Galois groups of topological covers
Galois theory is actually a general principle that is not restricted to the study of fields where it arose classically. In this video, we make an excursion into the Galois theory as it pertains to topology. We introduce firstly the notion of a covering space or map in topology. These are analogues of field extensions that arise in classical Galois theory. We give examples such as those arising from a group acting freely and discretely on a space. We then define the Galois group of such a covering map which, as in the field theory case, captures the symmetry of the situation. We end by looking at an example of the Riemann surface associated to the square root function, which produces both a Galois group corresponding to a topological cover as well as a Galois group corresponding to the field extension of meromorphic functions. This is an example where both the topological and algebraic theories are connected.