Smooth isotopies, diffeotopies and ambient isotopies

Let $M$ and $N$ be smooth manifolds (possibly with boundary, Smooth maps between manifolds with boundary) and let $I=[0,1]$. A smooth isotopy of $M$ in $N$ is a smooth map $F:M\times I\to N$ such that every slice $F_t:=F(\cdot ,t)$ is a smooth embedding (Smooth embeddings); if in addition $F_0=f_0$ and $F_1=f_1$, then $F$ is a smooth isotopy from $f_0$ to $f_1$, and $f_0,f_1$ are isotopic. A smooth diffeotopy of $N$, also called an ambient isotopy, is a smooth map $H:N\times I\to N$ with $H_0={\mathrm{i}\mathrm{d}}_N$ and every $H_t$ a diffeomorphism (Diffeomorphisms and local diffeomorphisms of manifolds); it extends an isotopy $F$ of $M$ when $H_t\circ F_0=F_t$ for all $t\in I$. A diffeotopy is compactly supported if there is a compact $K\subseteq N$ with $H_t={\mathrm{i}\mathrm{d}}_N$ outside $K$ for every $t$ (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right), and is stationary near the ends if $H_t$ is independent of $t$ near $t=0$ and near $t=1$; the same terms apply to isotopies.

The track of $F$ is the level-preserving map $\bar{F}:M\times I⟶N\times I,\bar{F}(x,t)=(F(x,t),t),$ and the support of $F$ is the closure of the set $\{x\in M:F(x,t)\mathrm{\ne }F(x,0)\text{ for some }t\in I\}$. Equivalently, $F$ is a smooth family of embeddings parametrised by $I$ in the sense of Smooth families of maps and their evaluation maps. Here smoothness is tested in product charts by local extension of the coordinate functions to Euclidean open sets; when both $M$ and $I$ have boundary this is the explicit product-corner convention. For boundaryless parameter manifolds it is exactly the cited smooth-family definition; the same evaluation convention is used for $I$. The track is a map of this kind, retaining the time coordinate: it is not the parametrised image surface $F(M\times I)$ alone, and every statement on this page about the track refers to the map $\bar{F}$ and to its image $\bar{F}(M\times I)\subseteq N\times I$.

On this page an isotopy is always a family of embeddings, as above. A family of immersions that need not be injective is a regular homotopy (Regular homotopy of immersions); the sources' occasional use of the word "isotopy" for a family of immersions is never imported here, and where a source means a regular homotopy the term regular homotopy is used. Smoothness of a time reparametrisation, compactness of $M$ or $N$, properness, orientability and any choice principle are not part of the definition: they are hypotheses of the theorems that use it, and each of those states its own hypotheses.