Rational cohomology of section spaces of one point compactification of tangent bundles over closed manifolds.

Question

Let M be a connected closed manifold ( oriented or nonoriented) of finite type (Betti numbers are finite) and $\Gamma(\mbox{TM}_{c})$ is the space of sections of fiberwise one point compactification of tangent bundles. In the paper ''Rational Betti numbers of configuration spaces'' Y. Félix and J.- C. Thomas give a model for rational cohomology of $\Gamma(\mbox{TM}_{c})$. In this paper the manifold is closed oriented and nilpotent. I am looking for general closed manifold (oriented or nonoriented). In particular for closed nonoriented surfaces. I need references for general closed manifold.