11
$\begingroup$

I am mostly interested in the case when you have a smooth degree $d$ algebraic surface $X$ over $\mathbb C$ and we can define three distinct groups: $\pi_0(\mathrm{Diff}^+(X))$, $\pi_0(\mathrm{Homeo}^+(X))$ which is isomorphic to $O(H_2(X;\mathbb Z),\langle\cdot,\cdot\rangle)$, and $\pi_0(\mathrm{Symp}(X))$.

Question: What is known about each of these groups? Are they ever finitely generated (for diffeo/symplectomorphisms)?

What about the relative mapping class group case, i.e. if $\Sigma\subset X$ is a smooth algebraic curve, then what is known about $\pi_0$ of the group of homeo/diffeo/symplectomorphisms which fix $\Sigma$ setwise?

$\endgroup$
3
  • 1
    $\begingroup$ The situation is in general fairly grim, see: mathoverflow.net/questions/161768/… $\endgroup$
    – Igor Rivin
    Nov 11, 2017 at 1:04
  • $\begingroup$ I don't think there has been much in the way of progress in the smooth case since the comments in the post Igor links. I think you'll find most of what's known in the smooth case by looking at Danny Ruberman's papers on invariants of diffeomorphism and at papers citing those. (There is a completely different literature on the symplectic mapping class group and its variants. You might start with Paul Seidel's thesis.) I have no idea but would like to know if it's possible to understand higher homotopy or cohomology groups of the space $B\text{Homeo}(M)$ for any $M$. $\endgroup$
    – mme
    Nov 11, 2017 at 1:35
  • 1
    $\begingroup$ For something new in a different direction, Dave Gabai's new preprint on the 4D Lightbulb theorem says something about the diffeomorphism group of a manifold with boundary, but it seems hard to get a statement about closed manifolds. $\endgroup$
    – mme
    Nov 11, 2017 at 1:47

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.