I've been reading Galatius's Park City notes on the Madsen-Weiss theorem (available [here](http://math.stanford.edu/~galatius/notes2.pdf)).

On page 8, he states the following theorem. Let $X$ be a space such that $\pi\_1(X)$ is abelian and acts trivially on the rational cohomology of the universal cover of $X$. Let $V = \oplus\_{n \geq 1} V\_n$ be a graded vector space and $V \rightarrow H^{\ast}(X;\mathbb{Q})$ a linear map such that the induced map $\mathbb{Q}[V] \rightarrow H^{\ast}(X;\mathbb{Q})$ is an isomorphism in degrees $\leq n$. Then there is a map $\mathbb{Q}[s^{-1} V] \rightarrow H^{\ast}(\Omega X;\mathbb{Q})$ which is an isomorphism in degrees $\leq n-1$. Here $s^{-1} V$ is the result of shifting the gradings on $V$ down by one.

He claims that this can be proved by "This can be proved using the Serre spectral sequence for the path-loop fibration over the universal cover $\tilde{X}$".

Despite a lot of work, I have been unable to fill in the details of this use of the Serre spectral sequence. It seems weird -- how on earth is the homology of the universal cover at all related to the homology of $X$?

Can anyone help me? Thank you very much.