Gluing two diffeomorphisms and then smoothing

This question did not get an adequate answer on math.stackexchange.

Let $M_1,M_2$ be two $n$-dimensional closed manifolds and suppose that $M_i=\bar{U}_i^+\cup \bar{U}_i^-$ where $\bar{U}_i^\pm$ are compact codimension zero sub-manifolds with boundary so that $\bar{U}_i^+\cap \bar{U}_i^-=\Sigma_i$ are smooth codimension one submanifolds in $M_i$.

If there are diffeomorphisms $\phi^\pm :\bar{U}_1^\pm \to \bar{U}_2^\pm$ so that $\phi^+|_{\Sigma_1}=\phi^-|_{\Sigma_1}$, then it is clear that there is a homeomorphism $\phi: M_1\to M_2$ which restricts to the $\phi^\pm$.

Can this homeomorphism be smoothed to give a diffeomorphism $\hat{\phi}:M_1\to M_2$?

This result is stated in "An Introduction to Morse Theory" by Yukio Matsumoto (it's Theorem 2.8) but is not proved there. Does anyone know a reference with a proof?

• Yes, it can, because you can pick two collars of the boundaries, and isotope each diffeomorphism $\phi^\pm$ so that it acts productwise on each collar. This is also the same technique one uses to prove that two smooth isotopies can be composed and transformed into a smooth isotopy. – Bruno Martelli Aug 21 '15 at 3:56
• I seem to recall that Milnor's notes on Smale's proof of the Poincare conjecture carefully treated this type of question. – Jim Conant Aug 21 '15 at 11:48
• @BrunoMartelli Do you have a reference for that? – foliations Aug 21 '15 at 23:24
• @BrunoMartelli I’m a student strugling with this question as well now. Would you provide a reference for the hint please ? – Sou Jan 28 at 13:51