# Gluing two diffeomorphisms and then smoothing

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 $\hat{\phi}:M_1\to M_2$?
• 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