Gluing isotopic smoothings

Question

Let $M$ be a topological manifold which can be written as $M = U \cup V$ where $U$ and $V$ are open. Suppose both $U$ and $V$ admit smooth structures. Also assume that on the overlap $U \cap V$ the smooth structures induced from $U$ and $V$ are isotopic, i.e., there is a diffeomorphism $$f: (U\cap V)_U \to (U\cap V)_V$$ which is isotopic via homeomorphisms to the identity (here the left-hand-side is the smooth structure on $U\cap V$ induced from $U$ and the right-hand-side is the one induced from $V$.

Question 1: is that true that one can "glue" the smooth structures on $U$ and $V$ together to obtain a smooth structure on $M$?

Question 2: if $K \subset M$ is a compact subset, can one produce a smooth structure on a neighborhood of $K$ using the smooth structures on $U$ and $V$?

If not, here is a weaker version.

Question 3: is that true that after stabilization one can glue the smooth structures together? This means: there exists $k\geq 0$ such that one can glue the induced smooth structure on $U \times {\mathbb R}^k$ and the induced smooth structure on $V \times {\mathbb R}^k$ to obtain a smooth structure on $M \times {\mathbb R}^k$? Similar question for a given compact subset $K \subset M$.

Answer 1

In dimension 6 and above, one can apply the concordance extension theorem to answer (1) positively. Isotopic smooth structures are concordant, and hence one can extend the concordance on $U \cap V$ to all of $U$ to obtain a smooth structure on $U$ which agrees with the smooth structure on $U \cap V$ induced by $V$. Now we can glue to obtain a smooth structure on $M$. Kupers has a nice summary of the situation in "Lectures on Diffeomorphism Groups of Manifolds", at the start of chapter 26.

Since (1) is positive, (2) is as well, since open subsets inherit a smooth structure. (3) is always true, because we can stabilize to have dimension $\geq 6$.