  A great open problem is: can one learn the topology of the non-smooth path
spaces with an L2 Hodge-deRham theory This one hopes to establish through a
suitable complex of differential forms. Since the space is a Banach manifolds,
and the Hodge theory is based on Hilbert spaces, it is trick to find such a
complex. The relatively simpler Bismut tangent spaces and their tensor
products, whose dual spaces are natural candidates for the complex, cannot be
used, because these spaces are not necessarily closed under the Lie bracket
operation if there is the effect of curvature. In this article we seek out a
class of nice vector fields whose brackets behaves are calculable and behave
nicely the damped tensor vector fields, using an It\^o mp, which is also used
by the authors in `Special It\^o maps and an L2 Hodge theory for one forms on
path spaces. Stochastic processes, physics and geometry: new interplays, I
(Leipzig, 1999), 145-162, CMS Conf. Proc., 28, Amer. Math. Soc., Providence,
RI, 2000').
