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Formality of the little $N$-disks operad

About this Title

Pascal Lambrechts and Ismar Volić

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 230, Number 1079
ISBNs: 978-0-8218-9212-1 (print); 978-1-4704-1669-0 (online)
Published electronically: November 14, 2013
Keywords:Operad formality, little cubes operad, Fulton-MacPherson operad, trees

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Table of Contents


  • Acknowledgments
  • Chapter 1. Introduction
  • Chapter 2. Notation, linear orders, weak partitions, and operads
  • Chapter 3. CDGA models for operads
  • Chapter 4. Real homotopy theory of semi-algebraic sets
  • Chapter 5. The Fulton-MacPherson operad
  • Chapter 6. The CDGAs of admissible diagrams
  • Chapter 7. Cooperad structure on the spaces of (admissible) diagrams
  • Chapter 8. Equivalence of the cooperads $\mathcal {D}$ and $\mathrm {H}^*(\mathrm {C}[\bullet ])$
  • Chapter 9. The Kontsevich configuration space integrals
  • Chapter 10. Proofs of the formality theorems
  • Index of notation


The little -disks operad, , along with its variants, is an important tool in homotopy theory. It is defined in terms of configurations of disjoint -dimensional disks inside the standard unit disk in and it was initially conceived for detecting and understanding -fold loop spaces. Its many uses now stretch across a variety of disciplines including topology, algebra, and mathematical physics. In this paper, we develop the details of Kontsevich's proof of the formality of little -disks operad over the field of real numbers. More precisely, one can consider the singular chains on as well as the singular homology of . These two objects are operads in the category of chain complexes. The formality then states that there is a zig-zag of quasi-isomorphisms connecting these two operads. The formality also in some sense holds in the category of commutative differential graded algebras. We additionally prove a relative version of the formality for the inclusion of the little -disks operad in the little -disks operad when . The formality of the little -disks operad has already had many important applications. For example, it was used in a solution of the Deligne Conjecture, in Tamarkin's proof of Kontsevich's deformation quantization conjecture, and in the work of Arone, Lambrechts, Turchin, and Volić on determining the rational homotopy type of spaces of smooth embeddings of a manifold in a large euclidean space, such as the space of knots in , .

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