Time and place of Lectures: TR 3:40- 5:00 pm, Physics P130

Instructor: Filip Zivanovic, current office hours: My SB card 

Course information and materials will be posted on this page.

Prerequisites: Linear algebra, including tensor products. Modules over commutative rings. General topology, including familiarity with CW complexes, topological and smooth manifolds.

Course Description:

This will be a course mainly on characteristic classes, which are particular invariants of vector bundles. These will include Stiefel Whitney classes, Chern classes, and Pontrjagin classes. We will also cover classifying spaces (in particular, Grassmannians) and Thom spaces. 

Grading: 

At the end of the course, students will present chosen topics that naturally lead from the lecture materials:

1. Immersions of Real Projective Spaces (C. Macmahon)
 Applications of Stiefel–Whitney classes to immersion problems, showing how characteristic classes give sharp obstructions to immersing real projective spaces into Euclidean space.

2. Parallelizable Spheres and Division Algebras (Z. Liu)
 A study of which spheres admit trivial tangent bundles, explaining why only S1, S3, and S7 are parallelizable and how division algebras enter the picture.

3. Characteristic Classes and Obstructions to Almost Complex Structures (S. Xu)
 An investigation of when a smooth manifold admits an almost complex structure, using Stiefel–Whitney, Chern, and Pontryagin classes as obstruction-theoretic invariants, with concrete examples.

4. Almost Parallelizable Manifolds and Stable Triviality of the Tangent Bundle (B. Wong)
 An introduction to manifolds whose tangent bundle becomes trivial after adding a trivial bundle, explaining stable parallelizability and how characteristic classes detect this phenomenon, with applications to spheres and exotic smooth structures.

5. The Hirzebruch Signature Theorem (N. Callahan)
 The relationship between the signature of a 4k-dimensional manifold and its Pontryagin classes, with emphasis on the L-genus and applications to smooth topology.

6. Milnor’s Exotic 7-Sphere (C. Luke Martin)
 The construction of a smooth manifold homeomorphic but not diffeomorphic to the 7-sphere, using sphere bundles over S4 and Pontryagin class computations.

7. The Kervaire–Milnor Classification of Exotic Spheres (E. Spingarn)
 A discussion of the classification of exotic 7-spheres, explaining how they form a finite abelian group and how cobordism and characteristic classes detect distinct smooth structures.

8. Framed Manifolds, Thom Spaces, and the J-Homomorphism (O. Milshtein)
 An introduction to framed manifolds and their relation to Thom spaces and cobordism, culminating in a conceptual explanation of the J-homomorphism and its role in exotic spheres.

Textbook: Characteristic Classes, Annals of Mathematics Studies 76, by John Milnor and James Stasheff.

Tentative lecture notes: Newest version.

Tentative schedule:             

Week. Lecture Dates.      Topics covered.