# Variants of isotopy extensions

I am interested in slight variations of the usual isotopy extension theorems.

In short, my question is the following : Can one extend isotopies of $C \subseteq M$, where $C$ is compact and $M$ is a manifold with boundary ?

Now to the details. The usual reference for these is the book Differential Topology by Hirsch, where I found the following theorems :

1.3 Let $V \subseteq M$ be a compact submanifold and $F:V \times I \rightarrow M$ an isotopy of $V$. If either $F(V \times I) \subseteq \partial M$ or $F(V \times I) \subseteq M \setminus \partial M$, then $F$ extends to a diffeotopy with compact support.

1.4 Let $U \subseteq M$ be an open set and $A \subseteq U$ a compact set. Let $F: U\times I \rightarrow M$ be an isotopy of $U$ such that $\widehat{F}(U \times I) \subseteq M \times I$ is open. Then there is a diffeotopy of $M$ having compact support, which agrees with $F$ on a neighborhood of $A \times I$.

I am quite confused by both the statements and the proofs. Here are my questions :

• Is the 'submanifold' hypothesis in Theorem 1.3 really needed, or can one relax it to any compact set ? More specifically, I am looking to apply it to a graph embedded on a surface, is there any obstruction against it ? I took a close look at the proof and did not see any trouble with that, but I'm puzzled since every reference I found seems to include this hypothesis as well.

• A way to obtain the result for any compact set would be to take an open tubular neighborhood and apply Theorem 1.4, but does it hold that any isotopy of a compact set extends to its open tubular neighborhood ? This seems obvious but I cannot find a straightforward proof.

• Does Theorem 1.4 hold for manifolds with boundary ? Looking at the proof, Hirsch does not seem to take the precautions he took in Theorem 1.3 to avoid getting an 'isotopy' that moves the boundary to the interior of the manifold. More generally, both results emane from Theorem 1.1 which does not seem to hold for manifolds with boundaries (but nothing is explicitly written), because integrating a vector field with vectors on the boundary pointing towards the interior does not give a homeomorphism; so I am a bit worried.

• And a last question, a bit disconnected from the others : These proofs use vector fields, and as such we need smooth isotopies. If two sets included on a surface (say for instance graphs) are isotopic, are they smoothly isotopic ? The usual way to tackle these problems is to use a Theorem by Munkres stating that homeomorphisms of a surface are homotopic to diffeomorphisms, but it does not seem to do the job here..?