Entry requirements

120 credits including 90 credits in mathematics. Participation in Real Analysis. Basic Topology is recommended. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

On completion of the course, 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.
• define the fundamental group and account for its connections with covering spaces.

Content

Manifolds, submanifolds, smooth maps, tangent space and tangent bundles. Implicit function theorem. Immersions and embeddings. Submersions. Transversality. Sard's theorem (sketch/proof of simple version). Brouwer's fix point theorem. Intersection theory modulo 2. Jordan-Brouwer's separation theorem. Orientation. Oriented intersection theory. The degree of a map. Hopf's degree theorem. Vector fields on manifolds and Poincaré-Hopf's theorem. The fundamental group and covering spaces. Morse functions. Sketch of classification of one- and two-dimensional manifolds.

Instruction

Lectures and problem solving sessions.

Assessment

Written assignments during the course combined with an oral follow-up examination at the end of the course (10 credits).

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.