We use differential forms on loop spaces to prove that the fundamental group of certain geometric transformation groups is infinite. Examples include both finite and infinite dimensional Lie groups. The finite dimensional examples are the conformal group of S^4k+1 for a family of nonstandard metrics, and the group of pseudo-Hermitian transformations of a compact CR manifold. Infinite dimensional examples include the group of strict contact diffeomorphisms of a regular contact manifold, and other groups coming from symplectic and contact geometry.
Maeda et al. (Thu,) studied this question.