These are the lecture notes for courses on differential topology, 2018-2020. Last updated: December 21st 2020. Please email me any corrections or comments.
Topics covered:
- Smooth manifolds.
- Smooth maps and their derivatives.
- Immersions, submersions, and embeddings.
- Whitney embedding theorem.
- Transversality and the pre-image theorem.
- Topological applications such as the Brouwer fixed point theorem, Borsuk-Ulam theorem, and Jordan-Brouwer separation theorem.
- Intersection theory.
- Knot theory and linking numbers.
- Differential forms and Stokes’ theorem.
- DeRham cohomology.
- Isotopy extension and the Ehresmann fibration theorem.
- Morse theory and Milnor’s exotic 7-spheres.