Prerequisites: Analysis in Varieties and Analysis in Rn

Varieties: definition and examples. Edge varieties. Oriented varieties. Partitions of the unit. Sard’s theorem.Cr topology (compact domain). Jet spaces and transversality on jets. Whitney’s theorems. Degree modulo two and Brower degree. Homotopy invariance.
Applications: Brower fixed point theorem, dimension invariance theorem. Hopf’s theorem for the homotopic classification of applications on the sphere. Intersection and degree theory. Homotopy invariance of the intersection number. Vector fields and Euler characteristic. Poincaré-Hopf index. Poincaré-Hopf theorem. Lefschetz theorem.

References:
LIMA. E. L. – Introduction to Differential Topology. Rio de Janeiro, IMPA, 2005.
MILNOR, J. – Topology from the Differentiable Viewpoint. Charlottesville, Princeton Univ. Press, 2nd (1969).
HIRSH, M. – Differential Topology. Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1994.

 

* Basic syllabus. The teacher has the autonomy to make any changes.