This is an introductory course in algebraic topology. We will begin by covering the basics of homotopy theory, fundamental groups and classifying spaces. We then move on to CW-complexes and 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 March 5th in class.

Lectures: Tuesday and Thursday 1:30 to 3:00 in Skiles 256.
Professor: John Etnyre
Office: Skiles 106
Phone: 404.385.6760
e-mail: etnyre "at" math .gatech.edu

Grading Policy

The course grade will be based on the following.

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 March 5. 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 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 April 30th from 2:50pm - 5:40pm.

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.

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: Read Chapter 0 and Sections 1.1 and 1.2 in Hatcher's book. Work the following problem. I have assigned many problems, but do not write up all of them. Carefully write up the stared ones and turn them in to me in class on Janurary 22. Please do try to work the other problems though. If you have any questions stop by my office.

• Section 0: 1, 2 , 10* , 17, 18*
• Section 1.1: 5, 6*, 7, 16*, 17*, 18*,
• Section 1.2: 8*, 9*, 11*, 14, 17*, 18, 21

Homework Assignment 2: Read Sections 1.3 and 1.A in Hatcher's book. Work the following problem. I have assigned many problems, but do not write up all of them. Carefully write up the stared ones and turn them in to me in class on February 5. Please do try to work the other problems though. If you have any questions stop by my office.

• Section 1.3: 4*, 5, 7, 9*, 10*, 12*, 14, 18*, 19, 20*, 21*
• Section 1.A: problems 3, 7*, 8*, 9, 11
• Extra Problem*: Describe all covering spaces of the torus (S^1 \times S^1) and all covering spaces of the Klein bottle up to equivalence.

Homework Assignment 3: Read Sections 2.1 in Hatcher's book. (Note we are taking a bit of a different approach to homology to start, so Section 2.1 will look different from the material in the lectures, but it is still good to read and try to compare with the lecture material.) Work the following problem. Carefully write up the stared ones and turn them in to me in class on February 19. Please do try to work the other problems though. If you have any questions stop by my office.

• Section 2.1: 11*, 12, 13, 14, 15*, 16*
• Extra Problem*: For an path connected space X show that any map f:X --> X indues the identity map on H_0(X).
• Extra Problem: In class we showed that there is a map h_X: \pi_1(X) --> H_1(X) for any path connected space X. Suppose f:X -->Y is a map between path connected spaces and f_* is the map induced on \pi_1 and f_1 is the map induced on H_1. Show that h_Y composed with f_* equals f_1 composed with h_X.
• Extra Problem*: Recall if p:Y-->X is a covering space the induced map p_* on \pi_1 is injective. Show that p_1 on H_1 need not be injective. Hint: Consider a wedge of circles.

Homework Assignment 4: Read Sections 2.2, 2.A and 2.B in Hatcher's book. Work the following problem. Carefully write up the stared ones and turn them in to me in class on March 10. Please do try to work the other problems though. If you have any questions stop by my office.

• Section 2.1: 20*, 21, 22*, 26, 28*, 29*
• Section 2.2: 6, 7*, 8, 9*, 10*, 11*, 12*, 13, 14*, 19, 22, 24, 26

Homework Assignment 5: Read Sections 2.3, 2.A, and 2.B in Hatcher's book. Work the following problem. Carefully write up the stared ones and turn them in to me in class on March 31. Please do try to work the other problems though. If you have any questions stop by my office.

• Section 2.2: 21*, 23*, 25*, 28, 29*, 30*, 31, 32*
• Section 2.3: 1*, 4
• Section 2.B: 1*, 2, 3, 4*, 9, 10*, 11

Homework Assignment 6: Read Sections 3.1 and 3.2. Work the following problem. Carefully write up the stared ones and turn them in to me in class on April 14. Please do try to work the other problems though. If you have any questions stop by my office.

Section 3.1: 5*, 6*, 7, 8*, 9*, 11
Section 3.2: 1*, 6*, 7*, 18*