Goals of the Lecture: - In the previous couple of lectures, the universal covering space was constructed for a given space as a Hausdorff topological space along with a natural map into the given space. That natural map was shown to be a covering map. In the first part (part A) of this lecture, we showed that the universal covering space we constructed is indeed simply connected and has a universal property. In this lecture (part B), we show that we can naturally identify the fundamental group of the base space with a subgroup of self-isomorphisms of the universal covering space called the Deck Transformation Group - For any covering space, we may define the so-called Deck Transformation Group. This is the subgroup of self-homeomorphisms of the covering space that respect the covering projection map. If the covering space is the universal covering space, then the fundamental group of the base space (the space whose coverings we are concerned with) gets naturally identified with the deck transformation group. Thus the fundamental group of the base acts on the universal covering via the so-called deck transformations. These act along the fibers of the covering projection map. This action is called the Monodromy Action Topics: Path, Fixed-end-point (FEP) homotopy equivalence class, fundamental group, pathwise or arcwise connected, Hausdorff, locally simply connected, universal covering, basic open set, base for a topology, sub-base for a topology, admissible neighborhood, isomorphism of covering spaces, universal property, deck transformation, deck transformation group, monodromy action
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.