A typical syllabus includes items such as:
- Cell complexes, simplicial complexes, manifolds
- Homotopy, homotopy equivalence, retracts, homotopy extension property
- Fundamental group: Seifert-Van Kampen, covering spaces and groups, lifting
- Fundamental groups and topological classification of 2d manifolds
- Homology: simplicial, singular, and cellular homology with coefficients, relative homology, long exact sequence
- Mayer-Vietoris sequence, excision, Euler characteristic, axioms for homology
- Applications: Brouwer fixed point theorem, Borsuk-Ulam theorem, Lefschetz fixed point theorem
- Cohomology: Simplicial, singular, and cellular cohomology with coefficients, universal coefficient theorem, ring structure, Kunneth formulae
- Cohomology rings of surfaces, real and complex projective spaces
- Orientations, degrees of maps. Poincare duality simplicial