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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?