Instructor: Ina Petkova

Course on canvas.dartmouth.edu.⇗

Course Description:

This course is an introduction to differential topology.

We won’t follow one book exclusively, but Spivak’s Differential geometry, Vol. 1 is the best single resource for this course. As everyone’s learning style is unique, I strongly encourage you to get familiar with all the books suggested above.

The following is an approximate list of topics that will be covered in this course, along with references to the relevant chapters in the above books.

  1. Smooth manifolds (and manifolds with boundary): charts, smooth maps, partitions of unity. S 1–2, W 1, GP 1–2, L 1–2.
  2. Tangent vectors, tangent spaces, differential of a smooth map, computations in coordinates. S 3, W 1, GP 1, L 2.
  3. Tangent bundles, smooth vector fields, velocity vectors of curves, alternative definitions of the tangent space; vector bundles, their sections and operations on vector bundles. S 3, W 1, GP 1, L 3, L 10.
  4. Inverse Function Theorem, Implicit Function Theorem, Rank Theorem. S 2, W 1, GP 1, L Appendix C, L 2.
  5. Submersions, immersions, embeddings. Immersed and embedded submanifolds, graphs of smooth functions, level sets, critical points, tangent spaces to submanifolds. S 2, W 1, GP 1, L 4–5.
  6. Sets of measure zero, Sard’s Theorem, Whitney Embedding Theorem. S 2, W 1, GP 1, L 6.
  7. The cotangent bundle, pullbacks. S 4, W 1, L 11.
  8. Tensors, differential of a function, differential forms S 4, S 7–8, BT 1, W 2, GP 4, L 11–12, L 14.
  9. Integration on Manifolds, Stokes’s Theorem S 4, S 7–8, BT 1, W 2, GP 4, L 15–16.
  10. De Rham cohomology. S 8, S 11, BT 1, W 4, GP 4.

    Additional topics that may vary from year to year.

  11. Mayer-Vietoris Theorem. S 8, S 11, BT 1, W 4, GP 4.
  12. Distributions, integral submanifolds. S 6, W 1.
  13. Lie groups and homogeneous spaces. S 10, W 3, L 7.
  14. Cohomology and the pairing between the bordism group and cohomology given by the Stokes Theorem. L 16.
  15. Riemannian Metrics. L 13.