# 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 theory.

**After this course.**

- The course
*Higher categories*(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 happen on *Mondays 10:45–12:45* and *Fridays 15:45–17:45*.
They will begin on January 11th and end on February 19th.

For now, lectures will be online. We will meet via Zoom. You will receive the link to the meeting by email; if you did not receive it, please contact me. The sessions will also be recorded and available on YouTube. To access the different videos, click on the playlist icon in the top right of the embedded player.

**The exam will happen on Friday, March 5th, 14:00–17:00, in room 207C of the Halle aux Farines (
note the room change).**
This will be an open-book exam: you will have access to printed and handwritten lecture notes.
Electronic devices will be forbidden.

##### Lecture 1

###### Monday January 11^{th}, 10:45–12:45

Analogies 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)

Blackboard Video##### Lecture 2

###### Friday January 15^{th}, 15:45–17:45

Definition of model categories and first examples (Section 1.3)

Blackboard Video##### Lecture 3

###### Monday January 18^{th}, 10:45–12:45

Definition and construction of the localization of a category with respect to a class of weak equivalence, definition of left homotopies, first properties, dual case of right homotopies (Sections 1.4.1 and 1.4.2).

Blackboard Video##### Lecture 4

###### Friday January 22^{nd}, 15:45–17:45

End of Section 1.4: explicit description of the homotopy category of a model category, Whitehead theorem.

Blackboard Video##### Lecture 5

###### Monday January 25^{th}, 10:45–12:45

Section 1.5.1: The proof of the existence of the projective model structure on the category of bounded-below chain complexes.

Blackboard Video##### Lecture 6

###### Friday January 29^{th}, 15:45–17:45

Existence theorem for cofibration generated model categories (Section 1.5.2), Quillen adjunctions (beginning of Section 1.6).

Blackboard Video##### Lecture 7

###### Monday February 1^{st}, 10:45–12:45

Definition of Quillen equivalences and characterization, homotopy limits and colimits.

Blackboard Video##### Lecture 8

###### Friday February 5^{th}, 15:45–17:45

Reminders on simplicial sets and beginning of the proof of the existence of the Quillen model structure on the category of simplicial sets (Sections 2.1, 2.2, 2.3, and beginning of 2.4).

Blackboard Video##### Lecture 9

###### Monday February 8^{th}, 10:45–12:45

Continuation of the proof of the existence of the Quillen model structure on simplicial sets, anodynes extensions, simplicial homotopy groups.

Blackboard Video##### Lecture 10

###### Friday February 12^{th}, 15:45–17:45

End of the proof of the Quillen equivalence between simplicial sets and topological spaces. Left Bousfield localization and rational homotopy theory.

Blackboard Video##### Lecture 11

###### Monday February 15^{th}, 10:45–12:45

Transferred model structure on CDGAs and Sullivan algebras.

Blackboard Video##### Lecture 12

###### Friday February 19^{th}, 15:45–17:45

Equivalence between CDGAs and rational homotopy types through the PL forms. Applications of Sullivan's theory: models of spaces, dichotomy theorem...

Blackboard Video## Bibliography

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:

**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-Verlag, 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 Verlag, 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.**2*Rational homotopy theory and differential forms*.^{nd}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]**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 \(\infty\)-categories.]*Higher topos theory*.

Reminders on algebraic topology and homological algebra:

**Glen E. Bredon.**Graduate Texts in Mathematics 139. New York: Springer-Verlag, 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-Verlag, 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*.