On the simplicity of the group of contactomorphisms

Abstract

We consider the group $\mathrm{Cont}_c^r (M^{2n+1}, \alpha)$ of $C^r$ contactomorphisms with compact support of a contact manifold $(M^{2n+1}, \alpha)$ of dimension $(2n+1)$ with the $C^r$ topology. We show that the first homology group of the classifying space $B \overline{\mathrm{Cont}}{}_c^r (M^{2n+1}, \alpha)$ for the $C^r$ foliated $M^{2n+1}$ products with compact support with transverse contact structure $\alpha$ is trivial for $1 \le r \lt n+(3/2)$. This implies that the identity component $\mathrm{Cont}_c^r (M^{2n+1}, \alpha)_0$ of the group $\mathrm{Cont}_c^r (M^{2n+1}, \alpha)$ of contactomorphisms with compact support of a connected contact manifold $(M^{2n+1}, \alpha)$ is a simple group for $1 \le r \lt n+(3/2)$.