# Topology I: smooth manifolds

## Course information

Code: MAT1300F
Instructor: Marco Gualtieri, office hours by appointment.
Class schedule: W12-2 and F11-12, starting September 19th, 2018
Schedule changes: I must be away on Sep 12 and 14 for a funding agency meeting. I’ll make up this time during term.
Teaching assistant: Lennart D\"oppenschmitt, TA office hour Tuesdays 2-3pm in PG 003 (Basement)
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 14, in class.
Final Exam: December 12, 11AM-2PM.

## Course notes

• Manifolds, Cobordism, Smooth maps (pdf)
• The tangent functor, derivatives of maps (pdf)
• Vector fields and flows (pdf)
• Local classification of smooth maps (pdf)
Inverse function theorem (screencast)
Constant rank theorem (screencast)
• Maps between manifolds with boundary (pdf)
• Transversality, stability and Sard’s theorem (pdf)
• Intersection theory (pdf)
• Partitions of Unity (pdf)
• Vector fields vs derivations (pdf)
• Vector bundles and differential forms (pdf)
• De Rham Cohomology (pdf)

## Assignments

Please discuss the problems, but avoid reading a written solution before you write your own, since these must be original. Also, do not share written solutions with anyone, even after the deadline.

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 5
• Assignment 2 distributed by Crowdmark, due October 19
• Assignment 3 distributed by Crowdmark, due November 2
• Assignment 4 distributed by Crowdmark, due November 23

## 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,