Assume $M$ is a non-compact smooth manifold without boundary (although I would also be interested in the case where $M$ is compact with non-empty boundary). I would like to find an explicit counter-example --- or a proof --- for the following statement: The inclusion $\textit{incl}:\text{Diff}(M)\to \iota:\text{Diff}(M)\to \text{Emb}(M,M)$ of the space of diffeomorphisms of $M$ into the space of smooth self-embeddings is a weak equivalence on each component of $\text{Diff}(M)$. In other words, every homotopy fibre of the inclusion $\textit{incl}$ \iota$is either empty or weakly contractible. I am considering the compact-open$C^1$-topology on the above spaces. Obviously, I would also be very interested in any known results relating the homotopy type of$\text{Diff}(M)$with that of$\text{Emb}(M,M)$. Edit: Tom Goodwillie has provided an answer for the case when$M$is compact or the interior of a compact manifold. I have placed a bounty for the case in which$M$is open (and not necessarily the interior of a compact manifold). Edit 2: I managed to adapt Agol's nice idea from my previous question to apparently resolve the present question. Since nobody has raised any major issues so far, and people are upvoting my answer, I will likely accept it. As a curious aside, I edited my answer so many times --- out of persnicketiness --- that it turned into community wiki. I was unaware that would happen. Oh well... 6 deleted 76 characters in body Previously, I asked a question on mathoverflow comparing smooth embeddings and diffeomorphisms, which received a very interesting and somewhat unexpected answer by Agol. I now ask a further question about the relation between the homotopy type of embedding spaces and diffeomorphism spaces. Assume$M$is a non-compact smooth manifold without boundary (although I would also be interested in the case where$M$is compact with non-empty boundary). I would like to find an explicit counter-example --- or a proof --- for the following statement: The inclusion$\textit{incl}:\text{Diff}(M)\to \text{Emb}(M,M)$of the space of diffeomorphisms of$M$into the space of smooth self-embeddings is a weak equivalence on each component of$\text{Diff}(M)$. In other words, every homotopy fibre of the inclusion$\textit{incl}$is either empty or weakly contractible. I am considering the compact-open$C^1$-topology on the above spaces. Obviously, I would also be very interested in any known results relating the homotopy type of$\text{Diff}(M)$with that of$\text{Emb}(M,M)$. Edit: Tom Goodwillie has provided an answer for the case when$M$is compact or the interior of a compact manifold. I have placed a bounty for the case in which$M$is open (and not necessarily the interior of a compact manifold).If the bounty falls through, I will probably mark Tom's answer as accepted. 5 fixed typo Previously, I asked a question on mathoverflow comparing smooth embeddings and diffeomorphisms, which received a very interesting and somewhat unexpected answer by Agol. I now ask a further question about the relation between the homotopy type of embedding spaces and diffeomorphism spaces. Assume$M$is a non-compact smooth manifold without boundary (although I would also be interested in the case where$M$is compact with non-empty boundary). I would like to find an explicit counter-example --- or a proof --- for the following statement: The inclusion$\textit{incl}:\text{Diff}(M)\to \text{Emb}(M,M)$of the space of diffeomorphisms of$M$into the space of smooth self-embeddings is a weak equivalence on each component of$\text{Diff}(M)$. In other words, every homotopy fibre of the inclusion$\textit{incl}$is either empty or weakly contractible. I am considering the compact-open$C^1$-topology on the above spaces. Obviously, I would also be very interested in any known results relating the homotopy type of$\text{Diff}(M)$with that of$\text{Emb}(M,M)$. Edit: Tom Goodwillie has provided an answer for the case when$M$is compact or the interior of an open a compact manifold. I have placed a bounty for the case in which$M\$ is open (and not necessarily the interior of a compact manifold). If the bounty falls through, I will probably mark Tom's answer as accepted.