Homotopy Theory

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

## Introduction to the subject

(From Wikipedia) In mathematics, **homotopy theory** is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry and category theory.

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

## Syllabus

### Course Overview

This course in homotopy theory covers the same topics as one would encounter in an undergraduate or graduate course in algebraic topology, except that these topics are explored at a far greater depth. In this case, there is a particular emphasis on the study of homotopy theory as opposed to homology or cohomology, though the latter topics are still discussed at length. In this course, concepts are introduced almost exclusively in the language of category theory.

Topics include obligatory category theory, CW complexes, homotopy groups, fibrations and cofibrations, Mayer-Vietoris sequences and the Fubini theorems, the Seifert-Van Kampen theorem, suspension, pushouts and pullbacks, cohomology, homology, chain complexes, and much more.

### Format and Scope

Lectures will be offered on the Discord every Saturday at 4pm EST. 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 homotopy theory with respect to the contents of the book is partitioned into five main phases:

#### Phase I - Category theory

The first phase, which is perhaps the shortest phase in our study, is entirely dedicated to fulfilling the essential prerequisite of category theory. The topics for this phase include discussion of important tools in category theory:

* Categories
* Functors
* Duality
* Groups and cogroups
* Limits and colimits
* Pushout and pullback squares
* Categories of topological spaces
* Pointed spaces

#### Phase II - Essential homotopy theory

The second phase focuses on building the foundation for our study in homotopy theory. This phase begins with discussion of the homotopy of maps, and ends with the famous Seifiert-Van Kampen theorem. Topics for this phase include:

* The homotopy category
* Fibrations and cofibrations
* Homotopy limits and colimits
* Homotopy pushout and pullback squares
* Long cofiber and fiber sequences
* Mayer-Vietoris sequences
* The Fubini theorems
* Group actions
* Homotopy type of joins and products
* Cone decompositions
* Phantom maps
* Model categories
* The J. H. C. Whitehead theorem

#### Phase III - Advanced homotopy theory

The third phase of our study explores the beauty of homotopy theory and its applications to active areas of research. Phase III begins with a deeper discussion on fibrations and ends with a buffet of special topics, including Hopf invariants and infinite symmetric products.

#### Phase IV - Algebraic topology

The fourth phase of our study discusses the foundational concepts in algebraic topology - cohomology and homology - but with a categorical twist. Phase IV discusses cohomology, homology, chain complexes, the Borsuk-Ulam theorem, the Wang sequence, filtered spaces, Serre filtrations, spectral sequences, and Bott periodicity.

#### Phase V - Additional Topics

The fifth and final (optional) phase of our study explores further topics in the theory, such as rational homotopy theory and Miller's theorem.

### Textbook

We will be using the textbook *Modern Classical Homotopy Theory* by Jeffrey Strom (Amazon).

### Coursework

Those who take part in studying homotopy theory 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. At the end of each part, a long-form test will be assigned, which may require up to 2 hours of work.

### Prerequisites and Difficulty

(From the Preface) This book was written with the idea that it would be used by students in their first year or two of graduate school. **It is assumed that the reader is familiar with basic algebraic concepts such as groups and rings. It is also assumed that the student has had an introductory course in topology**. It would be nice if that course included some mention of the fundamental group, but *that is not necessary*.