In this paper, 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)) $.

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 $M^n$.