Math 655: Differential Topology
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Revision as of 14:36, 13 July 2010 by AAEmpty (Talk | contribs) (→Courses for which this course is prerequisite)
Contents
Catalog Information
Title
Algebraic Topology 1.
Credit Hours
3
Prerequisite
Math 313, 342, and Math 553 and 554 or Instructor's consent.
Description
An introduction to manifolds and smooth manifolds.
Desired Learning Outcomes
Prerequisites
Knowledge of basic point topology from Math 553, 554 will be assumed. This includes topological spaces, basis and countability, metric spaces, quotient spaces, fundamental group, and covering maps. We also assume a course in linear algebra (Math 313) and one in introductory analysis (Math 341 and 342).
Minimal learning outcomes
- Manifolds
- Topological and smooth manifolds
- Manifolds with boundary
- Tangent vectors
- Tangent bundles
- Vector bundles and bundle maps
- Cotangent bundles
- Submanifolds
- Submersions, immersions, embeddings
- Inverse and implicit function theorems
- Transversality
- Embedding and approximation theorems
- Differential forms and tensors
- Wedge product
- Exterior derivative
- Orientations
- Stoke's Theorem
- DeRham cohomology
Additional topics
At the discretion of the instructor as time allows. Topics might include Lie groups and homogeneous spaces, Morse functions, Jordan curve theorem, Lefschetz fixed-point theory, degree, Gauss-Bonnet theorem, etc.
