This is an introductory course in differential topology. The aim of this course is to introduce the basic tools to study the topology and geometry of manifolds (and some other spaces too). (Smooth) Manifolds are "locally Euclidean" spaces on which we can "do calculus" and "do geometry". These spaces are at the center of a great deal of much of the most exciting current research in mathematics and are essential to many applications of mathematics into science and engineering. Throughout the year we will discuss the theory of manifolds and a way to generalize differential, integral and vector calculus. We will define and study manifolds, tangent and cotangent bundles, vector bundles, differential forms, vector fields and many other things.
The prerequisites for the course are vector calculus andsome basic knowledge of point set topology. Most topics beyond the most basic material will will be reviewed, at least briefly, when we need it. (Also I am more than willing to meet in office hours and discuss background material here and there should the need arise.) But to get started I have written up an introduction to the course with some of the most important ideas we will need from point set topology. In particular, a very important concept that many people have not seen much of before is quotient spaces. This is a very convenient way to rigorously describe how to build up complicated spaces from simple one.
- Our mid-term exam will be in class on October 19. It will cover the material in Sections I to VI of the class notes and Homework Assignments 1 to 3. The exam will consist of 3 or 4 proof/computation problems and several true/false questions. Here are some sample problems for the exam. The best way to prepare for the test is to (1) go through the class notes, (2) work through the sample problems, (3) read text book, (4) go through all the homework problems.
Tuesday and Thursday from 3:00 to 4:15 in Skiles 254.
Office: Skiles 106
Office Hours: 10:00-11:00 Thursdays (I will usually be available at the same time on Tuesday and feel free to stop by when you see my door open or set up an appointment.)
e-mail: etnyre "at" math .gatech.edu
The course grade will be based on the following.
The cutt-offs for grades my be reduced from what is indicated below, but they will not increase.
|less than 60
The homework assignments will be posted below and
will be due in class on the day indicated on the assignment. I encourage you
to work on these assignments with other students in the class and to
use whatever other resources you might have (like me and others in the department), but each problem must be written up in your own words by you. At some point everyone
needs to learn TeX or LaTeX so I
encourage you to write up your homework using one of these packages,
but this is not a requirement. If you would like help getting started
with TeX or LaTeX you are welcome to talk to me about it.
The midterm exam will be in class and I will announce it on this web page and in class at least 1 week before the exam. The tentative dates for the exam is October 19. If you need to miss a midterm exam you must talk to me about this in advance if possible. If you miss the midterm exam for an excused reason you will be given the option to take a makeup exam or skip the exam and have the homework and final exam count more towards your final grade.
The final exam is tentatively scheduled for December 14th from 11:30pm-2:20pm.
There are two textbook for this class:
- Introduction to Smooth Manifolds by Lee.
- Differential Topology by Guillemin and Pollack
The primary text is Lee, but Guillemin and Pollack is also a good reference and at times has a different perspective on the material. Neither text is required but I will sometimes assign homework out of Lee. You can download an electronic version of Lee from the library.
There are many good textbooks for differential topology. I mention several below. Some of them use a different perspective that we take in this class, but that can be useful to see!
- Topology and Geometry by Bredon
- An Introduction to Differential Manifolds by Barden and Thomas
- An Introduction to Manifolds by Tu
- A Comprehensive Introduction to Differential Geometry, Vol. 1 by Spivak
- All students are expected to comply with the Georgia Tech Honor Code.
- Students with Disabilities and/or in need of Special Accommodations: Georgia Tech complies with the regulations of the Americans with Disabilities Act of 1990 and offers accommodations to students with disabilities. If you are in need of classroom or testing accommodations, please make an appointment with the ADAPTS office to discuss the appropriate procedures.
I have assigned many problems, but do not write up all of them. Carefully write up the underlined ones and turn them in to me in class on September 5.
Please do try to work the other problems though. If you have any questions stop by my office.
- Lee Chapter 1, Problems 1-7, 1-9, 1-10
- Lee Chapter 2, Excercise 2.1 (on page 33)
- Lee Chapter 2, Problems 2-2, 2-3, 2-5, 2-6, 2-10
- Extra Problem: Recall the Grassmann of k-dimensional subspaces of \R^n is denoted G(k,n) (see Lee pages 22-24). Using the standard inner product on \R^n and denoting the orthogonal complement of a subspace V of \R^n by V^\perp, we can define a map f:G(k,n) --> G(n-k,n) by f(V) = V^\perp for every V in G(k,n). Show that f is a diffeomorphism.
Assignment 2: Due in class on September 19
Assignment 3: Due in class on October 5
Assignment 4: Due in class on October 26
Assignment 5: Due in class on November 9
Assignment 6: Due in class on
November 21, now November 28