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# basedfree loop space and invariant forms

Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact $G$-manifold: the idea is to "average" left invariant forms on $G$ using a Haar measure.

Can we extend this result for non compact infinite dimensional manifolds? In particular, consider the based free loop space $\Omega_bM$ LM$of a manifold$M$; this is an infinite dimensional$S^1$-manifold. Is there a way to compute the cohomology of$\Omega_bM$LM$ using a model of "invariant forms" and the idea of averaging?

By a result of Chen, we know that iterated integrals of differential forms in $M$ yield the correct cohomology of $\Omega_bM$. LM$. Is this model related to Cartan's story of invariant forms? These questions are a bit vague, but I guess how to make them precise is part of my question. 4 edited tags; edited tags 3 added 196 characters in body; added 11 characters in body; edited tags Cartan proved that for a connected compact Lie group$G$the let left invariant differential forms yield the correct cohomology of$G$. Is there The same argument works for a analogous connected compact$G$-manifold: the idea is to "average" left invariant forms on$G$using a Haar measure. Can we extend this result for$H$-spaces? non compact infinite dimensional manifolds? In particular, consider the based loop space$\Omega_bM$of a manifold$M$. M$; this is an infinite dimensional $S^1$-manifold. Is there a way to compute the cohomology of $\Omega_bM$ using a model of "invariant forms"forms" and the idea of averaging?

By a result of Chen, we know that interated iterated integrals of differential forms in $M$ yield the correct cohomology of $\Omega_bM$. Does Is this model fit in somehow with related to Cartan's story of invariant formsin compact Lie groups?

These questions are a bit vague, but I guess how to make them precise is part of my question.

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