Differential Topology

Varieties: definition and examples. Varieties with board. Steerable varieties. Drive partitions. Theorem of Sard. Cr Topology (compact domain). Transversality. Whitney’s theorems. Degree module two and Brower degree. Homotopy invariance.
Applications: Brower fixed point theorem, dimension invariance theorem. Hopf’s theorem of the homotopic classification of applications in the sphere. Theory of intersection and degree. Invariance by homotopy of the intersection number. Vector fields and characteristic of Euler. Index of Poincaré-Hopf. Poincaré-Hopf’s theorem. Lefshetz’s theorem.

References:
LIMA. E. L. – Introdução à Topologia Diferencial. 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.