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I would like to find an example, if one exists, of manifolds $M$ and $N$ with embeddings $f:M\to N$ and $g:N\to M$ such that $f\circ g$ and $g\circ f$ are both isotopic (i.e. homotopic through embeddings) to the respective identities, yet the interiors of $M$ and $N$ are not diffeomorphic. Obviously, $M$ and $N$ cannot be closed. You may assume that the manifolds have no boundary, but I would also be interested in compact examples.

By the way: in the other direction, are there conditions under which $M$ and $N$ are necessarily diffeomorphic if $f$ and $g$ as above exist?
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I would like to find an example, if one exists, of manifolds $M$ and $N$ with embeddings $f:M\to N$ and $g:N\to M$ such that $f\circ g$ and $g\circ f$ are both isotopic (i.e. homotopic through embeddings) to the respective identities, yet the interiors of $M$ and $N$ are not diffeomorphic. Obviously, $M$ and $N$ cannot be closed.

By the way: in the other direction, are there conditions under which $M$ and $N$ are necessarily diffeomorphic if $f$ and $g$ as above exist?
show/hide this revision's text 2 deleted 17 characters in body; edited body

I would like to find an example, if one exists, of manifolds $M$ and $N$ without boundary, and with embeddings $f:M\to N$ and $g:N\to M$ such that $f\circ g$ and $g\circ f$ are both isotopic (i.e. homotopic through embeddings) to the respective identities, yet $M$ and $N$ are not diffeomorphic. Obviously, $M$ and $N$ cannot be closed.

By the way: in the other direction, are there conditions under which $M$ and $N$ are necessarily diffeomorphic if $f$ and $g$ as above exist?
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