TITLE: Loop space: De Rham cohomology
QUESTION [8 upvotes]: How to calculate  the DeRham cohomology of the free loop space $LM= C^\infty(S^1,M)$ as a Frechet manifold?.   
Edit: It  will be enough for me to know: 
When $H^1_{DR}(LM)$ is not $\{0\}$. 
Bounty is ending within 5 hours.

REPLY [3 votes]: Assuming that de Rham cohomology is ordinary cohomology (which shouldn't be too hard by the standard sheaf-theoretic proof?), $H^1 = Hom(\pi_1,\mathbb{R})$. Now the base point fibration gives rise to a short exact sequence
$$
1 \rightarrow \pi_2(M) \rightarrow \pi_1(LM) \rightarrow \pi_1(M) \rightarrow 1
$$
Hence yes, if $H^1(M;\mathbb{R}) \neq 0$, then also $H^1(LM;\mathbb{R}) \neq 0$. Maybe more importantly, in the simply-connected case $H^1(LM) = H^2(M)$, something that one can also see by hand.