Geometry and Topology I
  
Math 6457, Fall 2008
    

This is an introductory course in algebraic and differential topology and geometry. 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

  • Algebraic topology: a way of studying topological spaces by associating algebraic objects (like groups, vector spaces ...). Specifically we will study the fundamental group, covering spaces, homology and cohomology.
  • Differential topology: the study 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.
  • Riemannian geometry: a way to measure distances and angles, areas and volumes on a manifold. We will introduce the Riemannian metrics and the basics of curvature, geodesics and covariant derivatives.

The prerequisites for the course are vector calculus, some basic knowledge of point set topology and very basic algebra. 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.

Lectures: Tuesday and Thursday 1:35 to 2:55 in Skiles 240.
Professor: John Etnyre
Office: Skiles 106
Phone: 404.385.6760
e-mail: etnyre "at" math .gatech.edu

Grading Policy

Grading for the class will be based on approximately six homework assignments.

The 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.

Textbook

There are two textbook for this class:

  • Algebraic Topology by Hatcher. The text is available on-line, but is is a fairly inexpensive book and having a hard copy can be a nice reference.
  • Introduction to Smooth Manifolds by Lee.

There are many good textbooks for algebraic and 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
  • Differential Topology by Guillemin and Pollack

Homework

Homework Assignment 1: I have assigned many problems, but do not write up all of them. Carefully write up the underlined ones (there are just three such problems) and turn them in to me in class on September 2. Please do try to work the other problems though. If you have any questions stop by my office.

  • Read Hatcher pages 1-14 (through example 0.14).
  • Chapter 0, problems 3, 5, 6, 18
  • Chapter 1.1, problems 5, 6, 7, 16, 20

Homework Assignment 2: Again you only need to turn in the underlined problems, but please try to work the others too. If you have any questions on these problems just stop by my office. This assignment is due on September 23.

  • Chapter 1.2, problems 7, 9, 14, 21
  • Chapter 1.3, problems 9, 10, 12, 15, 19, 29
  • Read Section 1.A
  • Chapter 1.A, problems 3, 8, 11

Homework Assignment 3: Due October 28.

Homework Assignment 4: Due December 4.