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

Let X and Y be simplicial sets and image 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 image-operad image for the mapping space image, 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 image is the closure image of the finite field image and the given mapping space is connected, then the finiteness assumption of the homotopy group of Y can be dropped in constructing the image-algebra model. Moreover, we give a spectral sequence converging to the cohomology of image with coefficients in image, whose E2-term is expressed via Lannes’ division functor in the category of unstable image-algebra over the Steenrod algebra.