Office hours: Wednesday 2:00-4:00 (may be rescheduled some weeks), 923 Evans.

- Introduction to homotopy groups and obstruction theory.
- Lecture notes (updated 2011-02-17). Comments, questions, and corrections are welcome.
- See also Hatcher, Algebraic Topology, Chapter 4, which has some overlap with the topics to be covered.

- Characteristic classes.
- Milnor and Stasheff,
*Characteristic classes*. - Notes on cup product and intersections (updated 2011-03-15)

- Milnor and Stasheff,
- Spectral sequences.
- Lecture notes (updated 2011-04-27, but still very incomplete).
- John McCleary, A user's guide to spectral sequences.

- Additional topics to be determined, depending on time remaining and interests of the class.

- Friday 4/22: first version of paper due (please email it to me).
- Friday 4/29: referee report due.
- Friday 5/6: revised paper due.

- (1/18) Introduction to the course. Definition and some basic properties of higher homotopy groups. See section 1 of the lecture notes.
- (1/20) Proved the Hurewicz isomorphism theorem. See section 2 of the lecture notes. (I updated this slightly after lecture to clarify the proof, but the old version was OK.)
- (1/25) Dependence of homotopy groups on the basepoint. Introduction to fiber bundles. See lecture notes sections 3 and 4.
- (1/27) Fun with fiber bundles. See sections 4 and 5 of the lecture notes. (I updated these slightly to better correspond to what we did in class.)
- (2/1) The long exact sequence in homotopy groups associated to a fiber bundle, or more generally a Serre fibration. See section 6 of the lecture notes.
- (2/3) Introduction to obstruction theory. See section 7 of the lecture notes.
- (2/8) Eilenberg-MacLane spaces. (After class I revised the lecture notes to improve the explanation of this topic, which now appears in section 8.)
- (2/10) Whitehead's theorem. Orientations of sphere bundles. See sections 9 and 10 of the lecture notes.
- (2/15) The Euler class of an oriented sphere bundle over a CW complex. See section 11 of the lecture notes.
- (2/17) Homology with twisted coefficients. See section 12 of the lecture notes.
- (2/22) Vector bundles. See Milnor-Stasheff chapters 2 and 3.
- (2/24) Stiefel-Whitney classes: axiomatic description, basic applications. Milnor-Stasheff chapter 4.
- (3/1) Universal vector bundles, CW structure of the Grassmannian. See M-S chapters 5,6.
- (3/3) Class cancelled, a makeup class will be scheduled at the end of the semester.
- (3/8) The cohomology ring of the Grassmannian, see M-S chapter 7. (We're skipping the construction of Stiefel-Whitney classes via Steenrod squares in M-S chapter 8 and will discuss an alternate construction later.) Introduction to the Thom isomorphism and Euler class, see M-S chapter 9. (We're skipping the proof of the Thom isomorphism theorem in M-S chapter 10 and will give a shorter proof later using spectral sequences.)
- (3/10) On a smooth manifold, cup product is Poincare dual to intersection of submanifolds. The Euler class of a smooth vector bundle is Poincare dual to the zero set of a generic section. See notes posted above.
- (3/15) The Lefschetz fixed point theorem on a manifold and the Poincare-Hopf index theorem. See the lecture notes.
- (3/17) Introduction to Chern classes. See Milnor-Stasheff chapter 14.
- (3/29) Connections and curvature. See Milnor-Stasheff Appendix C (and beware of funky sign conventions).
- (3/31) Chern classes from curvature. Principal bundles.
- (4/5) Classifying spaces. Connections on principal bundles.
- (4/7) Conclusion of discussion of characteristic classes and curvature.
- (4/12) More about Chern classes: splitting principle and Chern character. Introduction to K-theory.
- (4/14) Introduction to cobordism theory and Pontrjagin classes. See M-S chapters 15,17.
- (4/19) Thom-Pontrjagin construction. See M-S chapter 18.
- (4/21) The spectral sequence associated to a filtered complex. See lecture notes.
- (4/26) The Leray-serre spectral sequence for the homology of a Serre fibration.
- (4/28) Cohomological Leray-Serre spectral sequence, multiplicative properties, and applications. I only had time for a brief sketch, but the McCleary book explains all this in great detail with lots of nice examples.
- (5/3) [makeup class] Introduction to Morse homology.