Math 655: Differential Topology
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 and their topology.
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
Outlined below are topics that all successful Math 655 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.
- 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.