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

Let $X$ and $Y$ be simplicial sets and ${\mathbb{K}}$ a field. In [1], Fresse has constructed an algebra model over an $E_\infty{{\mathbb{K}}}$-operad ${\mathcal E}$ for the mapping space ${\mathcal F}(X, Y)$, 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 ${\mathbb{K}}$ is the closure $\overline{\mathbb{F}}_p$ of the finite field ${\mathbb{F}}_p$ and the given mapping space is connected, then the finiteness assumption of the homotopy group of $Y$ can be dropped in constructing the ${\mathcal E}$-algebra model. Moreover, we give a spectral sequence converging to the cohomology of ${\mathcal F}(X, Y)$ with coefficients in ${\overline{\mathbb{F}}}_p$, whose $E_2$-term expressed via Lannes' division functor in the category of unstable ${\overline{\mathbb{F}}}_p$-algebra over the Steenrod algebra.

[1] B. Fresse, Derived division functors and mapping spaces, preprint arXiv:math.At/0208091 (2002).