Relation between a universal cover, a fibre, and the fundamental group

Question

As the title states, I'm looking the relationship between the three. From what I've seen from other question, if $X$ is a space, and $X^*$ a cover, then $X = X^*/\pi_1(X)$, but I'm not entirely sure how this relates to the fibres of the cover.

Answer 1

The point is that $\pi_1(X)$ acts on the fibers of the covering map. If you want connected covers, then it acts transitively. A standard theorem is that for a sufficiently nice space $X$ (if I recall correctly, it must be locally connected and semi-locally simply connected), its connected covers are in natural correspondence with subgroups of $\pi_1(X)$. The correspondence sends the cover to the stabilizer of a point in one of the fibers.

Can you guess which subgroup corresponds to the universal cover? What does this say about the fibers of a universal cover?

Answer 2

The fibres of the universal cover are set theoretically equivalent to the fundamental group. See covering homotopy theorem...