Math 215b: Algebraic topology

Instructor

Here are the term papers.

Spacetime coordinates

About the course

• Higher homotopy groups and obstruction theory.
• Bundles and characteristic classes.
• Spectral sequences.
• Basic Morse theory and applications to differential topology.

Course requirements

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.

Textbook

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 go along.

Lecture summaries

• (1/18)
  • 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.
• (1/20)
  • Proved the Hurewicz isomorphism theorem.
  • Explained the extent to which homotopy groups depend on the base point.
• (1/25)
  • 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.
• (1/27)
  • 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.
• (2/1)
  • Introduced the homotopy lifting property and fibrations.
  • Explained the long exact sequence in homotopy groups in some detail.
• (2/3) Introduced obstruction theory and Eilenberg-Maclane spaces.
• (2/8) Explained orientations of sphere bundles and w_1.
• (2/10)
  • Discussed the Euler class of an oriented sphere bundle.
  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 me know.
• (2/15)
  • Show that oriented circle bundles are classified by the Euler class.
  • Briefly discussed (co)homology with local coefficients and its uses.
• (2/17)
  • Proved Whitehead's theorem. (Probably should have done this earlier.)
  • 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.)
• (2/24)
  • 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.)
• (3/3)
  • 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.)
• (3/8)
  • 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).
• (3/15-3/17)
  • 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 ch. 12.
  • 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.
• (4/7-4/14)
  • 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.
• (4/19--4/21)
  • 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.
• (5/2-5/4)
  • 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 Morse homology.
• (5/10)
  • 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 theorem".
  • 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 its applications".