TITLE: Cohomology of function spaces
QUESTION [6 upvotes]: Let $M,N$ be smooth manifolds and $C^\infty(M,N)$ be the function space with Whitney topology. 
If we know the cohomology groups of $M,N$, can we calculate the cohomology groups of $C^\infty(M,N)$?

REPLY [8 votes]: It seems there exists such a thing: There is a spectral sequence by Anderson that computes the homology of the mapping space out of the cohomology of $M$ and the homology of $N$. Here is a link to the announcement:
http://www.ams.org/journals/bull/1972-78-05/S0002-9904-1972-13034-9/S0002-9904-1972-13034-9.pdf
I have not been able to find the subsequent paper with the proof.