  We prove the triviality of the first L2 cohomology class of based path spaces
of Riemannian manifolds furnished with Brownian motion measure, and the
consequent vanishing of L2 harmonic one-forms. We give explicit formulae for
closed and co-closed one-forms expressed as differentials of functions and
co-differentials of L2 two-forms, respectively; these are considered as
extended Clark-Ocone formulae. A feature of the proof is the use of the
temporal structure of path spaces to relate a rough exterior derivative
operator on one-forms to the exterior differentiation operator used to
construct the de Rham complex and the self-adjoint Laplacian on L2 one-forms.
This Laplacian is shown to have a spectral gap.
