# Lectures on Homotopy Theory

### J.F. Jardine

The links below are to pdf files, which comprise the lecture notes for a course on Homotopy Theory.

This collection of files is the basic source material for the course, and the syllabus is listed on this page. All files are subject to revision as the course progresses.

More detail on topics covered here can be found in the Goerss-Jardine book *Simplicial Homotopy Theory*, which appears in the References file below.

The course will be given as a set of lectures at the University of Western Ontario, and will be available by video conference to students from other universities. Students from other sites can participate, from either traditional video conference rooms or by using personal computers. Please contact me if you wish to do so.

It would be quite helpful for a student to have a background in basic Algebraic Topology (our course Math 4152/9052) and/or Homological Algebra (our Math 9144) prior to taking this course.

Rick Jardine

Office: Middlesex College 118

Phone: 519-661-2111 x86512

E-mail: jardine@uwo.ca

This course will meet Monday, Wednesday and Friday, 12:30-13:30 EST, MC105C and online.

The first lecture will be on Monday, January 15.

**Course Outline for Western Students**: This is the OWL site for the course. Access it with your Western id. The syllabus is the same as below, but there is extra administrative information on the OWL site for Western students.

The main reference for the course is the Goerss-Jardine book "Simplicial Homotopy Theory". If your institution has the right kind of SpringerLink subscription (as does Western), you can download a pdf file for the book free of charge from the SpringerLink site, for example **at this link**. A (printed on demand) paper copy of the book is also available at subscribing institutions for a low fee.

**Recordings of the lectures** are available at** this link.**

The quality of the recordings is uneven. In particular, there is no sound for the first lecture, due to a hardware failure.

## Syllabus

## Part A: Homotopy theories

Lecture 01: **Homological algebra**

Lecture 02: **Spaces**

Lecture 03: **Homotopical algebra**

Lecture 04: **Simplicial sets**

Lecture 05: **Fibrations, geometric realization**

Lecture 06: **Simplicial groups, simplicial modules**

Lecture 07: **Properness, diagrams of spaces**

Lecture 08: **Bisimplicial sets, homotopy limits and colimits**

Section 21: Bisimplicial sets

Section 22: Homotopy limits and colimits (revisited)

Section 23: Some applications, Quillen's Theorem B

Lecture 09: **Bisimplicial abelian groups, derived functors**

Section 24: Bisimplicial abelian groups, derived functors

Section 25: Spectral sequence for a bicomplex

Section 26: Eilenberg-Zilber Theorem

Section 27: Universal coefficients, Künneth formula

Lecture 10: **Serre spectral sequence, path-loop fibre sequence**

Section 28: Fundamental groupoid, revisited

Section 29: Serre spectral sequence

Section 30: The transgression

Section 31: Path-loop fibre sequence

Lecture 11: **Postnikov towers, some applications**

**Cohomology: an introduction**

Section 35: Cohomology

Section 36: Cup products

Section 37: Cohomology of cyclic groups

## Part B: Stable homotopy theory - first steps

Lecture 13: **Spectra and stable equivalence**

Section 38: Spectra

Section 39: Strict model structure

Section 40: Stable equivalences

Lecture 14: **Basic properties**

Section 41: Suspension and shift

Section 42: Telescope construction

Section 43: Fibrations and cofibrations

Section 44: Cofibrant generation

Lecture 15: **Spectrum objects**

Section 45: Spectra in simplicial modules

Section 46: Chain complexes