A spectral sequence for string cohomology

Let X be a 1-connected space with free-loop space Λ X . We introduce two spectral sequences converging towards H * ( Λ X ; Z / p ) and H * ( ( Λ X ) h T ; Z / p ) . The E 2 -terms are certain non-Abelian-derived functors applied to H * ( X ; Z / p ) . When H * ( X ; Z / p ) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p = 2 .