MATH 225B

Main Course

Differential Topology

Description

Lecture, three hours; discussion, one hour. Sard's theorem and transversality, Whitney embedding theorem, tubular neighborhoods; intersection theory, degree, vector fields and Poincaré-Hopf theorem, Lefschetz fixed-point formula; compactly supported cohomology, Poincaré duality, Thom isomorphism, and the Künneth theorem from the point of view of de Rham theory; applications: homotopy types of self-maps of tori and spheres (Hopf degree theorem) and Lefschetz numbers of self-maps of spheres, real and complex projective spaces, and tori. S/U or letter grading.