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Formality of the Little \(N\)-disks Operad
Pascal Lambrechts, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, and Ismar Volić, Wellesley College, Massachusetts
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Memoirs of the American Mathematical Society
2013; 116 pp; softcover
Volume: 230
ISBN-10: 0-8218-9212-6
ISBN-13: 978-0-8218-9212-1
List Price: US$75
Individual Members: US$45
Institutional Members: US$60
Order Code: MEMO/230/1079
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The little \(N\)-disks operad, \(\mathcal B\), along with its variants, is an important tool in homotopy theory. It is defined in terms of configurations of disjoint \(N\)-dimensional disks inside the standard unit disk in \(\mathbb{R}^N\) and it was initially conceived for detecting and understanding \(N\)-fold loop spaces. Its many uses now stretch across a variety of disciplines including topology, algebra, and mathematical physics.

In this paper, the authors develop the details of Kontsevich's proof of the formality of little \(N\)-disks operad over the field of real numbers. More precisely, one can consider the singular chains \(\operatorname{C}_*(\mathcal B; \mathbb{R})\) on \(\mathcal B\) as well as the singular homology \(\operatorname{H}_*(\mathcal B; \mathbb{R})\) of \(\mathcal B\). 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. The authors additionally prove a relative version of the formality for the inclusion of the little \(m\)-disks operad in the little \(N\)-disks operad when \(N\geq2m+1\).

Table of Contents

  • Introduction
  • Notation, linear orders, weak partitions, and operads
  • CDGA models for operads
  • Real homotopy theory of semi-algebraic sets
  • The Fulton-MacPherson operad
  • The CDGAs of admissible diagrams
  • Cooperad structure on the spaces of (admissible) diagrams
  • Equivalence of the cooperads \(\mathcal{D}\) and \(\mathrm {H}^*(\mathrm{C}[\bullet])\)
  • The Kontsevich configuration space integrals
  • Proofs of the formality theorems
  • Index of notation
  • Bibliography
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