# Real Homotopy of Configuration Spaces: Peccot Lecture, Collège de France, March & May 2020

## Abstract

- Provides an in-depth discussion of the connection between operads and configuration spaces
- Describes a unified and accessible approach to the use of graph complexes
- Based on 4 lectures held in the framework of the Peccot Lecture and Prize by the Collège de France.

## About

This volume provides a unified and accessible account of recent developments regarding the real homotopy type of configuration spaces of manifolds. Configuration spaces consist of collections of pairwise distinct points in a given manifold, the study of which is a classical topic in algebraic topology. One of this theory’s most important questions concerns homotopy invariance: if a manifold can be continuously deformed into another one, then can the configuration spaces of the first manifold be continuously deformed into the configuration spaces of the second? This conjecture remains open for simply connected closed manifolds. Here, it is proved in characteristic zero (i.e. restricted to algebrotopological invariants with real coefficients), using ideas from the theory of operads. A generalization to manifolds with boundary is then considered. Based on the work of Campos, Ducoulombier, Lambrechts, Willwacher, and the author, the book covers a vast array of topics, including rational homotopy theory, compactifications, PA forms, propagators, Kontsevich integrals, and graph complexes, and will be of interest to a wide audience.

## Keywords

- Configuration Spaces of Manifolds
- Operad Theory
- Homotopy Invariants of Manifolds
- Semi-algebraic Forms
- Rational Homotopy Theory
- Sullivan Models of Spaces
- Poincaré Duality Models
- Fulton-MacPherson Compactifications
- Factorization Homology
- Kontsevich Integrals
- Propagators in Deformation Quantization