This course covers three areas of algebraic topology:
Each of these topics makes up about a third of the course.
The two tests will approximately cover the first two topics,
respectively, and will be scheduled in November and January.
The oral final exam covers the whole course.
- DeRham cohomology in the plane
- Path integrals, winding numbers, deRham cohomology, Mayer-Vietoris,
fixed point theorems, Jordan curve theorem. (cf. Fulton, Chap. 1-6,10)
- Covering spaces and the fundamental group
- Fundamental group, Hurewicz theorem, covering spaces, group actions,
deck transformations, classification and existence of covering spaces,
van Kampen Theorem. (cf. Fulton, Chap. 11-17)
- Singular homology
- Eilenberg-Steenrod axioms, homology and fundamental group of
spheres and tori, fixed point and separation theorems in higher dimensions.
(cf. Bredon, Chap. 4)