Math 655: Differential Topology
342 or equivalent.
An introduction to manifolds and smooth manifolds and their topology.
Desired Learning Outcomes
Knowledge of basic point set topology from Math 553, 554 will be assumed. This includes topological spaces, basis and countability, metric spaces, quotient spaces, fundamental group, and covering maps. Basic knowledge of linear algebra (Math 313) and introductory analysis (Math 341 and 342) will also be assumed.
Minimal learning outcomes
Outlined below are topics that all successful Math 655 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.
- Topological and smooth manifolds
- Manifolds with boundary
- Tangent vectors
- Tangent bundles
- Vector bundles and bundle maps
- Cotangent bundles
- Submersions, immersions, embeddings
- Inverse and implicit function theorems
- Embedding and approximation theorems
- Differential forms and tensors
- Wedge product
- Exterior derivative
- Stoke's Theorem
At the discretion of the instructor as time allows. Topics might include Lie groups and homogeneous spaces, Morse functions, de Rham cohomology and the de Rham theorem, Jordan curve theorem, Lefschetz fixed-point theory, degree, Gauss-Bonnet theorem, etc.