Instructor | Sheel Ganatra |

Office | Zoom |

sheel (dot) ganatra (at) usc (dot) edu (this is the best way to reach me) |

In Spring 2021, 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, etc.), Poincare duality, *vector bundles*, and their * 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 the base of a vector bundle measuring the non-triviality of the bundle.

Along the way, we will study properties of more general fibrations (and more specifically principal bundles) and classifying spaces. Time permitting, we will introduce * spectral sequences*, * homotopy groups*, and give applications of theory of characteristic classes to the study of * cobordism* and the classification theory of manifolds.

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.

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).

- Allen Hatcher's books:
*Algebraic Topology*[H-AT],*Vector bundles and K-theory*[H-VB] , and*Spectral sequences in algebraic topology*[H-SS]. - Glen Bredon's
*Topology and Geometry*. - Milnor and Stasheff's
*Characteristic classes*. - Bott and Tu's
*Differential forms in algebraic topology*. - Cohen's lecture notes,
*The topology of fiber bundles*. - MIT's OCW course notes for
*Algebraic Topology II*, taught by Haynes Miller. - Hutchings's lecture notes, titled
*Introduction to spectral sequences*. - McCleary's
*User's guide to spectral sequences*.

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 |
---|---|---|

2/17 | HW 1 | |

2/26 | HW 2 | |

4/5 | HW 3 | |

4/12 | HW 4 | |

4/23 | HW 5 | |

5/6 | HW 6 |

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 include:

- Milnor and/or Segal's construction of, and properties of BG.
- Cohomology calculations via spectral sequences of a series of manifolds (e.g., classical groups)
- Steenrod operations and Wu's formula / formula for Stiefel-Whitney classes in terms of Steenrod operations
- Eilenberg-Maclane spaces and/or their cohomology groups
- The Gysin sequence and applications
- K-theory and some applications (Bott periodicity, the study of division algebras, etc.)
- Model categories and some applications (too large of a topic, would need to specify a sub-topic)
- The Brown representability theorem, and/or spectra
- The Chern character and/or other multiplcative characteristic classes, and applications

Final papers will be due (by e-mail to the instructor) on **May 12**.

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 |
---|---|---|---|

Jan. 15 (Fri) |
Lecture 1: Welcome + an overview of the class | Notes | |

Jan. 18-22 (Wed, Fri classes) |
Week 1, Lectures 2-3: Cohomology and the universal coefficient theorem: Ext and Tor and projective/injective resolutions. Proof of the Universal coefficient theorem | Notes | |

Jan 25-29 (Mon, Wed classes) |
Week 2, Lectures 4-5: The Eilenberg-Zilber theorem (via the method of acyclic models). The algebraic Kunneth theorem. The Kunneth theorem for homology (which is a consequence of the previous two theorems). | Notes | |

Feb 1-3 (Mon, Wed classes) |
Week 3, Lectures 6-7: The Kunneth theorem in cohomology. The coproduct on homology and the cup product on cohomology (both via the abstractly defined cross product and a particular formula using the Alexander-Whitney map). Computations of cup product and applications (such as distinguishing new colllections of spaces, distinguishing homotopy classes of maps from S3 to S2, the Borsuk-Ulam theorem). | Notes | |

Feb 8-26 (Mon, Wed first week, Wed second week, Mon third week) |
Weeks 4-6, Lectures 8,9,10,11: Cap products, orientations, Poincare duality. | Notes | |

March 1 (Mon) |
Week 7, part a: More about Poincare duality. Poincare duality for manifolds with boundary. | Notes | |

March 3-10 (Wed, Fri, Mon, Wed lectures), |
Week 7, part b and week 8: Fiber bundles, principal bundles, and vector bundles. The equivalence between principle GL(k) bundles and rank k vector bundles. Various operations on such bundles (associated bundles of principal bundles, pullbacks, cartesian products, hom, tensor, and Whitney sum of vector bundles). Sections. Inner products on vector bundles. Homotopy invariance of pullbacks. Clutching functions. Classifying spaces for vector bundles. | Notes | |

