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

Let XX and YY be simplicial sets and KK 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∞KE∞K-operad EE for the mapping space F(X,Y)F(X,Y), whose source XX is finite, provided the homotopy groups of the target YY are finite. In this paper, we show that if the underlying field KK is the closure F¯p of the finite field FpFp and the given mapping space is connected, then the finiteness assumption of the homotopy group of YY can be dropped in constructing the EE-algebra model. Moreover, we give a spectral sequence converging to the cohomology of F(X,Y)F(X,Y) with coefficients in F¯p, whose E2E2-term is expressed via Lannes’ division functor in the category of unstable F¯p-algebra over the Steenrod algebra.