Homotopy II (2020–2021)

M2 Fundamental Mathematics (S2) • Lectures • 24h

The goal of this course is to introduce modern homotopy theory, its tools and applications. We will be particularly interested in two examples: chain complexes (see the previous Homology course) and topological spaces. We will present Quillen’s model categories, and we will explain the equivalence between topological spaces and simplicial sets. We will illustrate these methods with rational homotopy theory by showing that multiplicative structures of cochains encodes rational homotopy types of topological spaces.

Prerequisites. It is recommended to have taken the courses Homology (Emmanuel Wagner) and Homotopy I (Bruno Vallette). It will be useful to have a certain familiarity with categorical language and with basic notions of algebraic topology and homological algebra.

Plan of the course.

  1. Model categories
  2. Chain complexes
  3. Simplicial sets and topological spaces
  4. Rational homotopy

After this course.

Bibliography

Lecture notes are here.

I gave a similar course last year, you can in particular find past exams. Grégory Ginot gave a course in 2017–2018–2019 on the same subject. You can find on his page his lecture notes, as well as past exercise sheets and exams.

Works on homotopy theory:

Reminders on algebraic topology and homological algebra:

(The password is homotopie.)