# On homeomorphisms non-smooth along submanifolds

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 $2$?

• Without assuming that $\phi$ bi-Holder the answer is surely no. Let $M_1$ be an exotic $7$-sphere and $M_2$ be the standard $7$-sphere. Then after removing a point they become diffeomorphic because theire is only one smooth structure on $\mathbb R^7$. This diffeomorphism extends to a homeomorphism of one-point compactifications, which are $M_1$, $M_2$. For the last part note that in general if $C$ is a closed subset of a compact Hausdorff space $X$, thenthe one-point compactification of $X-C$ is $X/C$, see en.wikipedia.org/wiki/Alexandroff_extension#cite_note-rotman-1. – Igor Belegradek Jun 16 '17 at 20:53
• Igor, thank you for this comment! I was doubting that the first question is asking for too much. I hope still that under some restrictions the answer could be positive, and the case of the main interest for me is when $N_1$ has codimesnion $2$ in $M_1$. – aglearner Jun 16 '17 at 21:05
• Ben, you mean it is true in codimension 2, or that there is a counter-example similar to one, given by Igor? I would be especially interested to know how to prove this in dimension $4$. – aglearner Jun 22 '17 at 19:36
• The Alexander Trick is a high dimensional Lipschitz homeomorphism smooth away from a single point that easily yields a counterexample in codimension 1. Probably there are closely related counterexamples in all codimensions. Also, it makes Igor's example Lipschitz. Dimension 4 is always difficult. I think it is plausible that your hypotheses do yield diffeomorphism there. – Ben Wieland Jun 27 '17 at 15:37