Pullback de Rham cohomology of the free path fibration

Abstract:

Let M and N be smooth manifolds and let $\bar B (A)$ be the reduced bar construction on the de Rham complex $\Lambda (M)$ or a suitable subcomplex A of M. For every smooth map $f:N \to M \times M$, the tensor product $\Lambda (N) \otimes \bar B(A)$, equipped with a suitable differential, will yield the correct cohomology for the pullback of the free path fibration $P(M) \to M \times M$ via the smooth map F. Moreover, $\Lambda (N) \otimes \bar B(A)$ can be taken as a de Rham subcomplex of the pullback space.