Loop-fusion Cohomology and Transgression

‘Loop-fusion cohomology’ is defined on the continuous loop space of a manifold in terms of Čech cochains satisfying two multiplicative conditions with respect to the fusion and figure-of-eight products on loops. The main result is that these cohomology groups, with coefficients in an abelian group, are isomorphic to those of the manifold and the transgression homomorphism factors through the isomorphism. In this note we present a refined Čech cohomology of the continuous free loop space LM of a manifold M (or we could work throughout with the energy space instead). Compared to the standard theory, the cochains are limited by multiplicativity conditions under two products on loops, the fusion product (defined by Stolz and Teichner [ST]) and the figure-of-eight product (which appears implicitly in Barrett [Bar91] and explicitly in [KM13]). The main result of this paper is that the resulting ‘loop-fusion’ cohomology, Ȟ lf(LM ;A), recovers the cohomology of the manifold directly on the loop space.