Math 656: Algebraic Topology
Contents
Catalog Information
Title
Algebraic Topology 2.
Credit Hours
3
Prerequisite
Description
A rigorous treatment of the fundamentals of algebraic topology, including homotopy (fundamental group and higher homotopy groups) and homology and cohomology of spaces.
Desired Learning Outcomes
Prerequisites
Topology of manifolds, tensors, and orientation is assumed from Math 655. A basic knowledge of fundamental groups and covering spaces, as in Math 554, will also be required.
Minimal learning outcomes
Outlined below are topics that all successful Math 656 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems, including calculations.
- Fundamental group
- Constructions, homotopy
- Van Kampen Theorem
- Covering spaces and group actions
- Homology
- Simplicial, singular, cellular
- Exact sequences and excision
- Mayer-Vietoris sequences
- Homology with coefficients
- Homology and the fundamental group
- Cohomology
- Universal coefficient theorem
- Cup product
- Poincare duality
- Homotopy
- Constructions
- Whitehead's theorem
Additional topics
At the discretion of the instructor as time allows. Possible topics include de Rham cohomology and the de Rham theorem, additional duality theorems, including Alexander and Lefschetz duality, the Hurewicz theorem, fiber bundles, fibrations, obstruction theory.