Math 215B - Algebraic Topology - Spring 2012

D. Auroux - MWF 10-11am, Room 891 Evans

Instructor: Denis Auroux (auroux@math.berkeley.edu)

Office: 817 Evans.
Office hours: by appointment.

Course outline

The course will mainly focus on homotopy theory (homotopy groups, relations between homotopy and homology, fibrations, obstruction theory) and on characteristic classes. If time permits we will also give brief introductions to spectral sequences and to Morse homology.

Prerequisites: Math 215A or equivalent (Algebraic Topology); Math 214 (Differentiable Manifolds) recommended

Grading: based on homework

Recommended texts

• Homotopy theory: A. Hatcher, Algebraic Topology, chapter 4, available here.
• Characteristic classes: J. Milnor, J. Stasheff, Characteristic Classes, Princeton University Press.

Finally, here are some handwritten notes (not guaranteed to be always readable):

• Homotopy groups
• Whitehead's theorem and approximation
• Excision and applications
• The Hurewicz theorem
• Fiber bundles and fibrations
• Cohomology and homotopy
• Vector bundles
• Stiefel-Whitney classes
• Universal bundles on Grassmannians
• Euler class
• Poincaré duality and the Euler class
• Chern classes
• Connections, curvature, and Chern classes

Homework

Homework will be due every 2-3 weeks. Assignments will be posted here.

• Homework 1 (due Friday Feb 10)
• Homework 2 (due Wednesday Feb 29)
• Homework 3 (due Wednesday March 21)
• Homework 4 (due Friday April 27)

Topics covered in class

Part 1: Homotopy theory

• Wed 1/18: About the course; homotopy groups.
• Fri 1/20: Basic properties of π_n; relative homotopy groups.
• Mon 1/23: Long exact sequence in relative homotopy; Whitehead's theorem (statement).
• Wed 1/25, Fri 1/27: no class
• Mon 1/30: Whitehead's theorem (proof); cellular approximation
• Wed 2/1: Cellular approximation (continued), CW approximation
• Fri 2/3: CW approximation continued; Excision and applications (statements)
• Mon 2/6: Excision: proof.
• Wed 2/8: Excision: applications.
• Fri 2/10: Eilenberg-MacLane spaces.
• Mon 2/13: Hurewicz theorem.
• Wed 2/15: The Hurewicz map and action of π₁.
• Fri 2/17: Fibrations and the homotopy lifting property.
• Wed 2/22: Fiber bundles: examples, long exact sequence.
• Fri 2/24: Fiber bundles continued: more examples.
• Mon 2/27: Stable homotopy groups; more about fibrations.
• Wed 2/29: The homotopy description of cohomology.
• Fri 3/2: no class
• Mon 3/5: Suspensions, loop spaces and adjunction.

• Wed 3/7: Vector bundles: definition and examples.
• Fri 3/9: Vector bundles: constructions.
• Mon 3/12: Stiefel-Whitney classes: axioms.
• Wed 3/14: Stiefel-Whitney classes: examples (RPⁿ), relation to cobordism.
• Fri 3/16: Grassmannians and universal bundles.
• Mon 3/19: The classifying map and characteristic classes.
• Wed 3/21: The cell structure on the Grassmannian.
• Fri 3/23: The cohomology ring of the infinite Grassmannian.
• Mon 4/2: The Thom isomorphism theorem and the Euler class.
• Wed 4/4: The Euler class mod 2 and Stiefel-Whitney classes.
• Fri 4/6: Euler and Stiefel-Whitney classes as obstructions.
• Mon 4/9: Poincaré duality and the Thom class.
• Wed 4/11: Poincaré duality, cup product, and the Euler class.
• Fri 4/13: Chern classes.
• Mon 4/16: The cohomology ring of the complex Grassmannian.
• Wed 4/18: Classifying maps, axiomatic properties of Chern classes.
• Fri 4/20: Pontrjagin classes.
• Mon 4/23: Connections on smooth vector bundles.
• Wed 4/25: Curvature and Chern classes.
• Fri 4/27: Chern-Weil theory continued.
• Mon 4/30: Introduction to the Leray-Serre spectral sequence.
• Wed 5/2: Leray-Serre spectral sequence continued.