This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.
PrerequisitesAbility to work with topological spaces and continuous maps. Familiarity with basic topological notions including connectedness, path-connectedness, and the subspace, quotient and product topologies. Working knowledge of algebraic notions including group, ring, homomorphism, and isomorphism. (Duke Math 411 and 501 cover these prerequisites and more.)
- Homotopy and homotopy type, cell complexes, homotopy extension property.
- The fundamental group: homotopy of paths, homotopy-lifting property for paths through covering spaces, functoriality, Van Kampen's theorem, covering spaces, classification of covering spaces with the fundamental group, Deck transformations and group actions, K(G,1)s.
- Singular homology, cellular homology, Mayer-Vietoris Sequences, Homology with coefficients, relation with the fundamental group, relative homology, the long exact sequence of relative homology, applications to computing the homology of surfaces, projective spaces, Axioms for homology, Euler characteristic.
- The primary reference is: Algebraic Topology, by A. Hatcher, Chapters 0,1,2.
- Secondary references include: Algebraic Topology: A first course by Greenberg and Harper; Topology by Munkres (for fundamental group and covering spaces); Elements of Algebraic Topology by Munkres (for homology); A Basic Course in Algebraic Topology, by Massey; Elementary Topology, by Viro, Ivanov, Netsvetaev and Kharlamov.