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?



isotopy inverse embeddings vs. diffeomorphisms
Here I'll prove that $M$ and $N$ must be diffeomorphic under your hypotheses (I'll assume that $M$ and $N$ are open, i.e. no boundary). 


Consider the direct limit (see below) $X= M \overset{f}{\to} N \overset{g}{\to} M \overset{f}{\to} N \overset{g}{\to}\cdots$. By this, I mean the union $M \times \mathbb{N} \sqcup N\times \mathbb{N}$ modulo the identifications $(m,k)\sim (f(m),k), (n,k)\sim (g(n),k+1)$, for $m\in M, n\in N, k\in \mathbb{N}$. Then $X$ is a smooth manifold, since it is a nested union of smooth manifolds. Similarly, we have the direct limit $Y=N \overset{g}{\to} M \overset{f}{\to} N \overset{g}{\to}\cdots$. Clearly, $X \cong Y$, since the direct limit depends only on the tail of the sequence. Also, $X$ (and therefore $Y$) is diffeomorphic to the direct limits $ M \overset{g\circ f}{\to} M \overset{g\circ f}{\to} M \overset{g\circ f}{\to}\cdots$ and $N\overset{f\circ g}{\to} N\overset{f\circ g}{\to} N\overset{f\circ g}{\to}\cdots$. 


**Claim:** If $F:M\to M$ is isotopic to the identity, then the direct limit $M\overset{F}{\to} M\overset{F}{\to} M\overset{F}{\to} \cdots \cong M$. 


By the above discussion, this implies that $M\cong N$. 


To prove the claim, let $F\_t:M\to M, t\in [0,1]$ be an isotopy of $F$ to the identity, so $F\_1=F$, and $F\_0=Id$. Take an exhaustion of $M$ by smooth compact submanifolds with boundary $K\_1\subset K\_2\subset K\_3 \subset \cdots$, so that $F([0,1]\times K\_i)\subset int(K\_{i+1})$, $K\_i\subset int(K\_{i+1})$, and $\cup K\_i=M$. The we see that $X\_t = M\overset{F\_t}{\to} M\overset{F\_t}{\to}M\overset{F\_t}{\to}\cdots$ is diffeomorphic to $Y\_t= K\_2 \overset{F\_t}{\to} K\_4 \overset{F\_t}{\to} K\_6 \overset{F\_t}{\to} \cdots$ for all $t$ (where we really meant $F\_{t| K\_{2i}}$ in the maps of this direct limit and topologized by the inclusion of interiors). To see this, note that $X\_t$ is a quotient of $M\times \mathbb{N}$, which is exhausted by $\{(K\_k,i), k,i\in\mathbb{N}\}$. Since $Y\_t\subset X\_t$, we need only show that each point of $X\_t$ is in $Y\_t$. Suppose $x\in X\_t$, then $x$ is the image of some $x'\in (K\_k,i)$ for some $k,i$. If $k\leq 2i$, then $x'\in (K\_{2i},i)$, so we see that $x\in Y\_t$. Otherwise, $k>2i$ and we see that $(K\_k,i) \overset{F\_t}{\subset}(K\_{k+1},i+1) \overset{F\_t}{\subset} \cdots \overset{F\_t}{\subset} (K\_{2k-2i},k-i)$ after taking the quotient, so $x\in Y\_t$. 


By the [Isotopy Extension Theorem](http://books.google.com/books?id=iSvnvOodWl8C&lpg=PP1&dq=hirsch%2520differential%2520topology&pg=PA180#v=onepage&q&f=false), one may see that $Y\_0 \cong Y\_1$. This gives the desired diffeomorphism $M\cong M\overset{F}{\to} M\overset{F}{\to} M\overset{F}{\to} \cdots $. 


We actually use the isotopy extension theorem by induction. We want to find a sequence of diffeomorphisms $G^i: K\_{2i}\to K\_{2i}$ so that $G^{i+1}\_{|K\_{2i}} = F\circ G^i$. This immediately gives the diffeomorphism 


$$\begin{matrix} Y\_1= K\_2 & \overset{F}{\to} & K\_4 & \overset{F}{\to} & K\_6 & \overset{F}{\to} & \cdots\\
 G^1 \uparrow & &G^2\uparrow & &G^3\uparrow & & \\ 
Y\_0= K\_2 & \hookrightarrow & K\_4 & \hookrightarrow & K\_6 & \hookrightarrow & \cdots
\end{matrix}$$


We will actually construct a diffeotopy $G^i\_t:K\_{2i}\to K\_{2i}, t\in [0,1]$ such that $G^{i+1}\_{t|K\_{2i}}=F\_t\circ G^i\_t$, where $G^i\_0=Id$, and then set $G^i=G^i\_1$ for the desired diffeomorphism. 


Let $G^1\_t=Id$. Suppose we have constructed $G^i\_t$ by induction, so that $G^i\_0=Id$, $G^i\_{t|K\_{2i-2}}=F\_t \circ G^{i-1}\_t$, $t\in [0,1]$. Then $F\_t\circ G^i\_t: K\_{2i}\to int(K\_{2i+2})$ is an isotopy from $F\_0\circ G^i\_0=Id$ to $F\_1\circ G^i\_1=F\circ G^i\_1$. By the isotopy extension theorem, there exists a diffeotopy $G^{i+1}\_t: K\_{2i+2}\to K\_{2i+2}$ such that $G^{i+1}\_0=Id$, and $G^{i+1}\_{t|K\_{2i}}=F\_t\circ G^i\_t$, with compact support in $int(K\_{2i+2})$. This completes the proof. 


**Addendum:** I'll add some explanation of direct limits. Consider a sequence of smooth embeddings of open manifolds $X\_i \overset{f\_i}{\to} X\_{i+1}$. By the direct limit $X$ of $X\_1\overset{f\_1}{\to} X\_2 \overset{f\_2}{\to} X\_3 \cdots$, I mean the quotient of $X\_1\sqcup X\_2 \sqcup X\_3 \sqcup \cdots$ with respect to the equivalence relation generated by $x\_i \sim f\_i(x\_i)$ for all $x\_i \in X\_i, i\in \mathbb{N}$. Then there is a natural embedding of $X\_i \hookrightarrow X$ for all $i$, in such a way that $X= \cup\_i X\_i$, and one gives $X$ the structure of a smooth manifold by taking the atlas of charts generated by the charts of each $X\_i$. 


The space $X$ is determined by the direct limit of any subsequence $\{ X\_{i\_j}\}$, with maps $F\_j: X\_{i\_j}\to X\_{i\_{j+1}}$ defined by $F\_j=f\_{i\_{j+1}-1}\circ \cdots \circ f\_{i\_j}$. One may therefore verify that two direct limits are diffeomorphic if there are compatible diffeos. between subsequences, which justifies some of the arguments above. 


