@ YCor. I'm sorry I thought you're asking me to clarify the question. The topology on $Diff(M)$ is the $C^ \infty $-topology. $D^n_+ $ is the upper hemisphere and $\Gamma^{n+i+1}$ is identified with the set of diffeomorphism types of smooth manifolds that are homeomorphic to $S^n$, its construction is in page 12-13.

@ YCor. What I'm asking about in the first part of the question is if there are results about detecting non-triviality of some homotopy groups of Diff(M) when M is not a sphere. For the second part of the question, the homomorphism L sends homotopy groups of Diff(S,D) into Kervaire-Milnor group $Γ$. I asked if there is a similar homomorphism for general homotopy groups of Diff(M).

yes but this does not answer my question, I'm not asking if the group is locally compact. The diffeomorphism group is not locally compact.

I assumed the embeddings are proper and the spaces are smooth.

Thanks André, but I have sketched a proof that the induced map is a Kan fibration using Siebenmann's isotopy extension theorem. How far can that be right?

@Qiaochu Yuan; Yes, that's what I meant thanks, I re-edited the question, sorry I am not a native English speaker.

I'm so grateful to have an answer from you personally, thanks

Sorry, It's fixed now