Thinking about exotic 7-spheres, one can look at the maps $\cdots \rightarrow \Omega^2Diff(D^4, rel \space \partial) \rightarrow \Omega Diff(D^5, rel \space \partial) \rightarrow Diff(D^6, rel \space \partial)$. There are then homomorphisms $\cdots \rightarrow\pi_2Diff(D^4, rel \space \partial) \rightarrow \pi_1 Diff(D^5, rel \space \partial) \rightarrow \pi_0Diff(D^6, rel \space \partial)$.

The map $\pi_1 Diff(D^5, rel \space \partial) \rightarrow \pi_0Diff(D^6, rel \space \partial)$ is onto by an appeal to a well-known theorem of Jean Cerf, so "$\pi_1$ detects the exotic 7-sphere". But his theorem needs dimension at least 5, hence only applies to the right-most map.

Question: Can we lift up to $\pi_2$? What can be said about the map $\pi_2Diff(D^4, rel \space \partial) \rightarrow \pi_1 Diff(D^5, rel \space \partial)$?

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