Entry requirements

120 credits including 90 credits Mathematics. Differential topology participation. Modules and homological algebra participation. 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:

• define the various geometric and algebraic concepts introduced and apply and interpret them in concrete examples;
• formulate and apply central theorems in de Rham theory and present their proofs;
• use the theory methods and techniques of the course for problem solving.

Content

The de Rham Complex on Rn. Orientation and Integration, Stokes' theorem. Poincaré Lemmas. The degree of a proper map. The Mayer-Vietoris sequence. Poincare duality on an orientable manifold. The Kunneth formula and the Leray-Hirsch theorem. The Poincare dual of a closed oriented submanifold. The Thom Isomorphism. Vector bundles and cohomology. Poincare duality and the Thom class. The Euler class and the Thom class. Lefschetz fixed-point theorem. Singular homology and cohomology. Poincaré duality for singular (co)chains. Sketch of DeRham's theorem.

Instruction

Lectures and problem solving lessons.

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.

Other directives

The course may not be included in the same degree as Algebraic topology (1MA197).