Math 215b: Algebraic topology

This course is over. Some of the student term papers are available here.

Instructor

Spacetime coordinates

About the course

With the basics of algebraic topology established, 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. We will do as much of the following as time permits (not in exactly this order):

• Higher homotopy groups and obstruction theory.
• Bundles and characteristic classes.
• Basic Morse theory and applications to differential topology.
• Spectral sequences.
• As time permits, possible additional topics to be determined, such as:
  • Secondary invariants such as Reidemeister torsion, Alexander polynomial, Massey products, etc.
  • Equivariant cohomology.

Textbook

Course requirements

The only course requirement will be to write an expository paper on some specific topic of interest in algebraic topology. This should be about 10 pages or less. Inspired by Prof. Weinstein, I will ask students to anonymously referee each other's papers and then revise them. Click here for more details.

Lecture summaries

• 1/20.
  • Defined higher homotopy groups. (These are discussed in most algebraic topology textbooks.)
  • Sketched a geometric proof of the Hurewicz isomorphism theorem. (I don't know a reference for this proof unfortunately. Later we will see a much shorter proof using the Leray-Serre spectral sequence. I think that the latter proof actually has the same underlying geometric content as the former.)
  • Started discussing the dependence of pi_k on the choice of basepoint, by showing that a path between two base points induces an isomorphism between the corresponding pi_k's, which depends only on the homotopy class of the path etc. (If you want to be fancy, pi_k is a functor on the fundamental groupoid of X.)
• 1/22.
  • Introduced the mapping torus of a homeomorphism, an example where the action of pi_1 on pi_k can be nontrivial.
  • Introduced fiber bundles. Clutching construction, Hopf fibration. Started on homotopy properties of fiber bundles. (cf. Hatcher section 4.2. As I recall, Steenrod's book The topology of fiber bundles is beautiful and might be a good reference for what we are discussing.)
• 1/27.
  • Continued with homotopy properties, defining the pullback of a fiber bundle and proving that any fiber bundle over a contractible CW complex is trivial.
  • Introduced the long exact sequence in homotopy groups for a fiber bundle. Did basic examples. Discussed the Hopf invariant.
  • Defined homotopy lifting property and Serre fibrations, showed that every fiber bundle is a Serre fibration. Used this to define the connecting homomorphism in the long exact sequence.
• 1/29.
  • Proved exactness of the long exact sequence in homotopy groups associated to a Serre fibration. One more example: the path fibration of a topological space.
  • Proved that for any CW complex X, there is a canonical isomorphism [X,S^1]=H^1(X;Z). This is a prototypical obstruction theory argument; we will do several similar proofs later.
• Class was cancelled on Tuesday Feb. 3, because I had jury duty. Fortunately they called in more people than they needed and I was not selected. We will have a makeup lecture later at a time to be determined.
• (2/5)
  • Defined section of a fiber bundle and orientation of a sphere bundle.
  • For a sphere bundle E over a CW complex B, defined w_1(E) in H^1(B;Z/2), the obstruction to orienting E.
  • Started to define the Euler class of an oriented sphere bundle, which is the primary (but sometimes not the only) obstruction to finding a section.
• (2/10)
  • Discussed the Euler class in detail from the viewpoint of obstruction theory.
  • Proved that an oriented S^k bundle over a CW complex has a section over the k+1 skeleton iff its Euler class vanishes.
  • Showed that the Euler class gives an isomorphism between the set of oriented circle bundles over a given CW complex B (modulo orientation-preserving bundle isomorphism) and H^2(B;Z).
  • Made some brief remarks on what a characteristic class is, what the primary obstruction to finding a section of a general fiber bundle looks like, and Eilenberg-Maclane spaces.
• (2/12)
  • Whitehead's theorem: if a map between CW complexes induces an isomorphism on all homotopy groups, then it is a homotopy equivalence.
