Math 8803: Knots, 3-manifolds, and 4-manifolds

Email: hom [at] math.gatech.edu
Location: Skiles 005
Time: Tuesdays and Thursdays 12:30-1:45 pm
Office: Skiles 208B
Office hours: Mondays 3-4 pm, Thursdays 1:45-2:45 pm, and by appointment

Prerequisites

Math 6441 (Algebraic Topology) or consent of instructor

Assignments

Your grade will be based on classroom participation and assignments. At times throughout the lectures, I will leave parts of some proofs as exercises. There are also related exercises in the accompanying textbooks. At the end of each month (September, October, and November), you will turn in 5 exercises, which will be graded for completion. An assigment will be considered complete if most of the exercises are completed correctly (i.e., you have correctly conveyed the main idea, although there may be some minor errors in the details). For any exercises that you were unable to complete, you should describe what you tried, why it didn't work, what you tried next, and what you are still confused about.

Alternatively, instead of turning in 5 exercises, you may give a talk in the Wednesday student geometry-topology seminar.

Resources

We will start by roughly following Saveliev's Lectures on the topology of 3-manifolds. The following texts may be helpful as the semester progresses:

D. Rolfsen, Knots and links
R. Gompf and A. Stipsicz, 4-manifolds and Kirby calculus
R. Lickorish, An introduction to knot theory


Class Notes

Week 1: Heegaard splittings, lens spaces
Week 2: Seifert manifolds, Heegaard diagrams, links
Week 3: surgery
Week 4: Kirby moves
Week 5: Even surgeries, 4-manifolds, and intersection forms
Week 6: More intersection forms, handle decompositions
Week 7: Kirby diagrams
Week 8: Handleslides
Week 9: Handle cancellation
Week 10: 1-handles revisited
Week 11: No class, read the Kirby Problem List
Week 12: Homology cobordism, Dehn surgery, Seifert forms
Week 13: Alexander module, fibered knots, knot group
Week 14: Wirtinger presentation, genus is additive
Week 15: Knot concordance, ribbon knots, Heegaard Floer homology
Week 16: Knot Floer homology, applications to knot concordance and homology cobordism
Notes thanks to Sierra Knavel. All errors are my own.

Additional Information

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 accommodations, please make an appointment with the ADAPTS office to discuss the appropriate procedures.