Iterated Integrals of Differential Forms and Loop Space Homology

Abstract

We continue to define iterated integrals 5w ... w, where wl, *., w, are 1forms. Such iterated integrals will be defined for forms wi, *-*, w, of arbitrary degrees on a manifold or, more generally, a suitably defined differentiable The space P(X) of piecewise smooth paths on a manifold X turns out to be a space. If wi, *.., w, are forms of respective degrees P1, ..., Pr on X, then the iterated integral Wi .. Wr is a form of degree Pi + * -1+ Prr on P(X). Let P(X; x0, xl) denote the subspace of P(X) consisting of paths from a given point x0 to another given point xl. We are interested in those iterated integrals or linear combinations of iterated integrals, which give rise to de Rham cohomology classes of QSX = P(X; x0, x0) and shall prove an iterated integral version of a loop space de Rham theorem, which includes the following