Configurations spaces, algebraic topology and operads (2022−2023)

Abstract

I plan to explain how two classical objects in algebraic topology are related: configuration spaces, and operads. Configuration spaces, on the one hand, consist in collections of pairwise distinct points in a given manifold. The study of these spaces is a classical problem in algebraic topology. Operads, on the other hand, are algebraic objects whose representations are categories of algebras (e.g., associative algebras, commutative algebras, or Lie algebras). The intimate relationship between these two objects comes in the form of the little disks operads, a certain family of operads which is central in many applications and which involves the configuration spaces of the Euclidean spaces. I will show how ideas that come from the theory of operads – namely, the proof of the formality of the little disks operads – is useful in order to settle the question of the rational homotopy invariance of configuration spaces.

Plan of the lectures

A more detailed plan is available here.

1. Configuration spaces
   1. Definition and applications
   2. Homotopy invariance
   3. Rational homotopy theory
   4. Formality of $\mathrm{Conf}_{\mathbb{R}^n}(r)$
2. Lambrechts-Stanley model
   1. Model definition
   2. Fulton-MacPherson compactification
   3. Semi-algebraic sets and PA forms
   4. Graph complexes
   5. End of proof
3. Operads
   1. Motivation: Factorization homology
   2. Introduction to operads
   3. Operadic structure of $\mathrm{FM}_n$ and $\mathrm{FM}_M$
   4. Connection with algebraic models

There will also be tutorials held by Victor Roca i Lucio and Pedro Tamaroff. The exercise sheets are available here.

References

• Benoit Fresse. Homotopy of Operads and Grothendieck–Teichmüller Groups. Volume 1: “The Algebraic Theory and its Topological Background.” Mathematical Surveys and Monographs 217, AMS (2017). ISBN: 978-1-4704-3481-6. DOI:10.1090/surv/217.1
  Note: Parts Ia and Ib will be relevant for this course.
• Najib Idrissi. Real Homotopy of Configuration Spaces: Peccot Lecture, Collège de France, March and May 2020. Lecture Notes in Mathematics 2303. Springer (2023). ISBN: 978-3-031-04427-4. DOI:10.1007/978-3-031-04428-1.
  Note: a preprint of the book is available at https://hal.science/hal-03821309.
• Ben Knudsen. Configuration spaces in algebraic topology. arXiv:1803.11165.