Eilenberg-Moore spectral sequence calculation of function space cohomology

The purpose of this article is to introduce an Eilenberg-Moore spectral sequence converging to the cohomology algebra of a function space with an adjunction space as its source. Computability of the spectral sequence is shown by determining explicitly the mod $p$ cohomology algebra of the function space ${\mathcal F}(S, BG)$ of maps from a non-orientable surface $S$ to the classifying space of a simply connected Lie group $G$ whose homology is $p$-torsion free. Let $M$ be an orientable $3$-dimensional manifold. Applying the spectral sequence obtained from a Heegaard splitting of the manifold $M$, we also prove that $H_1(M; {\mathbb{Z}})$ is a direct summand of $H^3({\mathcal F}(M, BG); {\mathbb{Z}})$.