Syllabus for the Algebraic Topology Qualifying exam
Recommended book: Algebraic Topology, Hatcher
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.