I repeat this, which I posted in Math Stack, where it got some attention but no answer.


If two compact manifolds have diffeomorphic interiors and diffeomorphic boundaries, are they then diffeomorphic? Is it true for surfaces? Some context: there seems to exist an example by Barden and Mazur of a **nontrivial** cobordism between some oriented manifold $M$ and $-M$ whose interior is trivial (diffeo to $(0,1)\times M$). This would be a counterexample, but I cannot find the reference anywhere.



Diffeomorphisms of manifolds with boundary
This is definitely true for surfaces (easy) and for 3-manifolds (harder). The argument for surfaces goes via classification: Two smooth surfaces with boundary are homeomorphic if and only if they are diffeomorphic. For a compact connected surface $S$ the full set of topological invariants is the triple:


$t(S)$= (orientability, Euler characteristic, number of boundary components)


It is an easy exercise to see that if $S, S'$ have homeomorphic interiors then $t(S)=t(S')$, hence, $S$ is homeomorphic to $S'$.


For 3-manifolds the proof is harder, see


C. Edwards, *Concentricity in 3-manifolds*. Trans. Amer. Math. Soc. 113 (1964) 406–423.


