# Homotopy II (2021–2022)

## Introduction

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* (by Emmanuel Wagner) and *Homotopy I* (by 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

- Model categories.
- Chain complexes.
- Simplicial sets and topological spaces.
- Rational homotopy theory.

### After this course

- The course
*Higher categories*(by Muriel Livernet) is a natural continuation of this course and a conclusion to the “homotopy” sequence of the master’s degree (Homology – Homotopy I/II – Higher categories). - I gave in 2020 a Peccot lecture entitled
*Real homotopy of configuration spaces*at the Collège de France which gave concrete applications of the methods presented here.

## Organization

Lectures will start the week of January 10^{th} and end on the week of February 18^{th}.
They will be located in the Grands Moulins campus.
The time slots will be:

- Wednesdays from 11:15 to 13:15 in room 136 of the Olympe de Gouges building.
- Fridays from 16:15 to 18:15 in room 2016 of the Sophie Germain building.

^{th}and January 14

^{th}) will be held online on Zoom as I was tested positive for COVID on January 10

^{th}. The Zoom link will be made available to you by email and the lecture will be broadcast live in the room. On Wednesday 12, the room will be changed to room 0013 of the Sophie Germain building for technical reasons. You will have received a Zoom link by email. On Friday 14, the room will be room 2016 of the SG building as usual and the lecture will also be broadcast simultaneously.

My own office is in the Sophie Germain building. I can be contacted at najib.idrissi-kaitouni@imj-prg.fr.

## Lectures

### Lecture 1

1/12/2022 11:15-13:15Analogies between different homotopy theories: topological spaces, simplicial sets, chain complexes. Motivation for the definition of model categories. (Co)fibrations vs. injections/surjections. Definition of model categories. (Sections 1.1 and 1.2)

### Lecture 2

1/14/2022 16:15-18:15Definition of model categories and first examples (Section 1.3).

### Lecture 3

1/19/2022 11:15-13:15### Lecture 4

1/21/2022 16:15-18:15### Lecture 5

1/24/2022 11:15-13:15### Lecture 6

1/28/2022 16:15-18:15### Lecture 7

2/2/2022 11:15-13:15### Lecture 8

2/4/2022 16:15-18:15### Lecture 9

2/9/2022 11:15-13:15### Lecture 10

2/11/2022 16:15-18:15### Lecture 11

2/16/2022 11:15-13:15### Lecture 12

2/18/2022 16:15-18:15## Bibliography

I have written lecture notes which are available here. They were written two years ago and the content of the lecture has changed since then; in particular, the fourth chapter will not be covered in my lecture this year.

I gave a similar course last year, and also the year before that. You can in particular find past exams on these pages. 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:

**William G. Dwyer et Jan Spaliński. “Homotopy theories and model categories”.**In:*Handbook of algebraic topology*. Amsterdam: North-Holland, 1995, pp. 73–126. DOI:10.1016/B978-044481779-2/50003-1. MR1361887. Zbl:0869.55018. (Introduction to model categories.)**Yves Félix, Stephen Halperin et Jean-Claude Thomas.**Graduate Texts in Mathematics 205. New York : Springer, 2001, p. xxxiv+535. ISBN: 0-387-95068-0. DOI: 10.1007/978-1-4613-0105-9. (Reference book on rational homotopy theory.)*Rational Homotopy Theory*.**Paul G. Goerss et John F. Jardine.**Progress in Mathematics 174. Basel: Birkhäuser , 1999, pp. xvi+510. ISBN: 3-7643-6064-X. DOI:10.1007/978-3-0348-8707-6. MR1711612. Zbl:0949.55001. (Book on simplicial sets and their homotopical properties.)*Simplicial homotopy theory*.**Phillip Griffiths et John Morgan.**2nd ed. Progress in Mathematics 16. New York: Springer, 2013. 224 p. ISBN: 978-1-4614-8467-7. DOI: 10.1007/978-1-4614-8468-4. (Notes on rational homotopy theory)*Rational homotopy theory and differential forms*.**Kathryn Hess. “Rational homotopy theory: a brief introduction”.**In:*Interactions between homotopy theory and algebra*. Contemp. Math. 436. Providence, RI: Amer. Math. Soc., 2007, pp. 175–202. DOI:10.1090/conm/436/08409. arXiv:math/0604626. MR2355774. Zbl:1128.55010. (Introduction to rational homotopy theory)**Mark Hovey.**Mathematical Surveys and Monographs 63. Providence, RI: American Mathematical Society, 1999, pp. xii+209. ISBN: 0-8218-1359-5. MR1650134. Zbl:0909.55001. (Book on model categories.)*Model categories*.**Jacob Lurie.**Annals of Mathematics Studies 170. Princeton, NJ: Princeton University Press, 2009, pp. xviii+925. ISBN: 978-0-691-14049-0. MR2522659. Zbl:1175.18001 (Very complete book on ∞-categories.)*Higher topos theory*.

Reminders on algebraic topology and homological algebra:

**Glen E. Bredon.**Graduate Texts in Mathematics 139. New York: Springer, 1993, pp. xiv+557. ISBN: 0-387-97926-3. DOI:10.1007/978-1-4757-6848-0. MR1224675. Zbl:0791.55001. ((Algebraic) topology and (differential) geometry textbook.)*Topology and geometry*.**Allen Hatcher.**Cambridge: Cambridge University Press, 2002, pp. xii+544. ISBN: 0-521-79160-X. MR1867354. Zbl:1044.55001 (Algebraic topology textbook.)*Algebraic topology*.**Henri Paul de Saint-Gervais.**(Lecture notes on algebraic topology, in French.)*Analysis Situs*.**Pierre Schapira.**(Lecture notes on derived categories.)*Categories and homological algebra*.**Edwin Spanier.**Berlin: Springer, 1995, pp. xiv+528. ISBN: 978-1-4684-9322-1. DOI:10.1007/978-1-4684-9322-1. MR210112. Zbl:0145.43303. (Algebraic topology textbook.)*Algebraic topology*.**Charles A. Weibel.**Cambridge Studies in Advanced Mathematics 38. Cambridge: Cambridge University Press, 1994, pp. xiv+450. ISBN: 0-521-43500-5. MR1269324. Zbl:0797.18001. (Homological algebra textbook.)*An Introduction to homological algebra*.