Low-dimensional manifolds

Spring 2021 Tutorial

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

Course information

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

Wellness Days

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

Final paper topic suggestions

- 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.

References and Resources

Low-dimensional topology

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"

Differential topology

Milnor - Topology from the Differentiable Viewpoint
Guillemin, Pollack - Differential Topology
Kosinski - Differential Manifolds
Hirsch - Differential Topology
Milnor - Morse Theory

Algebraic topology

Hatcher - Algebraic Topology
Milnor, Stasheff - Characteristic Classes
Bott, Tu - Differential Forms in Algebraic Topology
Kupers - "Lectures on diffeomorphism groups of manifolds"

Databases and programs

Livingston, Moore - KnotInfo, LinkInfo
Bar-Natan, Morrison, et al. - The Knot Atlas
Culler, Dunfield, Goerner, Weeks - SnapPy