- The Fundamental Group. Definition and basic properties. The fundamental group of the circle. Van Kampen’s theorem. Further calculations and applications.
- Covering Spaces. Lifting properties. The universal cover. The classification theorem (Galois correspondence). Deck transformations and group actions.
- Homology Theory. Definitions of simplicial and singular homology. Homotopy invariance. Exact sequences and excision. Degree of maps. Cellular homology of CW complexes. Mayer-Vietoris sequences. Applications. The language of categories and functors. Axioms for homology.
Time permitting, there might also be a brief introduction to cohomology or to higher homotopy groups, but usually these topics are covered in MATH 7530.