The goal of this course is to provide the motivations, definitions and techniques for the translation of topological problems into algebraic ones, hopefully easier to deal with. On successful completion of this module, the student should understand the fundamental concepts of algebraic geometry and should be aware of potential applications of algebraic topological invariants in other fields as theoretical physics , including the computational fluid mechanics and electrodynamics.
General topology: topological spaces and continuous maps, induced, quotient and product topology, metric spaces, Hausdorff spaces, compact spaces, connected spaces, paths and path connected spaces
Manifolds and surfaces: the pancake problems, n-dimensional manifolds, surfaces and classification of surfaces.
Homotopy: Retracts and contractible spaces, paths and multiplication, the fundamental group, the fundamental group of the circle.
Covering spaces: the fundamental group of a covering space, the fundamental group of a orbit space, lifting theory and existence theorems, the Borsuk-Ulam theorem, the Seifert-Van Kampen theorem, the fundamental group of a surface.
Introduction to singular homology: standard and simplicial simplexes.
Czes Kosniowski, A first course in algebraic topology. Cambridge University Press. 1980.