## 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.