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.