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.)