Spring 1998
Instructor: Leslie Saper
Text: Differential Forms in Algebraic Topology, by Raoul Bott and Loring Tu
Prerequisites: Linear algebra, advanced calculus, and basic topology. Acquaintance with manifolds and basic algebraic topology (fundamental group and singular homology) would be helpful, but not essential.
About this course:
This is an introductory course in the use of differential forms in algebraic topology. It is recommended not only for students interested in topology, but also for those interested in eventually working in differential geometry, analysis on manifolds, algebraic geometry from a trancendental viewpoint, and the geometric analysis of noncompact and singular spaces.We will roughly cover the first two chapters of Bott and Tu, with some time at the beginning devoted to introductory material, and perhaps some time at the end devoted to a bridge to current research. Specifically, we will begin by introducing differential forms, first for Euclidean space, and then for smooth manifolds (which will be defined). Differential forms are ``dual'' to submanifolds, and this duality is realized by integration. For example, a 1-form can be integrated over a curve, a 2-form can be integrated over a surface, and so on. Differential forms allow one to construct an invariant of the smooth manifold, the de Rham cohomology, which will actually turn out to be a topological invariant. An important tool for computation will be the Mayer-Vietoris argument. This always one prove Poincaré duality and the Künneth formula. The de Rham cohomology can be connected to pure algebraic topology by using sheaf theory and Cech cohomology, which we will introduce. Finally we introduce vector bundles (and sphere bundles), a generalization of the idea of tangent vector fields, and study how we can associate a certain class of differential forms (the Euler class) to a vector bundle. Integrating over the manifold yields a number. In the case of the tangent vector bundle over a triangulated surface or a 2-cell complex, we recover the Euler number (#faces - #edges + #vertices).
Requirements: There will be weekly Problem Sets and a Final Exam. You may (and are encouraged) to discuss the problems with your classmates, however the paper you hand in must be composed and written by you individually.