ and $\mathrm{supp}\rho \subseteq \mathrm{\Omega }$.

Define $G$ on $\mathrm{\Omega }$ by $G(y,t):=\rho (y,t)\beta (t)\psi (y)\tilde{Y}(y,t)$, and define $G:=0$ on the complement of the closed set $\mathrm{supp}(\rho \beta \psi )\subseteq \mathrm{\Omega }$. The two definitions agree on the overlap, where $\rho \beta \psi =0$, so $G$ is a well-defined smooth map on all of $N\times I$: at a point outside $\mathrm{supp}(\rho \beta \psi )$ it vanishes on a whole neighbourhood, and on $\mathrm{\Omega }$ it is a product of smooth functions with the smooth map $\tilde{Y}$. It lies in $T_{y}N$ at $(y,t)$ by construction. Thus $H:I\times N\to TN$, $H(t,y):=G(y,t)$, is smooth by composition with the factor-swap map and has $H(t,y)\in T_{y}N$, so it is a time-dependent vector field on $N$ by [L4], with slices $H_t=G_t$.

Clause 2: let $(x,t)\in M\times I$. Since $\bar{F}(x,t)\in S$ and $\rho =1$ near $S$ and $\psi =1$ on $F(M\times I)$, one has $\rho \beta \psi \tilde{Y}(\bar{F}(x,t))=\beta (t){\mathrm{\partial }}_{t}F(x,t)$. The isotopy is constant near the ends, so ${\mathrm{\partial }}_{t}F(x,t)=0$ for $t\le \epsilon$ and for $t\ge 1-\epsilon$, while $\beta =1$ on $[\epsilon ,1-\epsilon ]$; in all cases $\beta (t){\mathrm{\partial }}_{t}F(x,t)={\mathrm{\partial }}_{t}F(x,t)$. Hence $G_t(F(x,t))={\mathrm{\partial }}_{t}F(x,t)$.

Clause 3: for $t\le \epsilon \mathrm{/}2$ and for $t\ge 1-\epsilon \mathrm{/}2$ one has $\beta (t)=0$, so $G_t=\rho \beta \psi \tilde{Y}(\cdot ,t)=0$ by step 3.1.

Put $K:={\mathrm{pr}}_N(\mathrm{supp}\rho )$. The support of $\rho$ is closed and contained in the compact set $\overline{{\mathrm{\Omega }}_0}$, so it is compact, and its continuous projection $K$ is compact and contained in $W$. Off $K$ the field $G_t$ vanishes for every $t$. Since $K$ is closed, every $\mathrm{supp}{G}_t$ lies in $K$. Thus their union and its closure lie in the compact subset $K\subseteq W$, proving clause 1. Containment alone would not prove that the union itself is closed.

Clauses 1, 2 and 3 are steps 4.3, 4.1 and 4.2; the field is smooth and horizontal, and countable choice was used only as declared in [A1].

The factor swap makes $V(t,y)=G(y,t)$ smooth, with $V_t=G_t$, so [F1] applies to $V$. If $N$ is boundaryless, [L1] gives a global smooth evolution on $I$. For the boundary case, in a boundary chart write the inward coefficient as $a(y^{\mathrm{\prime }},r,t)$, where $r\ge 0$. Tangency gives $a(y^{\mathrm{\prime }},0,t)=0$, and local smooth extension gives $a=rb$ with $b(y^{\mathrm{\prime }},r,t)={\int }_0^1{\mathrm{\partial }}_{r}a(y^{\mathrm{\prime }},ur,t)du$ smooth. Extend the coordinate field across $r=0$ and apply the local smooth ODE theory underlying [L1]. Uniqueness keeps solutions starting on $r=0$ there; for $r(s)>0$, the scalar equation gives $r(t)=r(s)\exp ({\int }_s^{t}b(y^{\mathrm{\prime }}(u),r(u),u)du)>0$ while the solution is in the chart. Thus local solutions and their reverse-time solutions preserve the half-space. Global continuation is the compact-support argument of [L1]: a solution meeting the complement of the common compact support set is constant by uniqueness; any other solution stays in that compact set. At a finite maximal endpoint a sequence of its values has a convergent subsequence there, and a local evolution around the limiting time and point extends the solution by uniqueness. This works also at a boundary point using the half-space solutions just established, and at $t=0,1$ using local smooth time extension. Hence solutions exist on all of $I$ with smooth dependence. Their initial-value identity and [L2] give properties 1 and 3.

Property 2: composing the cocycle law with $r=s$ gives ${\mathrm{\Psi }}_{s,t}\circ {\mathrm{\Psi }}_{t,s}={\mathrm{\Psi }}_{s,s}={\mathrm{i}\mathrm{d}}_N$, and with the roles of $s,t$ exchanged gives ${\mathrm{\Psi }}_{t,s}\circ {\mathrm{\Psi }}_{s,t}={\mathrm{i}\mathrm{d}}_N$; hence each ${\mathrm{\Psi }}_{t,s}$ is a bijection with inverse ${\mathrm{\Psi }}_{s,t}$, and both are smooth by [L1], so each ${\mathrm{\Psi }}_{t,s}$ is a diffeomorphism of $N$ (Diffeomorphisms and local diffeomorphisms of manifolds).

Property 4 and uniqueness: suppose $G$ vanishes identically on $[s,t]$. The constant curve $r\mapsto y$ solves the initial-value problem with value $y$ at time $s$, and so does $r\mapsto {\mathrm{\Psi }}_{r,s}(y)$; by uniqueness of integral curves [L3] the two agree, whence ${\mathrm{\Psi }}_{t,s}(y)=y$ for every $y$, i.e. ${\mathrm{\Psi }}_{t,s}={\mathrm{i}\mathrm{d}}_N$. Uniqueness of the evolution operator itself is the same statement: any evolution operator satisfying the initial-value problem has the same integral curves as the one constructed in step 1.1, so it agrees with it everywhere.

Setting $H_t:={\mathrm{\Psi }}_{t,0}$ gives a smooth family of diffeomorphisms with $H_0={\mathrm{i}\mathrm{d}}_N$ and $H_t^{-1}={\mathrm{\Psi }}_{0,t}$ by step 2.1; since ${\mathrm{\Psi }}_{t,0}$ is the identity outside the compact set containing ${\bigcup }_t\mathrm{supp}{G}_t$ (a point outside the supports has the constant curve as its integral curve, by [L3] as in step 2.2), $H$ is a compactly supported ambient isotopy, and it is stationary on every interval on which $G$ vanishes by step 2.2.

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.