Gereon Quick

Math 231br - Advanced Algebraic Topology

Course information

Time and place: MWF 2.30-3.30, Science Center 411
Instructor: Gereon Quick

Office hours: Wednesday 1.30-2.30, Science Center 341 (or by appointment)

Syllabus

Suggestions for topics of the final paper

Topics

Lecture notes

Lecture 1: Vector bundles

Lecture 2: Vector bundles and sections

Lecture 3: Families of sections

Lecture 4: Constructing new bundles out of old

Lecture 5: Euclidean bundles, orthogonal complements, and orientations

Lecture 6: Stiefel-Whitney classes and embedding problems (Guest lecture by Mike Hopkins)

Lecture 7: Stiefel-Whitney classes of projective spaces

Lecture 8: Existence and uniqueness of Stiefel-Whitney classes I

Lecture 9: Existence and uniqueness of Stiefel-Whitney classes II

Lecture 10: The splitting principle and the projective bundle formula

Lecture 11: The Grassmannian manifold and the universal bundle

Lecture 12: Representability and vector bundles

Lecture 13: Schubert cells and Schubert varieties

Lecture 14: A cell decomposition for the Grassmannian

Lecture 15: Cohomology of the Grassmannian

Lecture 16: Chern classes for complex vector bundles

Lecture 17: Complex K-theory

Lecture 18: Complex K-theory as a representable functor

Lecture 19: Complex K-theory as a cohomology theory

Lecture 20: K-theory of complex projective spaces (Guest lecture by Mike Hopkins)

Lecture 21: Splitting principle and the projective bundle formula in K-theory (Guest lecture by Mike Hopkins)

Lecture 22: Thom classes and the Thom isomorphism in K-theory (Guest lecture by Mike Hopkins)

Lecture 23: Proof of the Periodicity theorem I

Lecture 24: Proof of the Periodicity theorem II

Lecture 25: Adams operations in complex K-theory

Lecture 26: The Hopf invariant one problem via K-theory

Lecture 27: Applications of the Hopf invariant one problem

Lecture 28: The Chern character

Lecture 29: The e-invariant

Lecture 30: The e-invariant and the J-homomorphism

Lecture 31: The image of the J-homomorphism

Lecture 32: The image of the J-homomorphism and Thom classes

Lecture 33: Clifford algebras and vector fields on spheres (Guest lecturer: Mike Hopkins)

Lecture 34: The image of J and the Adams conjecture

Lecture 35: Sphere bundles and the Adams conjecture

Lecture 36: Sullivan's proof of the Adams conjecture

Problem Sets