An operadic model for a mapping space and its associated spectral sequence

Let X and Y be simplicial sets and K a field. In B. Fresse, Derived division functors and mapping spaces, 2002, Preprint arXiv:math.At/0208091], Fresse has constructed an algebra model over an E_∞K-operad E for the mapping space F(X,Y), whose source X is finite, provided the homotopy groups of the target Y are finite. In this paper, we show that if the underlying field K is the closure of the finite field F_p and the given mapping space is connected, then the finiteness assumption of the homotopy group of Y can be dropped in constructing the E-algebra model. Moreover, we give a spectral sequence converging to the cohomology of F(X,Y) with coefficients in , whose E₂-term is expressed via Lannes’ division functor in the category of unstable -algebra over the Steenrod algebra.