620-322 Topology

620-231 or 620-233; 620-311 and 620-321.

This subject introduces the basic concepts and examples of topological spaces, the definition of manifolds and the classification of surfaces, the idea of homotopy of mappings, the concept of covering spaces and their relationship with fundamental groups; and the basic ideas of homology theory. Students should develop the ability to work with the fundamental group and homology groups, to calculate and use the fundamental group, to convert problems involving topological spaces and continuous maps into problems in algebra, to distinguish between different topological spaces, and to construct homeomorphisms and homotopy equivalences between spaces. This subject investigates the basic questions in topology. It demonstrates the power of topological methods in dealing with problems involving shape and position of objects and continuous mappings, and shows how topology can be applied to many areas, including geometry, analysis, group theory and physics.

Topics include topological spaces and continuous maps; quotient spaces; homotopy and fundamental groups; surfaces; covering spaces; introduction to homology theory.

Up to 24 pages of written assignments and a 3-hour end-of-semester written examination.