| Instructor | Sheel Ganatra |
| Office | KAP 266D |
| sheel (dot) ganatra (at) usc (dot) edu (this is the best way to reach me) |
In Spring 2023, Math 641 will be an advanced algebraic topology course, roughly a continuation of Math 540.
The goals of this class will be to introduce and study: the cohomology ring of a topological space and its basic properties (universal coefficient theorem, Kunneth theorem, Steenrod operations, etc.), Poincare duality, vector bundles and characteristic classes. Vector bundles arise frequently in geometry and topology, particularly in the study of manifolds (as developed in Math 535a) through tangent bundles of manifolds and normal bundles to embeddings. Characteristic classes are certain cohomology classes associated to vector bundles which live in the cohomology of the base of a vector bundle and measure the non-triviality of the bundle.
Time permitting, we will also introduce and develop properties and computations in topological K-theory (a type of generalized cohomology theory built out of vector bundles), introduce spectral sequences and perform some computations involving homotopy groups, and study various applications.
We will require familiarity with the geometry of manifolds and algebraic topology at the level of USC's Math 535a and Math 540 courses or equivalent. Familiarity with homologial algebra at the level of the Math 510 series may also help at various points in the class.
This is a preliminary plan subject to some changes. A more detailed lecture plan (updated on an ongoing basis, after each lecture) will be posted below.
There is no official course textbook, but we will frequently refer to the following books and notes (and likely others). All of them are available online (either freely via the link provided or through USC libraries subscriptions).
We will assign homework exercises, to be collected once every 1-2 weeks. Almost all of these exercises are optional, and your submission will in fact only be graded on one problem per submission (of your choosing), and then also only for completeness. Your overall HW score will account for half of your grade
At the end of the course, there will be a final paper assignment, accounting for the other half of your grade.
| Due Date | Exercises | Notes |
|---|---|---|
| 2/15 | HW 1 | |
| 4/7 | HW 2 | |
| N/A | HW 3 (suggested problems, not for credit) |
As part of this course, you will write a 5-10 page expository paper on a topic of your choosing (with instructor approval), which develops some of the topics touched in our class. Some general options for expository paper topics will be e-mailed later in the semester.
Final papers will be due (by e-mail to the instructor) on May 10.
Lecture topics by day will be posted on an ongoing basis below. Future topics are tentative and will be adjusted as necessary.
| Day | Lecture topics | References and remarks | Notes |
|---|---|---|---|
| Mon 1/9 | Lecture 1: Welcome + an overview of the class. Introduction to cohomology. | Notes | |
| Wed 1/11 | Lecture 2: Statement of the Universal Coefficient theorems (UCTs) in homology and cohomology and examples. Short and split-exact sequences. Towards defining Ext and Tor of R-modules: the definition of projective resp. injective R modules. | Notes | |
| Wed 1/18 | Lecture 3: Projective (and injective) resolutions and the definition of Ext and Tor from them. Independence of choices and functoriality. | Notes | |
| Fri 1/20 (Friday class meeting) | Lecture 4: Computations of Ext and Tor. Proof of the UCT for cohomology. | Notes | |
| Monday 1/23 | Lecture 5: UCT over a field. The Eilenberg-Zilber theorem via the method of acyclic models. | Notes | |
| Wednesday 1/25 | Lecture 6: Finished proof of Eilenberg-Zilber theorem. The algebraic Kunneth theorem and the Kunneth theorem for homology. Sample computations. | Notes | |
| Monday 1/30 | Lecture 7: The Kunneth theorem for cohomology. The cup product on cohomology. | Notes | |
| Wednesday 2/1 | Lecture 8: Commutativity of the cup product. An explicit model for the cup product (using an explicit model of the Alexander-Whitney map), which is associative and unital on the chain level. The Kunneth isomorphism is a ring isomorphism. First computations of cohomology rings. | Notes | |
| Monday 2/6 | Lecture 9: The cup product on relative cochains. The cohomology rings of (real, complex, quaternionic projective spaces): statement of result and applications to distinguishing spaces not distinguished by cohomology or homology groups. Cap product. | Notes | |
| Wednesday 2/8 | Lecture 10: Properties of cap product. Orientations of vector spaces, and local orientations of manifolds at any given point. | Notes | |
| Monday 2/13 | Lecture 11: (Global) R-orientations of manifolds as coherently or continuously varying local R-orientations at each point of the manifold. Continuous in the sense that R-orientations can be thought of as sections of a suitable bundle of R-modules of local homology groups which pointwise generate (i.e., give local orientations at each point). The orientation double cover of a manifold, and the relationship between (Z-)orientability and R-orientability. | Notes | |
