Next, I suggest that you think about two special cases:

(1) When $\pi_1(X)$ is trivial. This will involve using the Serre spectral sequence for the contractible path fibration over $X4 with fiber $\Omega X$, including the multiplicative structure.

(2) When $\pi_k(X)$ is trivial for all $k>1$, i.e. when $\tilde X$ is contractible. For example, when $X$ is a circle. Oh, actually Galatius is saying something wrong in this case. He probably should be using homology instead of cohomology ...

Next, I suggest that you think about two special cases:

(1) When $\pi_1(X)$ is trivial. This will involve using the Serre spectral sequence for the contractible path fibration over $X$ with fiber $\Omega X$, including the multiplicative structure.

(2) When $\pi_k(X)$ is trivial for all $k>1$, i.e. when $\tilde X$ is contractible. For example, when $X$ is a circle. Oh, actually Galatius is saying something wrong in this case. He probably should be using homology instead of cohomology ...