contactomorphism of $S^{2n+1}$ for n>1

Question

Is there some result about contactomorphism groups of $S^{2n+1}$ or $T^{2n-1}$ for n>1? For example, do we know the rank of $\pi_{i}(Cont(S^{2n+1})) \otimes \mathbb{Q}?$ where "Cont" means the contactomorphism group, and $S^{2n+1}$ with standard contact structure.

Answer 1

I know of no technique capable of bounding above the homotopy groups of a symplectomorphism group in dimension $\geq 6$, nor of a contactomorphism group in dimension $\geq 5$.

There are, however, techniques for obtaining non-trivial elements in $\pi_i(\mathsf{Cont}(M))$. These were first explored, in the symplectic context, by Seidel, who showed that $\pi_1$ of the Hamiltonian automorphism group of a symplectic manifold has a natural representation on quantum cohomology. Analogues and extensions for contact manifolds, using linearized contact homology, have been developed by Bourgeois. He finds a $\mathbb{Z}^3$ inside $\pi_1 \mathsf{Cont}(T^5)$, for instance.

I think this is a promising research topic. For instance, one knows that $\pi_0\mathsf{Diff}(S^5)$ is trivial; what about $\pi_0\mathsf{Cont}(S^5)$? (Disclaimer: It's possible that this is already known.)

Answer 2

This might be of some help: arXiv:1304.5785. In Section 4, the evaluation map is used to detect a non--trivial element in higher homotopy groups of the contactomorphism group of the standard sphere.

Regarding the torus case, I am only aware of Tim's reference to Bourgeois work, Geiges paper on the 3--torus and its preceding work for torus bundles with J. Gonzalo. Yet again, this is 3--dimensional.

Note that the study of the contactomorphism group is tantamount to that of the space of contact structures, supposing that the diffeomorphism group is (or thought as if it were) known. This is the approach in the previous articles. In particular the contactomorphism group of the standard 3--sphere is known to retract to U(2) since the appearance of Eliashberg '92 "Contact 3--manifolds twenty years since J. Martinet's work".