March 15-19 (Mon, Wed, Fri lectures) |
Week 9: An introduction to characteristic classes. The first Chern and Stiefel-Whitney classes via the classification of line bundles and known cohomology of infinite projective space. The axiomatic characterization of Chern and Stiefel-Whitney classes. The construction of higher Chern and Stiefel-Whitney classes via the Leray-Hirsch theorem. | Hatcher's Vector bundles and K-theory, Hatcher Section 4.D. |
Notes |

March 22-26 (Mon, Wed, Fri lectures) |
Week 10: The splitting principle. The axioms of Chern and Stiefel-Whitney classes uniquely characterize these classes. Characteristic classes associated to manifolds (= the class of their tangent bundle). Computations of Stiefel-Whitney classes of real projective space and applications to immersions and embeddings. Stiefel-Whitney numbers and their cobordism invariance. Some computations of Chern classes, e.g., for complex projective spaces. | Notes | |

March 29-31 (Mon, Wed lectures) |
Week 11: Pontryagin classes, Pontryagin numbers, and oriented cobordism. The cohomology of Grassmannians via the splitting principle, and hence a classification of complex and real characteristic classes (with Z and Z/2 coefficients respectively). Orientations on vector bundles. | The computation of the cohomology of Grassmannians follows an argument in Husemoller's Fiber bundles. |
Notes |

April 5, 11 (Mon, Fri lectures) |
Week 12: Thom classes (and the proof they always exist and are unique for oriented vector bundles) and the Thom isomorphism theorem. Euler classes and their properties. Application: Gysin sequence for the unit sphere bundle of any vector bundle. The relationship between Euler number and signed count of zeroes of a transverse section on a smooth oriented vector bundle over a smooth oriented manifold (with rank equals dimension). | Notes | |

April 12, 14, 19 (Mon, Wed, Mon lectures) |
Week 13 and 14a: Relationship between Thom classes and Poincare duality for submanifolds. Cobordism rings and Thom's cobordism theorem. Thom spaces of bundles. A quick introduction to homotopy groups and some methods of computation (tailored towards learning information about homotopy groups of Thom spaces): Hurewicz and Whitehead theorems modulo the Serre class of finite abelian groups, and understanding the homotopy groups of Thom spaces in terms of homology in a range (mod this Serre class). Sketch of the map and proof appearing in Thom's theorem. The cohomology of the oriented Grassmannian (modulo 2-torsion) via the Gysin sequence. Applications. Classifying manifolds up to cobordism and numerical invariants of manifolds up to cobordism by Pontryagin numbers. The signuature of a 4k-dimensional manifold and the Hirzebruch signature theorem.
| Milnor-Stasheff | Notes |

April 21, 23 (Wed, Fri lectures) |
Week 14b: Introduction to spectral sequences. Filtrations and the spectral sequence associated to a filtration on a chain complex (i.e., a filtered chain complex). First examples (coming from bicomplexes such as the tensor product of two chain complexes, or from the chains associated to a skeletal filtration of spaces). The Leray-Serre spectral sequence of a fibration (including a detour describing local coefficient systems and their homology, necessary to describe page 2 in general, although we mostly focus on the case when the relevant system is trivial).
| Hutchings Introduction to spectral sequences. | Notes |

April 26, 28 (Mon, Wed lectures) |
Week 15: Sketch of proof of the Leray-Serre spectral sequence. First computations using the Leray-Serre spectral sequence. Naturality properties of the spectral sequence. Cohomological spectral sequences and spectral sequences of algebras. The cohomological Leray-Serre spectral sequence, and computations of cohomology rings from it. Edge homomorphisms, transgressions, and applications of the general machinery (sketch of proof of the Hurewicz theorem and the Wang long exact sequences). | Notes |