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.
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1Cross-posted at MO here. – Moishe Kohan Nov 27 '20 at 17:37