Moebius Transformations Make up Fundamental Groups of Riemann Surfaces

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 of the Lecture:

- Every good topological space possesses a unique simply connected covering
space called the Universal covering space

- The fundamental group of the topological space shows up as a subgroup of
automorphisms of its universal covering space

- The universal covering map expresses the target space as the quotient of
the universal covering space of the target, by the fundamental group of the target

- A covering map can be used to transport Riemann surface structures from source
to target and vice-versa, thus making it into a holomorphic covering map

- Any Riemann surface is the quotient of the complex plane, or the upper half-plane,
or the Riemann sphere by a suitable group of Moebius transformations isomorphic to
the fundamental group of the Riemann surface

- The study of any Riemann surface boils down to studying suitable subgroups of
Moebius transformations

Keywords:

Covering map, covering space, admissible open set or admissible neighborhood,
simply connected covering or universal covering, local homeomorphism, Riemann
surface structure inherited by a topological covering of a Riemann surface, uniformisation, fundamental group, Moebius transformation