Suppose $M_1\supset N_1$ and $M_2\supset N_2$ are two couples consisting of a smooth compact connected manifold $M_i$ with a smooth compact sub-manifold $N_i$.

Suppose there is a homeomorphism $\varphi: M_1\to M_2$ which has the following properties:

1) $\varphi$ sends $N_1$ to $N_2$ and $\varphi$ restricts to a diffeomorphism on $N_1$.

2) The the map $\varphi: M_1\setminus N_1\to M_1\setminus N_1$ is a diffeomorphism.

**Question.** Is it true that $M_1$ is diffeomorphic to $M_2$? What if we impose the condition, that $\varphi$ and $\varphi^{-1}$ are Holder maps? What if $N_1$ and $N_2$ are of codimension $2$?