# Formality of the Little \(N\)-disks Operad

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*Pascal Lambrechts; Ismar Volić*

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

# Table of Contents

## Formality of the Little $N$-disks Operad

- Acknowledgments vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. Notation, linear orders, weak partitions, and operads 918
- Chapter 3. CDGA models for operads 1322
- Chapter 4. Real homotopy theory of semi-algebraic sets 1928
- Chapter 5. The Fulton-MacPherson operad 2332
- 5.1. Compactification of configuration spaces in ℝ^{ℕ} 2433
- 5.2. The operad structure 2635
- 5.3. The canonical projections 2837
- 5.4. Decomposition of the boundary of 𝐶[𝑛] into codimension 0 faces 3039
- 5.5. Spaces of singular configurations 3241
- 5.6. Pullback of a canonical projection along an operad structure map 3443
- 5.7. Decomposition of the fiberwise boundary along a canonical projection 4352
- 5.8. Orientation of 𝐶[𝐴] 4453
- 5.9. Proof of the local triviality of the canonical projections 4655

- Chapter 6. The CDGAs of admissible diagrams 6372
- Chapter 7. Cooperad structure on the spaces of (admissible) diagrams 7382
- Chapter 8. Equivalence of the cooperads 𝒟 and ℋ*(𝒞[∙]) 8796
- Chapter 9. The Kontsevich configuration space integrals 91100
- Chapter 10. Proofs of the formality theorems 107116
- Index of notation 111120 free
- Bibliography 115124