Topics in Differential Topology

For master’s students; Semester 1, intensive course · Graduate School of Science and Technology, Niigata University

Note

This course is held online via Zoom.

Textbooks

1. Y. Matsumoto, “多様体の基礎” (in Japanese), University of Tokyo Press
2. I. Yokota, “多様体とモース理論” (in Japanese), Gendai-Sugakusha

References

1. John W. Milnor, Morse Theory, Princeton University Press

Course contents

1. What is Morse theory?; review of point-set topology (definitions, compactness, the Hausdorff property, continuous maps)
2. Manifolds I (definition and examples of differentiable manifolds)
3. Manifolds II ($C^{\infty}$ functions, tangent spaces, $C^{\infty}$ maps)
4. Manifolds III (critical points, vector fields, integral curves, one-parameter transformation groups, Riemannian metrics)
5. Some preliminaries from algebraic topology (homotopy equivalences, deformation retracts, CW complexes, spaces obtained by attaching cells)
6. Morse theory