5
$\begingroup$

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 $\pi_2Diff(D^4, rel \space \partial) \rightarrow \pi_1 Diff(D^5, rel \space \partial)$?

| cite | improve this question | |
$\endgroup$
  • 2
    $\begingroup$ 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. $\endgroup$ – Ryan Budney Sep 11 '10 at 5:50
  • $\begingroup$ 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... $\endgroup$ – Romeo Sep 11 '10 at 17:16
  • 3
    $\begingroup$ 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 $n$. $\endgroup$ – Ryan Budney Sep 11 '10 at 18:04
  • $\begingroup$ Thanks. Ryan also has some relevant remarks on this topic here: mathoverflow.net/questions/7892/… I'd mark this question as answered, but everything appears in the comments :( $\endgroup$ – Romeo Sep 14 '10 at 15:50
  • $\begingroup$ 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. $\endgroup$ – Ryan Budney Sep 15 '10 at 17:03
1
$\begingroup$

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).

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.