Semester 2

Unit(s) of Assessment	Weighting Towards Module Mark( %)
Class tests	25
Examination: two hours unseen	75
Qualifying Condition(s)  A weighted aggregate mark of 40% is required to pass this module.

MAT2004 Real Analysis 2. (recommended)
MAT3010 Function spaces (recommended)

This module introduces coordinate-free techniques for treating the topology of manifolds. These methods include Euler characteristics, fundamental groups, and differential forms.

At the end of the module a student should:

• understand what is meant by the topological structure of a space;
• understand the role of basic constructions such as quotient spaces;
• understand basic notions (e.g. continuity, dimension) in terms of topology;
• understand the notion of manifold and their simplest classifications (orientability, Euler characteristic);
• be able to determine the degree of differentiability of manifolds by means of charts;
be able to understand and integrate differential forms and the use Stokes’ theorem.

Introduction to Point Set Topology: continuity, compactness, quotient spaces and the Hausdorff property, homeomorphisms and diffeomorphisms.

Introduction to Manifolds: basic examples of manifolds, coordinate patches (charts), the definition of a manifold with boundary, orientation, Euler characteristic, covering spaces, homotopy and fundamental group.

Differential Forms: tangent spaces and bundles, differential forms, integration of differential forms, Stokes' Theorem, Brouwer's fixed-point theorem.

Additional topics will be selected from:
(1) The geometry of curves and surfaces.
(2) Topological dimension, levels of connectedness, the Cantor set.
(3) The role of curvature, Gauss-Bonnet theorem; Poincare's Theorem on singularities of vector fields.

Teaching is by lectures and tutorials, 3 hours per week for 11 weeks.
Learning takes place through lectures, tutorials, exercises and class tests.

9 May 2011