In this paper by Antonelli, Burghelea and Kahn (*Topology*, 1972), a homomorphism $L :\pi_i(\operatorname{Diff}(S^n, D_+^{n})) \rightarrow \Gamma^{n+i+1}$ was used as a tool to detect non-triviality of the homomorphism $ \pi_i(\operatorname{Diff}(S^n, D_+^{n})) \rightarrow \pi_i(\operatorname{Diff}(M^n)) $.

Here $D_n^+$ is the upper hemisphere in the $n$-sphere $S^n$. $\operatorname{Diff}(M^n)$ is the group of $C^\infty$ self-diffeomorphisms of $M^n$ with the $C^\infty$ topology; for $X\subset M^n$, $\operatorname{Diff}(M^n,X)$ is its subgroup of diffeomorphisms acting on $X$ as the identity. Finally, $\Gamma^{n+1}$ is the set of diffeomorphism types of smooth manifolds that are homeomorphic to $S^n$ its construction is in page 12-13 of the linked paper.

**My questions:**

Does there exist a generalization of this result that uses manifolds other than the spheres? In particular, I want to know if there is a homomorphism similar to $L$ that sends $\pi_i(\operatorname{Diff}(M^n))$ to a known space or group,for general $M^n$.

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 $\operatorname{Diff}(M^n)$ when $M$ is not a sphere. For the second part of the question, the homomorphism $L$ sends homotopy groups of $\operatorname{Diff}(S^n,D_n^+)$ into the Kervaire-Milnor group $\Gamma^{n+i+1}$. I'm asking if there is a similar homomorphism for general homotopy groups of $\operatorname{Diff}(M^n)$.