Cours fondamental 1 (TA)

Homotopy 1

Christian Ausoni (Travaux dirigés par Gabriel Angelini-Knoll)

Contact : ausoni à univ-paris13.fr

Notes de cours : https://www.math.univ-paris13.fr/~ausoni/m2-2022.html

Présentation

This lecture course serves as an introduction to the homotopy theory of topological spaces and simplicial sets.
We will introduce homotopy groups, compute them in some examples, and study their properties and the relationship to singular homology theory.
We will then define generalized (co)-homology theories and conclude with the example of stable homotopy.
This lecture course also provides prerequisites and motivating examples for the sequel lecture course "Homotopie 2" by Geoffroy Horel.

Contenu

Homotopy theory of topological spaces
Basic simplicial homotopy theory
(Stable) homotopy groups
Generalized (co-)homology theories

Prérequis

Basic point-set topology, as well as basic notions of category theory, homological algebra and singular homology (as covered in the cours introductif "Théorie de l'homologie" by Baptise Rognerud).

Bibliographie

Tammo tom Dieck. Algebraic Topology. EMS Textbooks in Mathematics, 2008 https://www.ems-ph.org/books/book.php?proj_nr=86
Peter May. A concise course in Algebraic Topology. Chicago Lectures in Mathematics, 1999 https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf
Allen Hatcher. Algebraic Topology. Cambridge University Press, 2001 http://pi.math.cornell.edu/~hatcher/AT/ATpage.html
Edward B. Curtis. Simplicial homotopy theory. Advances in Mathematics 6, 107-209, 197 https://www.sciencedirect.com/science/article/pii/0001870871900156