Published on Department of Mathematics (https://math.yale.edu)

Home > Printer-friendly > Syllabus for the Algebraic Topology Qualifying exam

Syllabus for the Algebraic Topology Qualifying exam

Recommended book: Algebraic Topology, Hatcher

Typical Syllabus

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.

Visit our web site at http://math.yale.edu for updates and special announcements