Math 215b: Algebraic topology
UC Berkeley, Spring 2005
[My last name with the last letter deleted]@math.berkeley.edu
Office phone: 510-642-4329.
Office: 923 Evans.
Office hours TBA.
TTh, 8:10 AM, 31 Evans.
About the course
I assume you have learned the basics of algebraic topology, in 215a or
elsewhere. In 215b we will study a selection of more advanced topics,
which nonetheless are in my opinion very important and useful in
geometry and topology. I will emphasize applications to the geometry
of smooth manifolds. Where we start in 215b will depend in part on
where 215a ends (which I will find out at the end of the fall
semester). In any case, I expect to do something similar to last year's 215b course, in which
we discussed the following topics:
- Higher homotopy groups and obstruction theory.
- Bundles and
- Spectral sequences.
- Basic Morse
theory and applications to differential topology.
I will suggest some homework exercises; these will not be graded,
although I am happy to discuss how to solve them.
The only official course requirement will be to write an expository
paper, of about 10 pages, on some specific topic of interest in
algebraic topology. Click here for more
detailed guidelines. Inspired by Prof. Weinstein, I will also ask
you to referee (i.e. make anonymous critical comments on) the paper of
one of your classmates, and then revise your paper in light of the
referee report you receive. Your final paper will (if you give
permission) be posted here for your classmates to read.
Term paper deadlines:
- (4/25) Last day to submit first draft of paper. If you submit
your paper earlier, you will have more time to referee.
- (5/2) Referee reports due.
- (5/11) Revised papers due.
You should already have a basic graduate level algebraic topology
text. One of my favorites is Topology and Geometry by
Bredon, in particular because of its emphasis on smooth manifolds.
The part of the course on bundles and characteristic classes will
roughly follow the beautiful book Characteristic classes by
Milnor and Stasheff. Other parts of the course will not follow any
particular book, but I will give references for specific topics as we
- We briefly reviewed what I am assuming you know from 215a or
elsewhere, and outlined what I plan to cover in the course. I can go
slower or faster depending on your background, so please feel free to
come to my office and introduce yourself.
- Introduced higher homotopy groups and the Hurewicz isomorphism.
- Proved the Hurewicz isomorphism theorem.
- Explained the extent to which homotopy groups depend on the base
- Introduced the mapping torus of a homeomorphism, which provides
examples where the action of pi_1 on pi_k is nontrivial.
- Introduced fiber bundles and some interesting examples thereof.
- Explained how fiber bundles over a CW complex B depend only on
the homotopy type of B.
- Stated the long exact sequence in homotopy groups of a fiber
bundle, and discussed some interesting examples.
- Introduced the homotopy lifting property and fibrations.
- Explained the long exact sequence in homotopy groups in some
Introduced obstruction theory and Eilenberg-Maclane spaces.
Explained orientations of sphere bundles and w_1.
Here are some rough lecture
notes (updated version here)
which should clarify what I said or meant to say in the first few
lectures. If you find mistakes or have further questions, please let
- Discussed the Euler class of an oriented sphere
Show that oriented circle bundles are classified by the Euler class.
Briefly discussed (co)homology with local coefficients and its uses.
- Proved Whitehead's theorem. (Probably should have done this
- Introduced vector bundles. (See the first couple of chapters of
Milnor and Stasheff.)
- (2/22) Introduction to Stiefel-Whitney classes, via axioms. (See
M-S chapter 4.)
- Brief review of some basic notions from differential topology.
(See e.g. Spivak, A comprehensive introduction to differential
geometry, volume 1.)
- How Stiefel-Whitney classes give obstructions to immersions and
cobordisms. (See M-S chapter 4.)
- (3/1) Grassmannians and the universal bundle. (See M-S chapter 5.)
Schubert cells. (M-S chapter 6. For more about the complex case, see
e.g. Griffiths-Harris, Principles of algebraic geometry, chapter 1.5.)
- Proof that the cohomology ring of BO(n) with Z/2 coefficients is
the polynomial ring generated by the Stiefel-Whitney classes, assuming
that the latter exist and satisfy the axioms. (M-S chapter 7.)
- Fun digression introducing Schubert calculus.
- The Thom isomorphism theorem; see M-S chapter 10. We'll revisit
this later using spectral sequences.
(3/10) We digressed to explain the duality between cup product and
intersections of submanifolds, and the Lefschetz fixed point theorem
on a manifold. Here are some notes on this
(updated version here).
- Redefined the Euler class of an oriented vector bundle using the
Thom isomorphism theorem; see M-S ch. 9.
- Introduced the Gysin
sequence and used it to show that the Euler class as defined above is
the primary obstruction to finding a nonvanishing section; see M-S
Introduced the Chern classes via axioms, used the Gysin sequence to
define them and prove their uniqueness, and discussed their
interpretation in terms of obstruction theory. See M-S ch. 14.
- (3/29) Introduced principal bundles and classifying spaces.
These are not really discussed in Milnor and Stasheff, but you should
know what they are.
- (3/31) Said more about principal bundles and classifying spaces,
introduced spin structures, and showed that w_2 is the obstruction to
the existence of a spin structure.
- (4/5) Stated the Leray-Hirsch theorem, used it to prove the
splitting principle, discussed the meaning of c_1, and introduced
Pontrjagin classes and the signature. We're going to depart for good
from the world of Milnor and Stasheff now, but I encourage you to read
the rest of the book.
- Introduced spectral sequences, as a gadget
for computing the homology of a filtered chain complex by successive
approximations. Here are some very incomplete
lecture notes on the subject (updated version here). For more about spectral
sequences, please see the references therein.
- Introduced the Leray-Serre spectral sequence for
the homology of a Serre fibration. Used it to compute the homology of
some fibrations, and to re-prove the Hurewicz isomorphism theorem.
- Explained the construction of the Leray-Serre spectral sequence
using cubical singular homology. As an example, related the
differential for an oriented sphere bundle to the Euler class.
- Explained the cohomological version of the Leray-Serre spectral
sequence, and its relation with the cup product.
- Used this to compute the cohomology rings of some fibrations, and
to prove the Thom isomorphism theorem and the Leray-Hirsch theorem.
- (4/26) Introduced Morse theory. For more details see the book
"Morse theory" by J. Milnor.
- (4/28) Introduced handle decompositions of smooth manifolds.
Used these to classify surfaces. For more on handle decompositions
and their uses in low-dimensional topology, see the book "4-manifolds
and Kirby calculus" by Gompf and Stipsicz.
- Explained how to compute the homology of a smooth manifold in terms of a
handle composition, in terms of intersection numbers of attaching
spheres and belt spheres.
- Explained the related but more general notion of Morse homology,
in which one can compute the homology of a smooth manifold by counting
gradient flow lines between critical points of a Morse function. I
have some lecture notes about this on my home page, although there are
some mistakes in there which I need to fix.
- Gave a nice proof of Poincare duality on a smooth manifold using
Outlined the proof of the Poincare conjecture in high dimensions. For
some more details, see Gompf-Stipsicz chapter 9, and for a lot more
details, see the book by Milnor, "Lectures on the h-cobordism
- Very briefly mentioned how one can define Morse theory
on the loop space of a Riemannian manifold and relate the topology of
the loop space to geodesics. More about this may be found in Milnor's
book on Morse theory mentioned above, and there are also several
excellent expository articles on this by R. Bott, such as
"The periodicity theorem for the classical groups and some of
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