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

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, levels of connectedness, dimension and compactness from topological point of view, 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 and variations, Brouwer's fixed-point theorem.

Additional topics will be selected from:
(1) De Rham Cohomology: sequences, exactness, the de Rham complex, homotopy, the Poincaré Lemma, de Rham cohomology, Betti numbers
(2) The role of curvature, Gauss-Bonnet theorem; Poincare's Theorem on singularities of vector fields;
(3) Hyperbolic manifolds, Poincare disk and hyperbolic plane, Kleinian groups, geodesic flow;
(4) Abstract topological spaces, infinite dimensional topological spaces, Cech-Stone and other compactifications.