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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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How far you can lift an element of $\pi_0 Diff(D^n, \text{ rel } \partial)$ up the pseudo-isotopy ladder to elements $\pi_k Diff(D^{n-k},\text{ rel }\partial)$ (as much as anyone gives these things names, which isn't much) tends to be called the `Gromoll degree' of that element. This comes from a 1966 paper of D.Gromoll in Math. Ann. In this particular case I don't believe anyone knows anything beyond Cerf's theorem, as $Diff(D^4)$ is very much an unknown object. – Ryan Budney Sep 11 '10 at 5:50
Does this mean that at the present state of knowledge nothing can be said at all here? Yikes. Something for me to work on then... – Romeo Sep 11 '10 at 17:16
There's very little known at present. Other than (1) the "Cerf-Morlet Comparison Theorem", that $Diff(D^n, \text{ rel } \partial) \simeq \Omega^{n+1}(Aut_{PL} \mathbb R^n/O_n)$ and (2) the argument that $Diff(D^n, \text{ rel } \partial)$ has the homotopy-type of the space of round metrics on $S^n$, which are both true for all $n$. – Ryan Budney Sep 11 '10 at 18:04
Thanks. Ryan also has some relevant remarks on this topic here:… I'd mark this question as answered, but everything appears in the comments :( – Romeo Sep 14 '10 at 15:50
Regarding your 1st comment -- if you were to get some nice structural results on $Diff(D^4)$ of any flavour I'm quite sure the mathematics community would be very appreciative. That said, it's pretty widely regarded as a difficult problem, closely related to problems like the 4-dimensional Schoenflies problem and the smooth 4-dimensional Poincare conjecture, which are also regarded as being quite difficult. – Ryan Budney Sep 15 '10 at 17:03
up vote 1 down vote accepted

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).

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