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 be able to

• give an account of central concepts and definitions in differential topology;
• state Sard's theorem and some of its applications;
• define and compute mapping degree and intersection number of two submanifolds;
• define index of a vector field and state the Poincaré-Hopf theorem;
• define Morse function and outline a proof of existence;
• state the classification of one- and two-dimensional manifolds.

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.