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## WARNING: from Wednesday November 21, the course will begin at 9am and finish at noon on Wednesdays and Thursdays.

**Abstract**

The goal of this lecture will be to present various
“concrete” homotopy theories. We will start with the classical homotopy
theory of topological spaces (higher homotopy groups, cellular
complexes, Whitehead and Hurewicz theorems, Eilenberg—MacLane spaces,
fibrations, and Postnikov towers). Then we will move to the homotopy
theory of simplicial sets (definitions, simplex category, adjunction
and cosimplicial objects, examples, fibrations, Kan complexes, and
simplicial homotopy). Finally, we will study the rational homotopy
theory via the homotopy theory of differential graded Lie or
commutative (co)algebras (Sullivan approach : minimal model, Quillen
approach : Whitehead Lie bracket, bar and cobar constructions, complete
Lie algebra-Hopf algebras-groups).

This course will directly follow the one of Chrisitan Ausoni on

Homology Theory (September-October 2018); it will open the doors to the one of Gregory Ginot on

Abstract Homotopy Theory (January-February, 2018) and to the one of Yonatan Harpaz on

Higher Algebra (March-April 2018)

**Lecture Notes**

The notes of the course are typed by Johan Leray: thank you very much!

(Version of 17/11/18: I still have to read them again).

**Layout****
**

- Homotopy theory of topological spaces

- Simplicial homotopy theory

- Rational homotopy theory

**Worksheet **

- Worksheet 1

- Worksheet 2 FINAL VERSION (UPLOADED ON NOV. 17)

**References **

- Algebraic Topology, Tammo tom Dieck, EMS Textbooks in Mathematics, 2008.
- A concise course in Algebraic Topology, Peter May, Chicago ectures in Mathematics, 1999.

- Algebraic Topology, Allen Hatcher, Cambridge University Press, 2001.

**Organisation **

The lectures will take place every Wednesday

9am-12am (room
2013, Sophie Germain building) and every Thursday 9-12am (room 2016,
Sophie Germain building) from November 7 to December 13 2018. Exercise
sessions will be organised every Wednesday

9am-10.30am.

**Prerequisistes**

From Christian Ausoni's course: category, functor, adjunction, (co)limits, topological space, homeomorphism.

**Professor**

Bruno Vallette (lectures/exercise sessions)