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.