  • Basic stuff about vector bundles. (See the first few chapters of Milnor and Stasheff.)
• (2/17)
  • Introduction to Stieffel-Whitney classes; see Milnor-Stasheff chapter 4.
  • A bit about spin structures and the obstruction to their existence.
• (2/19)
  • Discussed how Stieffel-Whitney numbers give obstructions to unoriented cobordisms. (See M-S ch. 4.)
  • Vector bundles modulo isomorphism over (a reasonable space) B are the same as homotopy classes of maps from B to the classifying space (the infinite Grassmannian). Therefore the ring of all characteristic classes is the cohomology ring of the classifying space. (See M-S ch. 5.)
• (2/24)
  • Discussed the cell decomposition of the real and complex Grassmannians in terms of Schubert varieties. (See M-S ch. 6.)
  • As a fun example, explained (in terms of intersection theory in the Grassmannian G_2(C^4)) why given four generic lines in CP^3 there are two lines meeting all four of them.
• (2/26)
  • We showed that the Z/2 cohomology of the real infinite Grassmannian is the polynomial ring over the Stiefel-Whitney classes, assuming that the Stiefel-Whitney classes exist. (See M-S chapter 7.)
  • Discussed one definition of Stiefel-Whitney classes in terms of obstruction theory. (See M-S chapter 12.) However it is hard to prove the Whitney sum formula from this definition.
• (3/2)
  • Introduced the Chern classes of a complex vector bundle, their basic properties, and their interpetation in terms of obstructions when the base is a CW complex. (See M-S ch 14.)
  • Discussed how in the smooth case, the Euler class of an oriented real vector bundle is Poincare dual to the zero set of a generic section. (I don't know a reference for this, but I think it is important to understand the basic intuition here. Maybe I'll try to write something.)
• (3/4)
  • Proved the Thom isomorphism theorem for an oriented real vector bundle over any base. (See M-S chapter 10 for the details that I omitted.)
  • Showed that (in the category of smooth compact oriented manifolds without boundary) the Thom class of the normal bundle (which is diffeomorphic to a tubular neighborhood) of a submanifold is Poincare dual to the fundamental class of the submanifold.
  • Used this to finally explain why given two transverse submanifolds, the Poincare dual of their intersection is the cup product of their Poincare duals. (This is in Bredon.)
• (3/9)
  • Used the Thom isomorphism to redefine the Euler class (over any base) and give simple proofs of its basic properties. (See M-S ch. 9.)
  • Introduced the Gysin sequence and used it to show that the above definition of the Euler class agrees with the obstruction-theoretic definition over a CW complex. (See M-S ch. 12.)
  • Showed how the Gysin sequence can be used to prove uniqueness of the Chern classes and give a definition of them from which all of the axioms can be proved. (See M-S ch. 14 for more about this.)
• (3/11)
  • More properties of Chern classes, the Chern character, and the Splitting Principle. (The proof of the latter that I presented is from Bott and Tu.)
  • Brief remarks about the Pontrjagin classes.
  Next week we will put characteristic classes aside and start on spectral sequences! If you want to learn more about characteristic classes, the rest of Milnor and Stasheff is good reading. For those of you with an interest in geometry, I especially recommend Appendix C, explaining the beautiful relation between characteristic classes and curvature.
• (3/16) Started discussing spectral sequences, which I will explain by successive approximations; hopefully it will eventually start to make sense.
  • Today we discussed how to compute the homology of a filtered chain complex by successive approximations. (See e.g. Griffiths and Harris pp. 438-442.)
• (3/18)
  • Clarified the basics of spectral seuqences from last time.
  • Introduced the Leray-Serre spectral sequence and did some simple computations using it.
  • Re-proved the Hurewicz isomorphism theorem using the Leray-Serre spectral sequence. (For this and many more applications of the Leray-Serre spectral sequence, see Bott and Tu.)
• (3/30)
  • Discussed the relation between the Euler class of a sphere bundle and the Leray-Serre spectral sequence.
