620-322 Topology

Credit Points

Coordinator

Prerequisites

620-231 (1997 Handbook 618-231), 620-311 (1997 Handbook 618-301) and 620-321 (1997 Handbook 618-321).

Semester

Contact

Subject Description

Students completing this subject should comprehend

• the basic concepts and examples of topological spaces;
• the definition of manifolds and the classification of surfaces;
• the idea of homotopy of mappings;
• how to calculate and use the fundamental group;
• the concept of covering spaces and their relationship with fundamental groups;
• the basic ideas of homology theory;

have developed the ability to

• work with the fundamental group and homology groups;
• convert problems involving topological spaces and continuous maps into problems in algebra;
• distinguish between different topological spaces;
• construct homeomorphisms and homotopy equivalences between spaces;

and appreciate

• the basic questions in topology;
• the power of topological methods in dealing with problems involving shape and position of objects and continuous mappings;
• 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 three-hour end-of-semester written examination.