Lectures on Homotopy Theory
The links below are to pdf files, which comprise the lecture notes for
a course on Homotopy Theory (Math 9151), last given at the
University of Western Ontario and UBC (teleconference) during the Winter term of 2016.
I expect to give this course again in the Fall of 2017, as Math 9151A, subject to student interest.
The 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. The indicated dates are
the most recent file updates.
More detail on topics covered here can be found in the
GoerssJardine 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 teleconference to students from other universities. Students from other sites can participate, from either traditional teleconference 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
(Math 4152/9052) and/or Homological Algebra (Math 9144) prior to taking
this course.
 Rick Jardine

Office: Middlesex College 118

Phone: 5196612111 x86512

Email: jardine@uwo.ca
Don't be misled by the link titles: each file is
worth at least two hours of class time.
Comments are welcome.
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 and a 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 for 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 groups
 Section 15: Simplicial modules
 Section 16: EilenbergMac 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 colimits and limits (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 sequences for a bicomplex
 Section 26: The EilenbergZilber Theorem
 Section 27: Universal coefficients, Kunneth formula

Lecture 10: Serre spectral sequence, pathloop fibre sequence
 Section 28: The fundamental groupoid, revisited
 Section 29: The Serre spectral sequence
 Section 30: The transgression
 Section 31: The pathloop fibre sequence

Lecture 11: Postnikov towers, some applications
 Section 32: Postnikov towers
 Section 33: The 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
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: Suspensions and shift
 Section 42: The 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