This course will cover classical and modern trends in the topology of 3-manifolds. We will cover constructions of 3-manifolds and classical decompositions of 3-manifolds into simple pieces. We will then consider more recent advances with a focus on Dehn surgery. Among other things will consider (1) the effects of Dehn surgery on a 3-manifold, for example when are incompressible surfaces in a 3-manifold constructed or destroyed vis surgery, (2) can surgery on two different knots yield the same 3-manifolds, and (3) can a knot be characterized by Dehn surgery. Our approach will largely be geometric/combinatorial, but on occasion we will need to quote results from other areas like geometry or Heegaard-Floer theory.
The prerequisites for the course are differential topology and algebraic topology. Specifically, I will be assuming you are familiar with smooth manifolds, fundamental groups, homology and cohomology, and homotopy groups.
You can find the class notes here.
Monday-Wednesday 9:30-10:45 in Skiles 308.
Office: Skiles 106
Office Hours: 10:00-10:55 Fridays (also feel free to set up an appointment via e-mail).
e-mail: etnyre "at" math .gatech.edu
Classes will be in person. I will record classes so that people can watch them if they are unable to attend class for some reason. I will be experimenting with various technology to achieve good recordings, so the first few might not be optimal, but I hope they will all be reasonable. I am also posting class notes that can be used to keep up with the class if you cannot attend all the classes in person.
It is highly recommended that you try to addend in person as much as possible, the live interaction with the easy ability to ask questions is very helpful in learning the material.
I highly encourage everyone in the classes to get vaccinated and wear a mask. Out of concern for your health and mine, I am choosing to wear a mask in the classroom and ask that you give me and your fellow classmates this same consideration. Masks have been shown to slow the spread of this virus. Your cooperation and understanding on this matter are much appreciated.
Our classroom should have enough seating to allow people to be somewhat distanced from one another and I encourage you to keep as much distance as possible from each other. Also, please use the same seats each day in class. This will help with contact tracking should someone come down with COVID.
All of these policies are subject to change as guidance for the CDC and Georgia Tech change.
Your grade will be based on class participation and some class assignments. For the class assignments, in the each of the months of September, October, and November, by the end of the month you need to either turn in 5 problems mentioned in class or in one of the text books (related to the material being covered in class) or alternatively give a talk in the Wednesday student geometry/topology seminars. The written assignments must be turned in by the end of month. The assignments will be graded on a pass/fail basis. A reasonable attempt will receive a pass. By reasonable attempt, I mean that the attempts for solutions to 4 out of 5 of the problems essentially correct even if there might be some minor errors. If you give a talk, that will count as a completed assignments for that month.
For solving the written problems you are welcome to consult with me or work with other students. But the problems must be written up by you, in your own words.
To receive passing grade you must complete and pass the monthly assignments. For an A you must also attend at least 90% of the classes (or have an excused absence), for an B you must attend at least 80% of the classes (or have an excused absence), for a C, 70%, and for a D, 60%. Due to the pandemic, I realize that some people might have difficulty participating in class as much as they would like (and to meet the cut-offs for the given grades). If you need to miss classes (maybe even all!), please just contact me and explain the situation and verify that you are keeping up with the class (thought recordings and class notes published on the website).
There are no official text book for this course, but the following books are quite useful for the material in the course:
- 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.