Lifting properties of covers and deck transformations
In this video, we how the fundamental group, defined as homotopy equivalence classes of loops acts as via deck transformations on covers. This links the theory to the Galois theory in topology. The key to defining the Galois action is the path-lifting property for covers. We go through slowly how this definition works and then look at the main result which show that given a cover of a path connected, locally path connected space X, the fundamental group of X acts transitively on the fibres of the cover and what's more, the stabiliser of the action is given by the fundamental group of the cover.