## Lectures on Homotopy Theory

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

The course material is widely applicable in Topology, Geometry, Number Theory, Mathematical Physics, and some forms of data analysis.

This collection of files is the basic source material for the course, and this page is an outline of the course contents. In practice, some of this is elective - I usually don't get much beyond proving the Hurewicz Theorem in classroom lectures.

Despite the titles, each of the files covers much more material than one can usually present in a single lecture.

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

J.F. Jardine

E-mail: jardine@uwo.ca

## 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: 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 colimits and limits (revisited)
• Section 23: Some applications, Quillen's Theorem B
• Lecture 09: Bisimplicial abelian groups

• Section 24: Derived functors
• Section 25: Spectral sequences for a bicomplex
• Section 26: The Eilenberg-Zilber Theorem
• Section 27: Universal coefficients, Kunneth formula
• Lecture 10: Serre spectral sequence

• Section 28: The fundamental groupoid, revisited
• Section 29: The Serre spectral sequence
• Section 30: The transgression
• Section 31: The path-loop 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