Entry requirements

120 credits including 90 credits in Mathematics. Basic Topology and Real Analysis recommended.

Learning outcomes

In order to pass the course (grade 3) the student should<br /><ul> <li> be able to give an account of important concepts and definitions in differential topology;</li> <li> be familiar with Sard's theorem and some of its applications, especially Brouwer's fixed point theorem and Whitney's simple embedding theorem;</li> <li> be able to define and compute the intersection index of two submanifolds, oriented as well as modulo 2;</li> <li> be able to define the degree of a map and to prove its invariance under homotopies, know Hopf's degree theorem and how to compute degrees in specific cases;</li> <li> know Lefschetz's fixed point theorem;</li> <li> be able to define the concept of Morse function and to outline a proof of its existence;</li> <li> know the classification of one- and two-dimensional manifolds.</li></ul>

Content

Manifolds, submanifolds, smooth maps, tangent space, tangent bundles. The constant rank theorem. Immersions and embeddings. Submersions. Transversality. Sard's theorem. Brouwer's fix point theorem. Intersection theory modulo 2. Jordan-Brouwer's separation theorem. Orientation. Oriented intersection theory. Lefschetz's fix point theorem. The degree of a map. Hopf's degree theorem. Vector fields on manifolds and Poincaré-Hopf's theorem. Morse functions. Classification of one- and two-dimensional manifolds.

Instruction

Lectures and problem solving sessions.

Assessment

Compulsory assignments. Oral examination may be given.