Syllabus for Topology Qualifying Exam

The Topology exam covers the material taught in MTH G121 (Topology 1), and the beginning of MTH G221 (Topology 2).

General Topology

  • Topological spaces, continuous maps, homeomorphisms, topological invariants.
  • Compactness, connectedness, path-connectedness.
  • Product topology, quotient topology, identification spaces.

Fundamental Group and Covering Spaces

  • Homotopy of paths, fundamental group, induced homomorphism, homotopy type, homotopy invariance, abelianization.
  • Seifert-van Kampen theorem, fundamental groups of two-complexes.
  • Classification of surfaces, cutting and pasting, fundamental groups of surfaces.
  • Covering spaces: lifting criterion, universal covering, classification of coverings.
  • Applications to problems in combinatorial group theory.

Simplicial and Cellular Homology

  • Finite simplicial complexes, simplicial chain complexes, homology groups, induced homomomorphism, homology with coefficients.
  • Finite CW-complexes, cellular homology, homology of product spaces, Euler characteristic, degrees of maps between spheres.
  • Borsuk-Ulam theorem, Leschetz number, Lefschetz fixed point theorem, other appplications.

References

  • James R. Munkres, Topology, 2nd Edition, Prentice Hall, 2000.
  • Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002.
  • Glen Bredon, Topology and Geometry, Springer-Verlag, GTM #139, 1997.