Isotopy extension needs compact source or proper support control

The isotopy extension theorem The isotopy extension theorem assumes a compact source $M$ (or, in the relative form, control on a compact set), and compactness is used at three separate places: the track is compact, so finitely many local velocity-extension charts suffice; the velocity field can be cut off with compact support; and the resulting time-dependent field is complete on the finite time interval. For an arbitrary isotopy of a noncompact manifold there need not be any ambient isotopy extending it, so the compact-source hypothesis is load-bearing and cannot simply be dropped.

The standard counterexample (Hirsch, Exercise 9, p. 183, reproduced in the UCR hand-out): a properly embedded copy $L\subseteq {\mathbb{R}}^3$ of the real line obtained from the $x$-axis by tying a small trefoil knot in a finite segment is smoothly isotopic to the straight line $\mathbb{R}$ through embeddings — roll the knot out to infinity along the line — but no ambient isotopy of ${\mathbb{R}}^3$ carries $L$ to $\mathbb{R}$: such an ambient isotopy would restrict to a diffeomorphism of the complements (Diffeomorphisms and local diffeomorphisms of manifolds, Smooth embeddings), whereas ${\pi }_1({\mathbb{R}}^3\setminus L)$ is the nonabelian trefoil knot group and ${\pi }_1({\mathbb{R}}^3\setminus \mathbb{R})\cong \mathbb{Z}$, so the complements are not even homotopy equivalent. The isotopy of embeddings here is a genuine smooth isotopy in the sense of Smooth isotopies, diffeotopies and ambient isotopies; only the ambient extension fails.

For a closed source submanifold, Hirsch Theorem 1.6 replaces compactness by bounded velocity in a complete Riemannian metric, provided the entire isotopy image lies either in $\mathrm{\partial }N$ or in $N\setminus \mathrm{\partial }N$. The extended time-dependent field is required to be boundary-tangent; bounded velocity and completeness of the metric then give a global ambient isotopy. The relative bounded-velocity form, Hirsch Theorem 1.7, instead assumes an isotopy of an open neighbourhood of a closed set whose track image is open. These boundary and open-track conditions are part of the respective substitutes, not consequences of bounded velocity alone. This remark asserts no new theorem; it records the exact hypothesis of The isotopy extension theorem that the counterexample tests and the substitute that restores the conclusion.