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Answer: For $M$ a non-compact manifold without boundary, it seems the homotopy fibre of the inclusion $\iota:\textrm{Diff}(M)\to\textrm{Emb}(M,M)$ \iota:\textrm{Diff}(M)\hookrightarrow\textrm{Emb}(M,M)$at$\textrm{id}_M$is weakly contractible. As hinted implicitly by Tom Goodwillie in his answer, it is not hard to conclude that all the every homotopy fibres fibre of$\iota$are is then either empty or weakly contractible. Before I continuecontinuing, I will clarify the structure of the homotopy fibre,$F$. F$, of $\iota:\textrm{Diff}(M)\to\textrm{Emb}(M,M)$ at $\textrm{id}_M$. Recall from the question above that $\textrm{Diff}(M)$ and $\textrm{Emb}(M,M)$ are both endowed with the compact-open (or weak) $C^1$-topology. As a set, $F$ consists of the continuous isotopies $(\varphi_t:M\to M)_{t\in I}$ through embeddings $M\to M$ for which $\varphi_0 = \textrm{id}_M$, and $\varphi_1$ is a diffeomorphism of $M$. An isotopy $(\varphi_t)_{t\in I}$ can will also be written as a map $\varphi: \varphi : M\times I\to I \to M$. The topology on As a topological space, $F$ makes it a is the subspace of the space of paths in $\textrm{Emb}(M,M)$, where $\textrm{Emb}(M,M)$ has consisting of the compact-open (or weak) paths which start at $C^1$ topology.\textrm{id}_M$and end at a diffeomorphism of$M$. Further, let me state that I will need a strong form of the isotopy extension lemma which I will require later: when$X$is a manifold without boundary, and$Y$is a compact manifold, the mapgiven by$(\textrm{eval},\textrm{proj})(u,v)=(u\circ v,v)$is a fibration. Here$\textrm{Diff}_c(X)$denotes the space of compactly supported diffeomorphisms of$X$, equipped with the strong (or Whitney)$C^1$topology --- the use of the strong topology is essential in the present answer, as remarked at a later point. In fact, the The map$(\textrm{eval},\textrm{proj})$is a locally trivial fibre bundle, as one can give local trivializations using tubular neighbourhoods; thus the above map is also a fibration, since the base is paracompact (even a metrizable space). The above statement implies most usual forms of the isotopy extension lemma. I will now sketch describe the argument that the homotopy fibre$F$of$\iota:\textrm{Diff}(M)\to\textrm{Emb}(M,M)$at$\textrm{id}_M$is weakly contractible. The argument below It is a modification of Agol's ingenious answer linked above. 10 deleted 4 characters in body $$H = ( H_0 \ast ( H_1 \ast ( H_2 \ast ( H_3 \ast \cdots ) ) ) )$$ 9 added details; [made Community Wiki] In conclusion, we have a compactly supported diffeotopy$\widetilde{\varphi} = g(x)$of$\textrm{int}\ M_{n+1}$(depending continuously on$x\in K$) which extends$\varphi|_{M_n \times I}$. Extend$\widetilde{\varphi} = g(x)$to all of$M$by making it the identity outside$\textrm{int}\ M_{n+1}$. Fundamentally, observe that this extension operation gives a continuous map $\textrm{Diff}_c(\textrm{int}\ M_{n+1}) \to \textrm{Diff}_c(M) \to \textrm{Diff}(M)$ thanks to the strong (or Whitney)$C^1$topology we placed on $\textrm{Diff}_c(\textrm{int}\ M_{n+1})$the space of compactly supported diffeomorphisms, so we are preserving continuity on$x\in K$. Then$\widetilde{\varphi} = g(x)$becomes a compactly supported diffeotopy of the whole manifold$M$depending continuously on$x\in K$. Moreover,$\widetilde{\varphi}_0 = \textrm{id}_M$. Condition (1) above then follows from the fact that$\widetilde{\varphi}$extends$\varphi|_{M_n \times I}$. Note that$f_{n+1}$is continuous since mapping a diffeomorphism to its inverse is continuous in the compact-open$C^1$topology ($\textrm{Diff}(M)$is a topological group); alternatively, we can instead use that$g$is actually continuous with respect to the strong$C^1$topology on$\textrm{Diff}_c(M)$, and that inversion is also continuous in the strong$C^1$topology.$$[H_n(x,s)]_t = (\widetilde{\varphi}_{\min\{s,t\}})^{-1} \circ \varphi_t = ([g(x)]_{\min\{s,t\}})^{-1} \circ [f_n(x)]_t$$As was done for$f_{n+1}$, we can show that$H_n$is continuous. It is also easy to check that$H_n$takes values in$F$:$[H_n(x,s)]_0 = (\widetilde{\varphi}_0)^{-1} \circ \varphi_0 = \textrm{id}_M$, and$[H_n(x,s)]_1 = (\widetilde{\varphi}_s)^{-1} \circ \varphi_1$is a diffeomorphism of$M\$. Moreover, the conditions (2) and (3) above are also straightforward: 
 
 
 
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