Topology I: smooth manifolds

Course information

• Code: MAT1300HF
• Instructor: Marco Gualtieri, office hours by appointment.
• Class schedule: W9-10 and F10-12
• Schedule changes: No class on 3 Oct and 10 Oct, instead have F9-12 on 5 Oct and 12 Oct.
• Teaching assistant: Kevin Luk
• Evaluation: The final grade is $$\tfrac{a}{2} + \tfrac{t}{4} + \tfrac{f}{4},$$ where $$a$$ is the average of the four best assignment grades, $$t$$ is the term exam grade, and $$f$$ is the final exam grade, all out of 100.
• Qualifying exam: If the average of your grades from MAT1300F and MAT1301S is at least A-, you will be exempt from the Topology qualifying exam.
• Term Exam: Friday, November 2 in class. Average was 75, std dev was 12.
• Final Exam: Monday, December 10, 2-5PM in BA6183

Course notes

• Week 1 (pdf)
• Week 2 (pdf)
• Week 3 (pdf)
• Week 4 (pdf)
• Week 5 (pdf)
• Week 6 (pdf)
• Week 1-7 cumulative with TOC (pdf)
• Week 8 (pdf)
• Week 9 (pdf)
• Cumulative (pdf)

Assignments

Please discuss the problems, but avoid reading a written solution before you write your own, since these must be original.

Late assignments are not be accepted: please hand in whatever you have at the deadline.

Assignments are marked for correctness, but also clarity. Keep your solutions concise, and make sure the structure of your argument is clear. I suggest that you type out your solutions in LaTeX.

Finally, the no B.S. bonus provides a 10% bonus for an assignment with no false statements.

• Assignment 1 (tex) (pdf) Due date: October 3, 2012.
• Assignment 2 (tex) (pdf) Due date: October 24, 2012.
• Assignment 3 (tex) (pdf) Due date: November 15, 2012.
• Assignment 4 (tex) (pdf) Due date: November 23, 2012.
• Assignment 5 (tex) (pdf) Due date: November 30, 2012.

Overview of topics

• Smooth manifolds and smooth maps
• Manifolds with boundary
• Differentiation and the tangent bundle
• Transversality, Sard's theorem and Whitney's embedding theorem
• Differential forms and the de Rham complex
• Integration and Stokes' theorem
• de Rham cohomology
• The Hodge star operator and Maxwell's equations

Suggested references

A textbook is not necessary, as notes and questions will be provided. However I recommend the book Introduction to Smooth Manifolds, GTM 218, by J. M. Lee. Other useful texts include:

• Warner's Foundations of differentiable manifolds and Lie groups, ISBN 0387908943,
• Bott & Tu's Differential Forms in Algebraic Topology, ISBN 0387906134.
• Milnor's Morse theory, ISBN 0691080089,
• Wells' Differential Analysis on Complex Manifolds, ISBN 0387904190,