Cobordism and Characteristic Classes
This graduate course develops the foundations and applications of cobordism theory with an emphasis on its deep connections to characteristic classes and manifold topology. Beginning with basic differential topology, students learn to reinterpret cobordism groups in homotopy-theoretic terms and compute basic examples, including unoriented, oriented, and complex cobordism rings. The course then introduces characteristic classes—Stiefel–Whitney, Chern, and Pontryagin classes—and explains how characteristic numbers serve as complete invariants for cobordism classes in many settings. Central results such as the Hirzebruch signature theorem are presented to illustrate the powerful interplay between topology and geometry. Depending on time, topics may include the structure of the complex cobordism ring, formal group laws and Quillen’s theorem, as well as geometric applications such as exotic spheres and the role of characteristic classes in distinguishing smooth structures.
I will write my own notes. Good references include: Characteristic classes (Milnor-Stasheff), Notes on cobordism theory (Stong), Topological methods in algebraic geometry (Hirzebruch).
Concepts of smooth manifolds, homology and cohomology