**Instructor:** David Nadler

**Office Hours:** by appointment, 815 Evans.

**Lectures:** Tuesdays and Thursdays 12:30-2:00pm, 31 Evans.

**Course Control Number:** 54399

**Prerequisites:** Math 215A or equivalent familiarity with algebraic topology. Also recommended: Math 214 or equivalent familiarity with differentiable manifolds.

**Primary sources:**

- A. Hatcher, Algebraic Topology, chapter 4, available here.
- J. Milnor, J. Stasheff, Characteristic Classes, Princeton University Press.

**Previous Math 215B course pages with useful lecture notes:**

- Auroux, Spring 2012.
- Hutchings, Spring 2011.

**Syllabus:** This will be an example-oriented introduction to homotopy theory and characteristic classes. If time permits, it will also include a brief introduction to spectral sequences.

- Homotopy groups.
- CW complexes.
- Whitehead's theorem.
- Hurewicz isomorphism.
- Calculations.
- Fibrations and fiber bundles.
- Cohomology and homotopy theory.
- Vector bundles and principal bundles.
- Stiefel-Whitney classes.
- Grassmannians and universal bundles.
- Thom isomorphism.
- Euler class.
- Chern classes.
- Chern-Weil theory.
- Spectral sequences.

**Evaluation:**
Each week there will be a mandatory homework assignment.

- HW 1, due Tuesday, January 28. Hatcher 4.1, 3, 4.
- HW 2, due Tuesday, February 11. Hatcher 4.1, 11, 15, Show pi_n(S^n) = Z.
- HW 3, due Tuesday, February 25. Hatcher 4.2, 2, 22, 32.
- HW 4, due Tuesday, March 11. Hatcher 4.2, 23, 33.