Etnyre: Differential Topology

Differential Topology

Math 6452, Fall 2018

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 semesterwe will discuss the theory of manifolds and a way to generalize differential, integral and vector calculus. By the end of the course, students should be understand and be able to work with manifolds, tangent and cotangent bundles, vector bundles, differential forms, vector fields and many other things listed in the official course description.

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.

Introductory notes, recollections from point set topology and quotient spaces.

Announcements:

Please take part in the Course/Instructor Opinion Survey (CIOS). Written comments are most appreciated and will help shape future versions of this course (and others). Please comment on specific things you liked and did not like about the course. Your feedback is most appreciated!
Our final exam coming up Wednesday, December 12, 2017 from 8:00 am to 10:50 am in our normal class room Skiles 257. It will be comprehensive and be in the form of a comprehensive exam. That is there will be 8 questions of which you will be asked to solve 5 (of your choice). The best way to prepare for the test is to (1) go through the class notes, (2) go through all the homework problems and sample test problems (from the midterm), and (3) read text book. Please let me know if you have any questions about the exam.
Our mid-term exam will be in class on October 12. 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.

Lectures: Monday-Wednesday-Friday from 10:10 to 11:00 in Skiles 257.
Professor: John Etnyre
Office: Skiles 106
Office Hours: Wednesdays 11:00-12:00 (I will usually be available at the same time on Friday and feel free to stop by when you see my door open or set up an appointment.)
Phone: 404.385.6760
e-mail: etnyre "at" math .gatech.edu

Grading Policy

The course grade will be based on the following.

Homework: 40%
1 Midterm: 30% each
Final Exam: 30%

The cutt-offs for grades my be reduced from what is indicated below, but they will not increase.

Average	Grade
in [90,100]	A
in [80,90)	B
in [70,80)	C
in [60,70)	D
less than 60	F

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 10. 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 12th from 8:00pm-10:50pm.

Textbook

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

Miscellaneous matters

Academic Integrity. 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.
Intent for Inclusivity. As a member of the Georgia Tech community, I am committed to creating a learning environment in which all of my students feel safe and included. Because we are individuals with varying needs, I am reliant on your feedback to achieve this goal. To that end, I invite you to enter into dialogue with me about the things I can stop, start, and continue doing to make my classroom an environment in which every student feels valued and can engage actively in our learning community.