homology ring of loop space is group ring with loop space of universal cover

Let be an anima and

be a fiber sequence for the universal cover.

Then the homology ring of is given by the group ring

using loop space is a coproduct of the loop space of the universal cover we get

This shows the additive isomorphism

Now we want to identify the multiplication:
Here under the above equivalence the connected corresponding to acts (under the identification of used by connected components of an anima valued group equivalent) by mapping the -th component to the resp. -th component - depending whether we multiply from the left or the right.

Thus

Therefore we get the multiplication

and analogously

Furthermore the inclusion

is a map of loop spaces, hence is a map of groups.

Thus the multiplication of since

is map of graded rings (and inclusion of connected components).

Therefore we get in general

which is precisely the presentation of a group ring :)