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\begin{document}

\begin{center}
{\Large \bf Eilenberg-Moore spectral sequence calculation 
of function space cohomology}
\end{center}

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 $\F(S, BG)$ 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 $H^3(\F(M, BG); {\mathbb Z})$. 

\end{document}
