Math 215b: Algebraic topology

UC Berkeley, Spring 2011

Instructor

Michael Hutchings
[My last name with the last letter removed]@math.berkeley.edu
Office hours: Wednesday 2:00-4:00 (may be rescheduled some weeks), 923 Evans.

Course outline and general references

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)
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.

Term paper

The only course requirement is that each student is expected to write a short (5-10 page) expository paper on a topic of interest in algebraic topology, to referee another student's paper, and to revise their paper based on the referee's comments. I am happy to help you find a topic. Collaborations are possible. Timeline:

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.

Lecture summaries and specific references

(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.