MAT 566: Differential TopologyFall 2006Course InformationCourse InstructorName: Aleksey Zinger Office: Math Tower 3-117 Phone: 632-8618E-mail: [email protected] Website: http://math.sunysb.edu/∼azinger/mat566Office Hours: Wed. 9-12Course WebsiteAll homework assignments and various updates will be posted on the course website,http://math.sunysb.edu/∼azinger/mat566.Please visit this website regularly.GradingYour grade will be based on class participation, light homework assignments (roughly two problemsevery two weeks), and class presentation (see below).ReadingsThe textbook is Characteristic Classes, Annals of Mathematics Studies 76, by John Milnor andJames Stasheff. We will cover most of the book plus some of the readings listed below (perhapsonly the S7-paper) and/or some other papers. You should acquire the book, but copies of thesupplementary readings will be provided as needed.About the CourseCharacteristic classes are certain cohomology classes associated to real and complex vector bundles.While they can be used to distinguish between non-isomorphic vector bundles, this is certainly notthe main point of them. Characteristic classes of vector bundles often provide a convenient way fordoing computations on manifolds which may lead to deep results in topology of smooth manifolds.Most of the papers listed below are essentially clever applications of these classes. Characteristicclasses, as well as vector bundles, are indispensable in modern-day geometry and topology.Homework AssignmentsEach section in the book is followed by a few exercises. These are generally directly related tothe section and are thus not very hard. Two or so of these exercises will be assigned every twoweeks or so. However, you should figure out all (or at least most) of the exercises for yourself.When writing solutions to the assigned exercises, you should take the statements of all precedingexercises as given. Feel free to discuss any of the exercises with anyone else, but do write your ownsolutions.You should also read (and study in detail) every section of the book (as well as additional readings)covered in class. The book is relatively leisurely written, more like Spivak than Warner.This is a second-year course, and the formal requirements are fairly light. However, the more effortyou put into this course, the more you are likely to benefit from it. If you are interested in algebraicgeometry, at the minimum you should have a firm grasp of chern classes and Grassmanians. If youare interested in algebraic topology, you should try to master the entire bo ok, as many modernconstructions build up on those in the bo ok. If you are interested in geometry in general, thiswould be somewhere in between.Class PresentationYou will be asked to give a presentation during one of the classes on either a geometric (i.e. veryconcrete) application of characteristic classes or a related geometric topic. You will prepare thepresentation with another student; the class time and the topic will be split between the two ofyou. You will need to meet with me before your presentation.Here are some of the potential topics for in-class presentation, time-permitting:• Schubert Calculus, based on Griffiths&Harris, pp197-207. How many lines pass through 4lines in 3-space? [Cohomology of Grassmanians]• How many lines lie on a cubic surface? on a quintic threefold?• J. Milnor, On Manifolds Homeomorphic to the 7-Sphere, Annals of Math. 64 (1956), 399–405[existence of more than one differentiable structure on S7]• Hirzebruch’s Signature Theorem• M. Kervaire, A Manifold Which Does not Admit Any Differentiable Structure, Comment. Math.Helv. 35 (1961), 1–14• M. Atiyah and R. Bott, The Moment Map and Equivariant Cohomology, Topology 23 (1984),1–28 [computation of characteristic classes by localization to fixed lo ci of group actions]• J. Milnor, Construction of Universal Bundles, II, Ann. of Math. (2) 63 (1956), 430–436[construction of classifying spaces]• J. Milnor, Topology from the Differentiable Viewpoint, Princeton Landmarks in Mathematics[Sard’s