MATHEMATICS 721 A1: Differential Topology I

Fall 1998

Instructor: Takashi Kimura

Lectures: MWF 11:00-12:00pm in MCS B29

Phone: (617)353-1486

Office: MCS 234

Text: Differential Topology, by V. Guillemin and A. Pollack; Prentice Hall, Inc. (c)1974

Supplemental Reference: Differential Geometry, by M. Spivak; Publish or Perish, Inc. (c) 1979.

Office Hours: WF 10:30-12pm

Web Page: This class has a Web page which can be viewed with a Web browser (such as Netscape). Documents distributed in class, homework assignments, and related information will be posted on the Web page. The URL is http://math.bu.edu/people/kimura/Fall98/721/index.html

Content: A topological n-dimensional manifold (or n-manifold) is a topological space which is a collection of copies of Rn which are glued together using homeomorphisms. On the other hand, using differential calculus on Rn, one can define a smooth n-manifold which is a topological n-manifold which is obtained from gluing together copies of Rn by using smooth maps.

Smooth n-manifolds are then objects on which one can generalize various notions from calculus on Rn. Differential topology is essentially the study of the topology of smooth manifolds using calculus. Calculus is a powerful tool in analyzing the topology of manifolds.

This course (and its sequel) will introduce the notions of a smooth manifold, smooth maps, inverse function theorem for manifolds, immersions and submersions, Sard's theorem, transversality, winding numbers, intersection theory, differential forms, and integration. Special topics such as Morse theory, Riemannian geometry, fiber bundles and connections may also be covered.

Homework: Homework will be assigned periodically. Late homework will not be accepted. Students may discuss homework with each other (and are encouraged to do so) but all written work must be prepared independently.

Exams: There will be a take home midterm and final exam.

File translated from TEX by TTH, version 1.60.