Topological Spaces
- Compactifications, various characterizations of paracompactness in terms of coverings and partitions of unity
- Metrization, completion, uniform spaces
- General topological constructions such as induced, coinduced topologies, adjunction spaces, etc.
Fundamental Notions of Algebraic Topology
- Homotopy (deformation, homotopy type, fundamental group), (Universal) covering space
- Mapping cone, mapping cylinder, suspension
- Eilenberg-Steenrod axioms for (co) homology theories, uniqueness theorems
- Simplicial sets, singular, simplicial, and Cech Theories
- Derived functors, EXT, TOR, Universal Coefficient theorem, Kunneth formula
Computation of (Co) homology and Fundamental Groups
- Graphs, compact 2-manifolds (sphere, torus, projective plane, sphere with handles and cross caps), adjunction spaces, topological spaces
Applications
- Invariance of dimension, existence of extensions and retractions
- Fixed point theorems, fundamental theorems of algebra
Past Exams
Below are some Doctorate Algebraic Topology exams from previous years. They should give you an idea of the range of topics covered by the exams.
References
Lin and Lin: Set Theory
J.R. Munkres: Topology
Dugundji: Topology
Greenberg & Harper: Algebraic Topology
Hatcher: Algebraic Topology
Hu: Homology Theory
Spanier: Algebraic Topology
