Differential Topology

Notes will appear here as we progress through the term.

• Topological manifolds (pdf)
• Smooth manifolds (pdf)
• The derivative, the tangent functor, and the local classification of smooth maps (pdf)

• Inverse function theorem (screencast)
  
• Constant rank theorem (screencast)
  

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.

All assignments must be submitted in LaTeX, I suggest using Overleaf if you need to get up to speed quickly.

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

• 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

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,