Lectures on Homotopy Theory


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




Part A: Homotopy theories


Lecture 01: Homological algebra

Section 1: Chain complexes
Section 2: Ordinary chain complexes
Section 3: Closed model categories  


 Lecture 02: Spaces

Section 4: Spaces and homotopy groups
Section 5: Serre fibrations, model structure for spaces 


Lecture 03: Homotopical algebra

Section 6: Example: chain homotopy
Section 7: Homotopical algebra
Section 8: The homotopy category 


Lecture 04: Simplicial sets

Section 9: Simplicial sets
Section 10: The simplex category and realization
Section 11: Model structure of simplicial sets 


Lecture 05: Fibrations, geometric realization

Section 12: Kan fibrations
Section 13: Simplicial sets and spaces 


Lecture 06: Simplicial groups, simplicial modules

Section 14: Simplicial group
Section 15: Simplicial modules
Section 16: Eilenberg-Mac Lane spaces

Lecture 07: Properness, diagrams of spaces

Section 17: Proper model structures
Section 18: Homotopy cartesian diagrams
Section 19: Diagrams of spaces
Section 20: Homotopy limits and colimits

 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

Section 32: Postnikov towers
Section 33: Hurewicz Theorem
Section 34: Freudenthal Suspension Theorem
Lecture 12: 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