Entry requirements

BSc, Topology

Learning outcomes

In order to pass the course (grade 3) the student should

be able to give an account of important concepts and definitions in differential topology;

be familiar with Sard's theorem and some of its applications, especially Brouwer's fixed point theorem and Whitney's simple embedding theorem;

be able to define and compute the intersection index of two submanifolds, oriented as well as modulo 2;

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;

know Lefschetz's fixed point theorem;

be able to define the concept of Morse function and to outline a proof of its existence;

know 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

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.