This is an introductory course in algebraic topology. We will begin by covering the basics of homotopy theory, fundamental groups and covering spaces. We then move on to homology theory from various perspectives. Finally we will develop as much cohomology theory as possible before the end of the semester.

The only prerequisite for the course is a basic understanding of point set topology and group theory. Here are some basic notes on quotient spaces that might help refresh this important idea (we will be using this to "glue" spaces together all the time).

Announcements:

• Our midterm exam will be on February 26th in class. The exam will cover everything up to and including covering spaces. That is Chapters 0 and 1 in Hatcher's book and Homework Assignments 1 through 4. As this class is supposed to help prepare students for the comprehensive exam it will have a little of the format of a comprehensive an exam: specifically, the exam will have 4 questions on it, and you will need to answer 3 of the 4 questions. At least 2 will be questions will be very similar to homework problems or a result in our class notes. So the best way to prepare for the exam is make sure you can work all the problems on the homework sheets and read through the class notes on t-square. If you have any questions about the exam or the material please come see me.

### Textbook

The textbook for this class is

• 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.

There are many good textbooks for algebraic topology, but I just mention one other book you might find useful:

• Topology and Geometry by Bredon.

### Course Content

This is a first course in Algebraic Topology. The official syllabus for the course can be found at

Algebraic Topology I

In brief we will be covering homotopy theory, CW complexes, the fundamental group, and covering spaces in the first third or so of the course. This will more or less be the contents of Chapters 0 and 1 in Hatchers book. We will then move to homology theory and cohomology theory for the rest of the course, hopefully ending with Poincare duality. This will more or less be the contents of Chapters 2 and 3 in Hatchers book. We might tough on part of Chapter 4 here and there in the course as well as other topics as time permits.

The course grade will be based on the following.

Homework: 60%
Midterm: 20%
Term Paper: 20%

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 date for the exam is February 26. If you need to miss the midterm exam you must talk to me about this in advance if possible. If you miss a midterm exam for an excused reason you will be given a makeup exam.

### Term Paper

Purpose: In graduate school it is important to learn how to research a topic independently and also present mathematics clearly in a written form. The term paper will give you a chance to practice and get feedback on these skills.

Each student will write a term paper for the class. The paper will cover some topic in topology, algebra, algebraic topology, or its application to some other area (physics, biology, data analysis, etc.). The idea is for you to explore some area you find interesting related to the class.

The paper will need to have

1. a significant mathematical component (that is proofs, computations, or the like) and
2. have a good exposition (that is, written well enough to other students to learn something from the paper).

The target audience for these papers is other students in the class. In fact part of the grade on the paper will be your providing helpful and constructive feedback to other students. Knowing who your audience is will help you while writing the paper and by seeing what others are doing well and not so well, you will be able to get a better idea how to be a better writer yourself.

• 5% Consulting with me by the end of January on a topic for your paper
• 15% Draft of your paper turned in by March 28
• 20% Constructive feed back you provide on 2 other students papers (by April 9)
• 60% Final paper turned in by April 23

When you turn in the draft of the paper, you will turn in three copies. I will then assign 2 other students to read your paper and provide feedback using this form. The feedback will be constructive and kind. (If you make negative comments that are un-helpful and un-constructive. Then I will not give the comments to the student and you will get a 0 for this 20% of the grade.) After you complete your evaluation of other student's papers, you will fill out this form about your own paper. I will collect the student feedback and return it to you shortly after April 9. You can use this to make the final version of your paper to be turned in April 23.

The paper will need to be 5 to 12 pages (you can talk to me to get approval if you have a good reason for the paper to be shorter or longer) and be turned in as a pdf document. You should try to write the paper in TeX or LaTeX (ask me if you do not know about this), but I will accept any pdf document (so you can use Word or some other program to create the paper if you like). I will post the final versions of the paper on the T-Square site so other students can read them if they would like to do so.

Your paper can be on anything you like, but here are some thoughts to get you started.

### Miscellaneous matters

• 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.

### Homework

Homework Assignment 1: Due in class Janurary 19 Janurary 22

Homework Assignment 2: Due in class Janurary 29

Homework Assignment 3: Due in class February 7

Homework Assignment 4: Due in class February 16

Homework Assignment 5: Due in class March 9

Homework Assignment 6: Due in class April 13