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)
Layout
- Homotopy theory of topological spaces
- Simplicial homotopy theory
- Rational homotopy theory
Worksheet
- Worksheet 1
NEW VERSION UPLOADED ON FRIDAY NOV. 9 (11PM)
The priority for the correction of these exercises should be: 1, 4, 3, 5, 2. So you are advised to look at them in this order.
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 8.30-11.30am (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 8.30-10am.
Prerequisistes
From Christian Ausoni's course: category, functor, adjunction, (co)limits, topological space, homeomorphism.
Professor
Bruno Vallette (lectures/exercise sessions)