M.S. Level Material

M.S. Topology Syllabus

Algebraic Topology

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

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