On the Simplicity of the Group of Contactomorphisms

We consider the group Contc(M 2n+1, α) of Cr contactomorphisms with compact support of a contact manifold (M2n+1, α) of dimension (2n + 1) with the Cr topology. We show that the first homology group of the classifying space BContc(M 2n+1, α) for the Cr foliated M2n+1 products with compact support with transverse contact structure α is trivial for 1 ≤ r < n + (3/2). This implies that the identity component Contc(M 2n+1, α)0 of the group Cont r c(M 2n+1, α) of contactomorphisms with compact support of a connected contact manifold (M2n+1, α) is a simple group for 1 ≤ r < n+ (3/2).