MA3F1 Introduction to Topology
Lecturer: Saul Schleimer
Term(s): Term 1
Status for Mathematics students: List A
Commitment: 30 one-hour lectures
Assessment: 85% 3 hour examination, 15% assignments
Formal registration prerequisites: None
Assumed knowledge:
- Topological spaces
- Continuous functions
- Homeomorphisms
- Compactness
- Connectedness
MA136 Introduction to Abstract Algebra:
- Groups
- Subgroups
- Homomorphisms and Isomorphisms
Useful background:
- Interest in geometry e.g., MA243 Geometry
- More experience with groups e.g., MA249 Algebra II: Groups and Rings
Synergies: The following modules go well together with Introduction to Topology:
Leads to: The following modules have this module listed as assumed knowledge or useful background:
- MA3H6 Algebraic Topology
- MA475 Riemann Surfaces
- MA4H4 Geometric Group Theory
- MA4J7 Cohomology and Poincare Duality
- MA4M6 Category Theory
- MA4M7 Complex Dynamics
Content: Topology is the study of properties of spaces invariant under continuous deformation. For this reason it is often called "rubber-sheet geometry''. The module covers: topological spaces and basic examples, compactness, connectedness and path-connectedness, identification topology, Cartesian products, homotopy and the fundamental group, winding numbers and applications, an outline of the classification of surfaces.
Aims: To introduce and illustrate the main ideas and problems of topology.
Objectives:
- To explain how to distinguish spaces by means of simple topological invariants (compactness, connectedness and the fundamental group)
- To explain how to construct spaces by gluing and to prove that in certain cases that the result is homeomorphic to a standard space
- To construct simple examples of spaces with given properties (e.g. compact but not connected or connected but not path connected).
Books:
Chapter 1 of Allen Hatcher's book Algebraic Topology
For more reading, see the Moodle Pages (link below). MA Armstrong, Basic Topology Springer (recommended but not essential).