Is Ham($S^2$) homeomorphic to SO(3)?

Question

Is it true that Ham($S^2$,\omega), the group of Hamiltonian diffeomorphisms of the sphere $S^2$ with its standard symplectic structure $\omega$, is homeomorphic to SO(3)?

Answer 1

The correct statement is:

Claim. The group $Ham(S^2)=Ham(S^2,\omega)$ of Hamiltonian diffeomorphisms of the $2$-sphere $S^2$ (endowed with the standard symplectic structure $\omega$) is homotopy equivalent to the group of rotations $SO(3)$.

Notice that, while $SO(3)$ is a finite-dimensional Lie group, the group $Ham(S^2)$ (i.e. the time-$1$ maps of Hamiltonian flows) is an infinite dimensional topological manifold with the $\mathcal{C}^\infty$ topology, i.e. that of "uniform convergence of all derivatives".

Probably you can reconstruct a complete proof of the above claim by looking at the details in the following references (which I just found online by a google search):

1. In general, there are inclusions $$Ham(M,\omega)\subseteq Symp_0(M,\omega)\subseteq Symp(M,\omega)\subseteq Diff^{+}(M)$$ with notations as in this short survey by Dusa McDuff.¹ On page 2 it's stated that, if $H^1(M,\mathbb{R})=0$ (which is the case of $S^2$) then $Ham(M,\omega)=Symp_0(M,\omega)$.

2. In the same survey, on page 3, it's stated that in dimension 2 "Moser's argument" implies that $Symp_0(M,\omega)$ is homotopy equivalent to $Diff^+(M)$.

3. Theorem (Smale): The group of orientation preserving diffeomorphisms of $S^2$ deformation-retracts onto $SO(3)$, so it's homotopy equivalent to the latter.

¹McDuff, Dusa, A survey of the topological properties of symplectomorphism groups, Tillmann, Ulrike (ed.), Topology, geometry and quantum field theory. Proceedings of the 2002 Oxford symposium in honour of the 60th birthday of Graeme Segal, Oxford, UK, June 24–29, 2002. Cambridge: Cambridge University Press (ISBN 0-521-54049-6/pbk). London Mathematical Society Lecture Note Series 308, 173-193 (2004). arXiv:math/0404340, ZBL1102.57013, MR2079375.