Difference between revisions of "Math 656: Algebraic Topology"
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Revision as of 15:46, 3 April 2013
Contents
Catalog Information
Title
Algebraic Topology.
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. Students should be able to demonstrate mastery of relevant vocabulary, and use the vocabulary fluently in their work. They should know common examples and counterexamples, and be able prove that these examples and counterexamples have properties as claimed. Additionally, students should know the content (and limitations) of major theorems and the ideas of the proofs, and apply results of these theorems to solve suitable problems, or use techniques of the proofs to prove additional related results, or to make calculations and computations.
 Fundamental group and homotopy
 Constructions
 Van Kampen Theorem
 Covering spaces and group actions
 Higher homotopy groups
 Homology
 Simplicial, singular, cellular
 Exact sequences and excision
 MayerVietoris sequences
 Homology with coefficients
 Homology and the fundamental group
 Cohomology
 Universal coefficient theorem
 Cup product
 Poincare duality
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, additional homotopy theory, including Whitehead's theorem, the Hurewicz theorem, fiber bundles, fibrations, obstruction theory.