| Wednesday 2/15 | Lecture 12: Main theorem about the computation of the n-dimensional homology of an n-dimensional manifold with R-coefficients (depending on whether M is compact and R-orientable or not). A more general technical Lemma about identifying homology localized along a closed set with (compactly supported) sections of the bundle of local homologies along points of that closed set. Proof of the technical Lemma modulo two technical inductive claims. | Notes | |
| Wednesday 2/22 | Lecture 13: Direct limits of direct systems. Proof of the remaining two technical claims about the isomorphism betweeen nth homology localized along a compact set A and the space of sections of the bundle of local homologies along points of a. (one of which requires direct limits) | Notes | |
| Monday 2/27 | Lecture 14: Statement of Poincare duality for compact manifolds, and consequences: the intersection pairing homologically and cohomologically, cohomology rings of projective spaces. | Notes | |
| Wednesday 3/1 | Lecture 15: Relative cap product. The statement of Poincare duality for arbitrary (not necessarily compact) manifolds, involving compactly supported cohomology. Outline of proof, and verification for Euclidean spaces. | Notes | |
| Monday 3/6 | Lecture 16: Completion of proof of Poincare duality (modulo a few technical verifications). Manifolds with boundary, collar neighborhoodsd, and Poincare duality for manifolds with boundary. | Notes | |
| Wednesday 3/8 | Lecture 17: Reduction of Poincare duality for manifolds with boundary to the case of non-compact manifolds. Fiber bundles and first examples of fiber bundles. | Notes | |
| 3/13-3/17 | No class (spring break). | ||
| Monday 3/20 | Lecture 18: More examples of fiber bundles. Vector bundles and examples. Principal bundles and the example of the frame bundle of a vector bundle. | Notes | |
| Wednesday 3/22 | Lecture 19: Operations on principal bundles, and in particular, vector bundles associated to principal G-bundles + G-representations. Operations on vector bundles. Sections of fiber bundles, and the implications of having sections for principal and vector bundles. Inner products on vector bundles, as an example of a structure inducing a reduction of structure group of the associated principal frame bundles. Other examples of structures on vector bundles and the associated reductions of structure group. | Notes | |
| Monday 3/27 | Lecture 20: Homotopy invariance of pullbacks. Vector bundles via clutching functions. | Notes | |
| Wednesday 3/29 | Lecture 21: Classifying spaces for vector bundles. | Notes | |
| Monday 4/3 | Lecture 22: Characteristic classes. The first Stiefel-Whitney and first Chern classes. Additivity of first Chern (and Stiefel-Whitney classes) under tensor product for line bundles. | Notes | |
| Wednesday 4/5 | Lecture 23: Axiomatic characterization of higher Stiefel-Whitney and Chern classes. Statement of the Leray-Hirsch theorem, and construction of Chern and Stiefel-Whitney classes using the Leray-Hirsch theorem. | Notes | |
| Monday 4/10 | Lecture 24: Verification that the Chern and Stiefel-Whitney classes as constructed (using Leray-Hirsch) satisfy stated axioms. The splitting principle, and the proof of the axiomatic characterization of Chern and Stiefel-Whitney classes, modulo proof of the splitting principle. | Notes | |
| Wednesday 4/12 | Lecture 25: Sketch of proof of the Leray-Hirsch theorem and the splitting principle. First computations of Stiefel-Whitney classes. | Notes | |
| Monday 4/17 | Lecture 26: The tanget space to Grassmannians in terms of the tautological bundle and its complement. The Stiefel-Whitney classes of real projective spaces. Applications to immersions and embeddings: the Whitney duality formula for immersions in Euclidean space, and immersion constraints for real projective spaces in Euclidean spaces (e.g., RP10 can only embed in Rn if n is greater than or equal to 15). Stiefel-Whitney numbers (not covered in lecture). | Notes | |
| Wednesday 4/19 | Lecture 27: A source of complex vector bundles: tangent bundles to complex manifolds. The tangent bundle to the complex Grassmannian. The Chern classes of complex projective space. The Chern classes of linear dual vector bundles (and conjugate vector bundles, which are non-canonically isomorphic). The complexification of a real vector bundle, and the fact that its odd Chern classes vanish or are 2-torsion. Pontryagin classes of a real vector bundle as the even Chern classes of its complexification. | Notes | |
| Monday 4/24 | Lecture 28: Whitney sum formula for Pontryagin classes (holds modulo 2-torsion). The Pontryagin classes of complex projective space. Pontryagin numbers and Stiefel-Whitney numbers and applications. The cohomology ring of BU(k) = the infinite Grassmannian of complex k-planes is a polynomial ring in the Chern classes of the tautological bundle. Consequentially all characterstic classes of complex rank k bundles are polynomials in Chern classes. | Notes | |
| Wednesday 4/26 | Lecture 29: Analogous computations for the cohomology BO(k) mod 2 in terms of polynomials in w_i of the tautological bundle (and statement of an integral calculation mod 2-torsion in terms of p_i). Invariants of oriented vector bundles: the Thom and Euler class. The Thom isomorphism theorem. The Euler class vanishes on bundles that have a non-zero section, so (given an analogous Whitney sum formula) cannot be invariant unde adding a trivial bundle. Relationship between the Euler class and the Euler characteristic for a compact oriented manifold. | Notes |