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

Description

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. Foreword Preface Acknowledgments Contents 1 Overview of the Volume 1.1 Configuration Spaces of Manifolds 1.2 Configuration Spaces of Closed Manifolds 1.3 Configuration Spaces of Manifolds with Boundary 1.4 Configuration Spaces and Operads 1.4.1 Conventions 2 Configuration Spaces of Manifolds 2.1 Configuration Spaces 2.2 Homotopy Invariance 2.3 Rational Homotopy Invariance 2.4 Configuration Spaces of Euclidean Spaces 3 Configuration Spaces of Closed Manifolds 3.1 The Lambrechts–Stanley Model 3.1.1 Definition of the Model 3.1.2 Statement of the Theorem and Proof Strategy 3.2 Fulton–MacPherson Compactifications 3.2.1 Case of Euclidean Spaces 3.2.2 Case of Closed Manifolds 3.3 Semi-algebraic Sets and PA Forms 3.3.1 Semi-algebraic Sets 3.3.2 Piecewise Semi-algebraic Forms 3.4 Graph Complexes 3.4.1 Informal Idea 3.4.2 Definition of the Unreduced Graph Complex 3.4.2.1 A Formal Definition Through Operadic Twisting 3.4.3 An Issue with the Homotopy Type 3.4.4 Propagator 3.4.5 Partition Function as a Maurer–Cartan Element 3.4.6 Simplification of the Partition Function 3.5 End of the Proof 4 Configuration Spaces of Manifolds with Boundary 4.1 Motivation 4.2 Poincaré–Lefschetz Duality Models 4.3 Graphical Models 4.3.1 Compactifications 4.3.2 Propagators 4.3.3 Graph Complexes 4.3.4 Simplification of the Partition Functions 4.3.4.1 Partition Function of the Cylinder on the Boundary 4.3.4.2 Partition Function of the Whole Manifold 4.3.5 Quasi-Isomorphism 4.4 Perturbed Lambrechts–Stanley Model 4.4.1 Computation of the Homology 4.4.2 Perturbed Model 5 Configuration Spaces and Operads 5.1 Motivation: Factorization Homology 5.1.1 Historical Remarks 5.2 Introduction to Operads 5.2.1 Definition of Operads 5.2.2 Algebras over an Operad 5.2.3 Modules over Operads 5.3 Configuration Spaces and Operads 5.3.1 Little Disks Operads 5.3.2 Relationship with Configuration Spaces 5.3.3 Operadic Structures on Compactifications 5.4 Models for Configuration Spaces and Their Operadic Structure 5.4.1 Formality of the Little Disks Operads 5.4.2 Operadic Module Structures on Models for Configuration Spaces 5.4.3 Swiss-Cheese 5.5 Example of Calculation References Index