*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