Goals of Lecture 10: - To explore the reasons for the fundamental group occurring both as the inverse image of any point under the universal covering map as well as a subgroup of automorphisms of the universal covering space - To understand the notions of lifting property, unique-lifting property and uniqueness-of-lifting property - To understand the Covering Homotopy Theorem - To note that surjective local homeomorphisms have the uniqueness-of-lifting property - To note that a surjective local homeomorphism is a covering iff it has the path-lifting property - To deduce that covering maps have the unique path-lifting property Topics: Lifting of a map, lifting of a path, lifting property, unique-lifting property, uniqueness-of-lifting property, Covering Homotopy Theorem, local homeomorphism, unique path-lifting property, existence of lifting, fundamental group, universal covering
The subject of algebraic curves (equivalently compact Riemann surfaces) has its origins going back to the work of Riemann, Abel, Jacobi, Noether, Weierstrass, Clifford and Teichmueller. It continues to be a source for several hot areas of current research. Its development requires ideas from diverse areas such as analysis, PDE, complex and real differential geometry, algebra---especially commutative algebra and Galois theory, homological algebra, number theory, topology and manifold theory. The course begins by introducing the notion of a Riemann surface followed by examples. Then the classification of Riemann surfaces is achieved on the basis of the fundamental group by the use of covering space theory and uniformisation. This reduces the study of Riemann surfaces to that of subgroups of Moebius transformations. The case of compact Riemann surfaces of genus 1, namely elliptic curves, is treated in detail. The algebraic nature of elliptic curves and a complex analytic construction of the moduli space of elliptic curves is given.