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.