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 $\pi\_2Diff(D^4, rel \space \partial) \rightarrow \pi\_1 Diff(D^5, rel \space \partial)$?



Exotic spheres detected in higher homotopy 
My answer here is to just point to Ryan Budney's comments above - they seem to cover all that is known at present (ie, very little).


