Topology I: smooth manifolds

Course information

Code: MAT1300F
Instructor: Marco Gualtieri, office hours by appointment.
Class schedule: Th2-4 and F2-3, starting September 14th, 2017
Schedule changes: I must be away on Oct 5 and 6 to give a colloquium. I will record a makeup class and provide the link below. My postdoc Dr Joey Van der Leer Duran will hold office hours on Thursday Oct 5 from 2-3:45 in room PG205B (Physical geography building).
Teaching assistant: Josh Lackman
Evaluation: The final grade is a/2 + t/4 + 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: November 16, in class.
Final Exam: Dec 13 1-4PM, BA6183.

Course notes

  • Part 1 (pdf)
  • Part 2 (pdf)
  • Part 3 (pdf)
  • Inverse Function Theorem (pdf, screencast)
  • Constant Rank Theorem (pdf, screencast)
  • Part 4 (pdf) (Flow of a vector field)
  • Part 5 (pdf) (Local structure of smooth maps, regular value theorem)
  • part 6 (pdf) (Transversality)
  • part 7 (pdf) (Stability)
  • part 8 (pdf) (Sard’s theorem)
  • part 9 (pdf) (Application of Sard’s theorem: Brouwer’s fixed point theorem)
  • part 10 (pdf) (Application of Sard’s theorem: Genericity and intersection theory)
  • part 11 (pdf) (Partitions of unity, Whitney embedding, and Tubular neighbourhoods)

Assignments

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

Late assignments will 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.

The No B.S. bonus provides a 10% bonus for an assignment with no false statements.

  • Assignment 1 (tex) (pdf) Due date: October 6
  • Assignment 2 (tex) (pdf) Due date: October 23, by 4 pm at main desk or via email to me and Josh
  • Assignment 3 (tex) (pdf) Due date: November 10
  • Assignment 4 (tex) (pdf) Due date: November 24

Overview of topics

  • Smooth manifolds and smooth maps
  • Manifolds with boundary
  • Differentiation and the tangent bundle
  • Vector fields and flows
  • Transversality, Sard’s theorem and Whitney’s embedding theorem
  • Differential forms and the de Rham complex
  • Integration and Stokes’ theorem
  • de Rham cohomology
  • (bonus topic) The Hodge star operator and Maxwell’s equations

Suggested references

A textbook is not absolutely necessary, as notes and questions will be provided. However I recommend the book Introduction to Smooth Manifolds, 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,