On the equivarant de-Rham cohomology for non-compact Lie groups
  Let $G$ be a connected and non-necessarily compact Lie group acting on a connected manifold $M$. In this short note we announce the following result: for a $G$-invariant closed differential form on $M$, the existence of a closed equivariant extension in the Cartan model for equivariant cohomology is equivalent to the existence of an extension in the homotopy quotient. 
Camilo Arias Abad and Bernardo Uribe
math.DG