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 held in November and January.
The take-home 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)