Qualifying exams in topology

The topology qualifying exam covers topics in algebraic topology.
  1. Some point set topological concepts: basic definitions, compactness, separation axiom, connectedness/path-connectedness, retractions, contractibility, quotient topologies.
  2. Homotopy theory: CW complexes, homotopic maps, properties of homotopy in CW complexes, homotopy equivalence, homotopy extension.
  3. Some group theory: free groups, free products, universal properties, presentations of groups.
  4. Brief overview of basic category-theoretic definitions: categories, functors, natural transformations, examples.
  5. The fundamental group: a brief discussion of homotopy groups and their functoriality, long exact sequences of homotopy groups, definition of the fundamental groupoid and the fundamental group, calculation of the fundamental group of a circle, winding numbers, the van Kampen theorem together with examples of fundamental group calculations, K(π, 1) spaces and their properties.
  6. Covering spaces: basic definitions, lifting properties, deck group actions, the Galois correspondence between covers and subgroups of the fundamental group.
  7. Brief overview of (co)homological algebra: (co)chain complexes, (co)chain maps, exact sequences, (co)homology, long exact sequences induced by short exact sequences.
  8. Homology: cellular and singular homology and their equivalence, reduced homology, relative homology, excision, Mayer–Vietoris sequences, the Künneth formula, examples, first homology and the fundamental group, homology with coefficients, definition of cohomology and calculation of examples.
  9. Applications of homology: orientability, degrees of mappings, Lefschetz fixed point theorem, Brouwer fixed point theorem, invariance of domain Borsuk–Ulam theorem.
  10. Structures on cohomology rings: universal coefficients theorem for homology/cohomology, the cup and cap products, calculating cohomology rings, intersection numbers, duality theorems.
The following textbooks are recommended:
  • Bredon, Topology and geometry
  • Hatcher, Algebraic topology
  • Spanier, Algebraic topology
Sample Exams