Math 215b: Algebraic topology
UC Berkeley, Spring 2011
Instructor Michael Hutchings
[My last name with the last
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.
- Characteristic classes.
- 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.
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
- (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
- (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
- (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
- (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.