Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions

An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/

Goals: * To ask the question as to when the quotient of a space, by a subgroup of automorphisms (self-isomorphisms) of that space, becomes again a space with good properties. For example: when does the quotient of a Riemann surface, by a subgroup of holomorphic automorphisms, again become a Riemann surface?

* To define properly discontinuous (free) actions and note that they are fixed-point-free

* To see that the action of the Deck transformation group on the covering space is properly discontinuous

* To define Galois (or) Regular (or) Normal coverings and characterize them precisely as quotients by properly discontinuous actions


Keywords: upper half-plane, biholomorphism class (or) holomorphic isomorphism class, complex torus, projective special linear group, unimodular group, quotient by a subgroup of automorphisms, quotient by the Deck transformation group, orbits of a group action, quotient topology, properly discontinuous action, action without fixed points, transitive action, admissible neighborhood, Galois covering (or) Normal covering (or) Regular covering, covariant functor, normal subgroup, equivalence relation induced by a group action, open map, unique lifting property, covering homotopy theorem, Riemann sphere, stabilizer (or) isotropy subgroup, ramified (or) branched covering