Differential Topology (Math 6250)
Section 100, Fall 2012
General Information and Syllabus

Course Description. A course in differential topology. Topics include differentiable manifolds, differentiable maps, embeddings, immersions, manifolds with boundary, vector bundles, the tubular neighborhood theorem, transversality, the Morse-Sard theorem, isotopy.

Instructor. David A. Richter

Coordinates of Class Meetings. 1:00-2:15 on Tuesdays and Thursdays in room 6620 of Everett Tower.

Basis of Evaluation.

Homework Assignments. Expect an assignment about every 2 weeks. You must type all of your homework solutions. Your instructor prefers the use of TeX over all other typesetting systems.

50-Minute Exam. There is one midterm exam, scheduled for Friday October 26.

Lecture. At some point during the last few weeks of the semester, you are expected to present some auxiliary material to the rest of class. This presentation should take one or two class periods. A list of suggested topics appears below.

Final Exam. The final exam is comprehensive over all material covered throughout the semester, although there may be greater emphasis on the material covered during the last few weeks of class. Unless otherwise announced or arranged, the final exam lasts from 10:15 until 12:15 on Wednesday December 12.

Required Text. Morris W. Hirsch. Differential Topology. Springer-Verlag, 1976.

Supplemental Texts.
Antoni A. Kosinski. Differential Manifolds. Dover Publications, New York, 2007. (Inexpensive, highly recommended.)
Victor Guilleman and Alan Pollack. Differential Topology. Prentice-Hall, 1974. (Another highly cited book on differential topology.)
John W. Milnor and James D. Stasheff. Characteristic Classes. Princeton University Press, 1974. (Provides solid motivation for this course.)

Student Conduct. Your instructor assumes that you are enrolled in this class because, at the very least, you want to participate in the University Community. In case there is any doubt, there is a code of rules of conduct which you must follow and which your instructor enforces. In particular, the use of any radio communication device during class violates this code (unless it's a life-or-death emergency). These policies and procedures are described in more detail at the website for the Office of Student Conduct.

Expectations. Attend and participate in every class meeting. Spend at least 10 hours per week outside of class on this subject. Attempt to improve your writing skills. Take pride in your work. Maintain a positive attitude. Understand your assumptions instead of assuming you understand.

Course Outline.

Differentiable Manifolds and Maps
   implicit function theorem
   differential structures
   tangent and cotangent bundles
   embeddings and immersions
   Whitney embedding theorema
   manifolds with boundary
Function Spaces
   weak and strong topologies
   approximation theory
Transversality
   Morse-Sard Theorem
Vector Bundles
   tubular neighborhood theorem
Elective Topics (Intended for student presentations)
   Degree of a Map
   Euler Characteristic
   Morse Theory
   Cobordism/Thom Homomorphism
   Isotopy
   Gauss-Bonnet Theorem

Homework.

Due September 28: Any 10 exercises from chapter 1 or assigned during class. You must work the exercises and submit solutions in teams of 2 or 3 students per team.

Due October 26: Any 10 exercises from sections 1.3, 1.4, 2.1, 2.2, 3.1 or assigned during class. You must work the exercises and submit solutions in teams of 2 or 3 students per team.

Due November 30: Any 10 exercises from chapter 4 or assigned during class. You must work the exercises and submit solutions in teams of 2 or 3 students per team.

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