  • Explained homology with twisted coefficients, discussed the Leray-Serre spectral sequence over a non-simply-connected base, and considered the example of a mapping torus.
• (4/1)
  • If a filtered cochain complex has a product respecting the filtration such that the differential is a derivation, then the differentials in the associated spectral sequence are derivations with respect to induced products. In particular, the differentials in the Leray-Serre spectral sequence are derivations with respect to the (twisted) cup product.
  • For an oriented sphere bundle, the differential in the spectral sequence is cup product with the Euler class.
  • We proved the Leray-Hirsch theorem.
• (4/6)
  • Showed how the Leray-Serre spectral sequence can give some information about homotopy groups of spheres. For more on this topic see Bott-Tu or Spanier.
  • Discussed the Mayer-Vietoris spectral sequence and its uses.
• (4/8)
  • Explained the "universal coefficient spectral sequence". As an example, used it to give a somewhat more difficult than usual computation of the homology of a two-torus or a wedge of two circles. I don't know a good reference for this but it is a special case of more general homological algebra.
  • *** MASSIVE SHIFT OF GEARS HERE ***
  • Introduced Morse theory.
• (4/13) Discussed how classical Morse theory gives rise to handle decompositions of manifolds. For more details see:
  • Milnor's book "Morse theory", chapters 1,2,3.
  • "4-manifolds and Kirby calculus" by Gompf and Stipsicz, early parts of chapters 4, 5, and 9 and the references therein. (The other parts of this book are full of fun stuff, too.)
  • Some parts of my lecture notes on modern Morse theory might also be useful (modulo the mistakes therein).
• (4/15)
  • Used handle calculus to classify compact oriented surfaces.
  • Going up a dimension, discussed Heegard splittings of 3-manifolds.
  • Going up another dimension, introduced Kirby calculus for 3-manifolds (and 4-manifolds that they bound).
  (Isn't this stuff fun? Maybe I should have started this earlier in the course. I think all these explicit low-dimensional pictures really bring algebraic topology to life.)
• (4/20)
  • Explained how to compute the homology of a manifold in terms of a handle decomposition. This is closely related to (and with more technical work can be realized as a special case of) the cellular homology of a CW complex.
  • Started to sketch the proof that if a closed smooth n-manifold with n at least 6 is simply connected and has the homology of S^n, then it is homeomorphic (although not necessarily diffeomorphic) to S^n. The argument is essentially the h-cobordism theorem; see Gompf-Stipsicz chapter 9 for an outline, and Milnor's book "Lectures on the h-cobordism theorem" for full details.
• (4/22)
  • Explained Whitney's cancellation lemma for getting rid of intersection points of submanifolds in sufficiently high dimension. (Cf. Milnor's Lectures on the h-cobordism theorem.)
  • Mostly finished the explanation of the proof of the higher-dimensional Poincare conjecture.
• (4/27)
  • Discussed Dehn surgery, examples thereof, and why every oriented 3-manifold can be obtained from Dehn surgery on a framed link in S^3.
  • Introduced Morse homology, which we will explain in more detail next time.
• (4/29)
  • Explained the definition of Morse homology and the signs in the differential, and did simple examples.
  • Proved that the differential is well-defined and has square zero, modulo the technical theorem about the compactification of the moduli space using broken flow lines.
• (5/4)
  • Explained the a priori proof that Morse homology is a topological invariant, in terms of continuation maps and chain homotopies.
  • Showed how Poincare duality is very easy in Morse homology.
• (5/6) Explained the isomorphism between Morse homology and singular homology.
• (5/11) Discussed classical applications of Morse theory relating geodesics to the topology of the loop space, including:
  • the Freudenthal suspension theorem
  • existence theorems for geodesics
  • Bott periodicity for the unitary groups
  (For details see Milnor's book "Morse theory".) There are lots of other things I would have liked to do in this course but didn't have time for. Some of these are covered in the term papers, which I will post here (given authorial permission).