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 this lecture, we show that the universal covering space we constructed is indeed simply connected and has a universal property - We show that the universal covering space we have constructed is also a covering space for any other covering space. We further show that any covering space which is simply connected is homeomorphic to the universal covering space we have constructed. It follows that any two simply connected covering spaces thereby are not only just homeomorphic, but homeomorphic by a map that respects the covering projections, i.e., are isomorphic as covering spaces; in fact, even the isomorphism becomes unique if a point of the source and one of the target are fixed. These results show the universality of a simply connected covering space, which is why such a space is called "the" universal covering space 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
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.