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Rational Homotopy Theory

Alexander A. Voronov

TL;DR

This survey articulates how rational homotopy theory (RHT) recasts topological spaces into algebraic data via Sullivan minimal models and Quillen’s DGLA/L∞-models, enabling complete rational invariants and a bridge to physics. It emphasizes model-categorical foundations, Quillen–Sullivan dualities, and the role of A_PL in connecting spaces to DGCAs, highlighting how minimal models encode rational homotopy groups and Whitehead products. The text also explains how L∞-algebras provide a modern, streamlined framework for Quillen-type models and their nerves, with the Maurer–Cartan construction linking to rational spaces. In physics, Hypothesis H and cyclic loop space constructions reveal deep ties between Sullivan models of S^4 and the equations of 11D supergravity, including dimensional reductions and the emergence of E_k root data. Overall, the article presents a cohesive picture of how algebraic models capture rational homotopy types and inform contemporary physical theories.

Abstract

This is a survey of Rational Homotopy Theory, intended for a Mathematical Physics readership.

Rational Homotopy Theory

TL;DR

This survey articulates how rational homotopy theory (RHT) recasts topological spaces into algebraic data via Sullivan minimal models and Quillen’s DGLA/L∞-models, enabling complete rational invariants and a bridge to physics. It emphasizes model-categorical foundations, Quillen–Sullivan dualities, and the role of A_PL in connecting spaces to DGCAs, highlighting how minimal models encode rational homotopy groups and Whitehead products. The text also explains how L∞-algebras provide a modern, streamlined framework for Quillen-type models and their nerves, with the Maurer–Cartan construction linking to rational spaces. In physics, Hypothesis H and cyclic loop space constructions reveal deep ties between Sullivan models of S^4 and the equations of 11D supergravity, including dimensional reductions and the emergence of E_k root data. Overall, the article presents a cohesive picture of how algebraic models capture rational homotopy types and inform contemporary physical theories.

Abstract

This is a survey of Rational Homotopy Theory, intended for a Mathematical Physics readership.
Paper Structure (21 sections, 11 theorems, 55 equations)

This paper contains 21 sections, 11 theorems, 55 equations.

Table of Contents

  1. Rational homotopy category of spaces
  2. Differential graded commutative algebras
  3. Minimal Sullivan models of DGCAs
  4. Sullivan minimal models of spaces
  5. Real algebraic models of manifolds
  6. Rational algebraic models of spaces
  7. Realizations
  8. Rationalization
  9. Model categories
  10. The fundamental theorem of RHT
  11. Quillen minimal models of spaces
  12. Quillen's DGLA model $\lambda X$ of a space $X$
  13. $L_\infty$-Algebras
  14. Minimal $L_\infty$-models
  15. The $L_\infty$ version of the fundamental theorem of RHT
  16. ...and 6 more sections

Key Result

Theorem 1

Theorems & Definitions (19)

  • Example 1.1
  • Example 1.2
  • Theorem 1: RHT
  • Theorem 2: RHT
  • Example 4.1: The real homotopy type of an odd-dimensional sphere
  • Example 4.2: The real and rational homotopy type of an even-dimensional sphere
  • Example 4.3: Realization of $g_3$ for $S^2$ as a Chern-Simons form
  • Theorem 3: Hovey, May.Ponto
  • Theorem 4: Bousfield.Gugenheim and Holstein
  • Theorem 5: Bousfield.Gugenheim and Holstein
  • ...and 9 more