Differential Topology

*This is a meta page for the ongoing study in differential topology, to access the resources for this subject (files, lectures, and homework) click here.*

## Introduction to the subject

In mathematics, **differential topology** is the study of differentiable (smooth) manifolds (topological spaces which locally resemble

R

n

{\displaystyle \R^n}

 on which one may do calculus) and properties of which are invariant under *diffeomorphism* (a smooth bijection with a smooth inverse).

For a more detailed overview of the subject, click here.

## Syllabus

### Course Overview

This course in differential topology covers topics as one would encounter in a graduate course in differential topology and/or geometry, except that these topics are explored with a more topological flavor.

Course topics include smooth manifolds, smooth maps, the tangent bundle, submanifolds, Sard's theorem, Lie theory, integral curves and flows, vector bundles, Riemannian manifolds, integration on manifolds, de Rham cohomology, and symplectic manifolds.

### Format and Scope

Lectures will be offered on an on-demand basis. Math discussion will be facilitated on the private Stack Exchange (for formal, on-topic discussion) and on the Element server (for informal on-topic discussion). Collaborative notes and on-topic information sharing (examples, explanations for concepts, etc.) will be facilitated here on the Fandom.

The scope of our study in differential topology with respect to the contents of the book is partitioned into four main phases:

#### Phase I - Foundational differential topology

The first phase is entirely dedicated to discussing the essential concepts in differential topology, of which there are many (it is a large subject). Topics of discussion include

* Smooth manifolds
* Smooth maps
* Partitions of unity
* Tangent vectors
* The differential
* The tangent bundle
* Submersions, immersions, and embeddings
* Smooth covering maps
* Submanifolds, embedded and immersed
* Sard's theorem

#### Phase II - Lie theory and vectors

The second phase focuses on Lie theory and vector fields on manifolds.

* Lie groups
* Vector fields on manifolds
* Lie algebras and the Lie derivative
* Integral curves and flows
* Partial differential equations
* Vector bundles
* Fiber bundles
* The cotangent bundle
* Tensors
* Riemannian metrics
* Riemannian manifolds

#### Phase III - Theory of differential forms

The third phase of our study explores the fundamental consequences of differential forms on smooth manifolds. Topics include differential forms, the exterior derivative, orientation, integration and Stokes' theorem, and de Rham cohomology.

#### Phase IV - Additional topics

The fourth and final phase of our study explores additional topics in differential topology. Topics include the de Rham theorem and symplectic manifolds.

### Textbook

We will be using the textbook *Introduction to Smooth Manifolds* by John M. Lee (Amazon).

### Coursework

Those who take part in studying differential topology are highly encouraged to solve problems. Homework will be "assigned" each week, along with a reading "assignment". At the end of each chapter, a short, 75 minute chapter quiz will be assigned to be completed over the weekend. Every three chapters, a long-form test will be assigned, which may require up to 2 hours of work.

### Prerequisites and Difficulty

Competency with multivariable calculus, linear algebra, and point-set topology are ***strongly*** recommended. Results about the fundamental group, covering spaces, and ordinary differential equations may be reviewed in the appendix as needed.