Students should be able to demonstrate mastery of relevant vocabulary, and use the vocabulary fluently in their work. They should know common examples and counterexamples, and be able prove that these examples and counterexamples have properties as claimed. Additionally, students should know the content (and limitations) of major theorems and the ideas of the proofs, and apply results of these theorems to solve suitable problems, or use techniques of the proofs to prove additional related results, or to make calculations and computations.  For more detailed information visit the Math 656 Wiki page.

1. Fundamental group and homotopy
   • Constructions
   • Van Kampen Theorem
   • Covering spaces and group actions
   • Higher homotopy groups
2. Homology
   • Simplicial, singular, cellular
   • Exact sequences and excision
   • Mayer-Vietoris sequences
   • Homology with coefficients
   • Homology and the fundamental group
3. Cohomology
   • Universal coefficient theorem
   • Cup product
   • Poincare duality