Meetings: MF 9:00-10:15am
Zoom Meeting ID: 952 8269 3329
Instructor: Joshua Wang
Email: jxwang@math.harvard.edu
Office Hours: W 9:00-10:15am and by appointment
The goal of this tutorial is to introduce the subject of low-dimensional topology and to illustrate the basic machinery of algebraic and differential topology. We will begin the tutorial with a warm-up: the classification of 1- and 2-dimensional manifolds. Next, we'll explore 3-manifolds and the knots and surfaces they contain. We'll meet 2-bridge knots, torus knots, the Hopf fibration, lens spaces, and the Poincaré homology sphere, among many others. A beautiful and important aspect of this theory is that the pictures we draw accurately reflect the mathematical objects of study. Finally, we will turn to 4-manifolds and their rich interplay with manifolds of lower dimension. Here we'll meet slice knots, plane curves, Brieskorn spheres, the K3 surface, and potentially exotic 4-spheres. The course will focus on careful constructions rather than sophisticated invariants.
First courses in algebraic topology and differential topology are required.
Exercises are given in class. Homework consists of completing all but one exercise from each class. Extra credit will be given for completing all exercises. The exercises for any given week are due on Wednesday at midnight the following week. Feel free to collaborate. Think about each exercise before going to the texts or online references. If you use external references, cite them.
75% homework, 25% final paper
If Monday or Friday is a Wellness Day, there will be no class. If Tuesday or Wednesday is a Wellness Day, homework is due Thursday at midnight.
January 25: introduction
January 29: gluing and cutting, Morse theory basics
February 1: attaching handles, classification of 1-manifolds
February 5: no class (Wellness Day)
February 8: isotopy extension theorem
February 12: the disc theorem of Palais
February 15: no class (Presidents' Day)
February 19: classification of 2-manifolds
February 22: smooth Jordan-Schoenflies theorem
February 26: simple closed curves on the torus
March 1: no class (Wellness Day)
March 5: Dehn twists, mapping class group of a torus
March 8: the 3-sphere, the Hopf fibration, lens spaces
March 12: Heegaard splittings, Dehn filling
March 15: 3-manifolds of Heegaard genus 1, Dehn surgery
March 19: Wirtinger presentation, knot groups
March 22: Seifert surfaces
March 26: cyclic branched covers
March 29: bridge position and the double branched cover
April 2: links of singularities, Brieskorn spheres
April 5: linking number, modification of surgery instructions
April 9: Poincaré homology sphere, surgery diagrams for cyclic branched covers
April 12: 4-manifolds and the intersection form
April 16: homology cobordism and Kirby calculus
April 19: slice genus, concordance
April 23: Thom conjecture, Milnor conjecture, spun 2-knots
April 26: Gluck twists, homotopy 4-spheres
- generating the mapping class group by Dehn twists
- additivity of genus and unique prime factorization of knots
- rational tangles
- links of singularities
- the Poincaré homology sphere
- Thurston norm
- Lickorish-Wallace theorem
- Thom conjecture implies the Milnor conjecture
- Mazur manifolds
May 5: final paper due at midnight.
Rolfsen - Knots and Links
Saveliev - Lectures on the Topology of 3-Manifolds
Gompf, Stipsicz - 4-Manifolds and Kirby Calculus
Kirby, Scharlemann - "Eight faces of the Poincaré homology 3-sphere"
Milnor - Topology from the Differentiable Viewpoint
Guillemin, Pollack - Differential Topology
Kosinski - Differential Manifolds
Hirsch - Differential Topology
Milnor - Morse Theory
Hatcher - Algebraic Topology
Milnor, Stasheff - Characteristic Classes
Bott, Tu - Differential Forms in Algebraic Topology
Kupers - "Lectures on diffeomorphism groups of manifolds"
Livingston, Moore - KnotInfo, LinkInfo
Bar-Natan, Morrison, et al. - The Knot Atlas
Culler, Dunfield, Goerner, Weeks - SnapPy