620-322 Topology

Credit Points

Coordinator

Prerequisites

620-231, 620-311 and 620-321.

Semester

Contact

Subject Description

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.

Introduction to topology: homotopy and the fundamental group of a space; covering spaces; simplicial homology. Introduction to manifolds: manifolds, tangent vectors, differential forms. Selection of additional topics: connections, Riemannian metrics, curvature, Gauss-Bonnet theorem; integration on manifolds, de Rham's theorem.

Assessment

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