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Higher structures in rational homotopy theory

Alexander Berglund, Robin Stoll

TL;DR

These notes present a concise, operad-centered introduction to rational homotopy theory, showing that the rational type of simply connected spaces of finite type is encoded by higher algebraic structures such as $C_ty$-algebras on cohomology and $L_ty$-algebras on shifted homotopy groups. They develop the unified Koszul framework for $ty$-algebras, relate formality and coformality to Koszulness, and illustrate the computational power of Koszul duality with classical and geometric examples. The third lecture extends these ideas to automorphisms of high-dimensional manifolds, using Kontsevich graph complexes and modular operads to model classifying spaces and to connect rational homotopy to graph-homology. Throughout, the text emphasizes minimal models, homotopy transfer, and $ty$-morphisms as practical tools for translating between topological problems and algebraic computations. The resulting picture unifies Sullivan–Quillen formalisms with modern operadic methods, enabling explicit calculations of rational invariants and deepening the connection between topology and higher algebra.

Abstract

These notes are based on a series of three lectures given (online) by the first named author at the workshop "Higher Structures and Operadic Calculus" at CRM Barcelona in June 2021. The aim is to give a concise introduction to rational homotopy theory through the lens of higher structures. The rational homotopy type of a simply connected space of finite type is modeled by a $C_\infty$-algebra structure on the rational cohomology groups, or alternatively an $L_\infty$-algebra structure on the rational homotopy groups. The first lecture is devoted to explaining these models and their relation to the classical models of Quillen and Sullivan. The second lecture discusses the relation between Koszul algebras, formality and coformality. The main result is that a space is formal if and only if the rational homotopy $L_\infty$-algebra is Koszul and, dually, a space is coformal if and only if the cohomology $C_\infty$-algebra is Koszul. For spaces that are both formal and coformal, this collapses to classical Koszul duality between Lie and commutative algebras. In the third lecture, we discuss certain higher structure in the rational homotopy theory of automorphisms of high dimensional manifolds, discovered by Berglund and Madsen. The higher structure in question is Kontsevich's Lie graph complex and variants of it.

Higher structures in rational homotopy theory

TL;DR

These notes present a concise, operad-centered introduction to rational homotopy theory, showing that the rational type of simply connected spaces of finite type is encoded by higher algebraic structures such as -algebras on cohomology and -algebras on shifted homotopy groups. They develop the unified Koszul framework for -algebras, relate formality and coformality to Koszulness, and illustrate the computational power of Koszul duality with classical and geometric examples. The third lecture extends these ideas to automorphisms of high-dimensional manifolds, using Kontsevich graph complexes and modular operads to model classifying spaces and to connect rational homotopy to graph-homology. Throughout, the text emphasizes minimal models, homotopy transfer, and -morphisms as practical tools for translating between topological problems and algebraic computations. The resulting picture unifies Sullivan–Quillen formalisms with modern operadic methods, enabling explicit calculations of rational invariants and deepening the connection between topology and higher algebra.

Abstract

These notes are based on a series of three lectures given (online) by the first named author at the workshop "Higher Structures and Operadic Calculus" at CRM Barcelona in June 2021. The aim is to give a concise introduction to rational homotopy theory through the lens of higher structures. The rational homotopy type of a simply connected space of finite type is modeled by a -algebra structure on the rational cohomology groups, or alternatively an -algebra structure on the rational homotopy groups. The first lecture is devoted to explaining these models and their relation to the classical models of Quillen and Sullivan. The second lecture discusses the relation between Koszul algebras, formality and coformality. The main result is that a space is formal if and only if the rational homotopy -algebra is Koszul and, dually, a space is coformal if and only if the cohomology -algebra is Koszul. For spaces that are both formal and coformal, this collapses to classical Koszul duality between Lie and commutative algebras. In the third lecture, we discuss certain higher structure in the rational homotopy theory of automorphisms of high dimensional manifolds, discovered by Berglund and Madsen. The higher structure in question is Kontsevich's Lie graph complex and variants of it.
Paper Structure (17 sections, 27 theorems, 129 equations, 1 figure)

This paper contains 17 sections, 27 theorems, 129 equations, 1 figure.

Table of Contents

  1. Lecture 1: L∞- and C∞-models for spaces
  2. A∞-algebras
  3. C∞-algebras
  4. L∞-algebras
  5. Unified theory of ∞-algebras via algebraic operads
  6. ∞-morphisms
  7. Homotopy transfer theorem
  8. Minimal P∞-algebra model
  9. Sullivan's rational homotopy theory
  10. Quillen's rational homotopy theory
  11. Nothing new under the sun: ∞-algebras vs. minimal models
  12. Lecture 2: Koszul algebras, formality, and coformality
  13. Formality and coformality
  14. Koszul algebras and simultaneous formality and coformality
  15. Koszul C∞- and L∞-algebras
  16. ...and 2 more sections

Key Result

Theorem 1.2

For each simply connected space $X$ of finite ${\mathbb{Q}}$-type, there exists a $C_\infty$-algebra structure on the rational cohomology groups ${\operatorname{H}^{*}_{}}(X;{\mathbb{Q}})$ that provides a complete invariant of the rational homotopy type, in the sense that $X$ and $Y$ are rationally

Figures (1)

  • Figure 1: Leibniz's harmonic triangle

Theorems & Definitions (64)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Proposition 1.5
  • proof : Proof sketch
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Proposition 1.9
  • ...and 54 more