eBook ISBN:  9781470416690 
Product Code:  MEMO/230/1079.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 
eBook ISBN:  9781470416690 
Product Code:  MEMO/230/1079.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 230; 2013; 116 ppMSC: Primary 55; Secondary 18
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 zigzag of quasiisomorphisms 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

Chapters

Acknowledgments

1. Introduction

2. Notation, linear orders, weak partitions, and operads

3. CDGA models for operads

4. Real homotopy theory of semialgebraic sets

5. The FultonMacPherson operad

6. The CDGAs of admissible diagrams

7. Cooperad structure on the spaces of (admissible) diagrams

8. Equivalence of the cooperads $\mathcal {D}$ and $\mathrm {H}^*(\mathrm {C}[\bullet ])$

9. The Kontsevich configuration space integrals

10. Proofs of the formality theorems

Index of notation


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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 zigzag of quasiisomorphisms 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 \(N\geq2m+1\).

Chapters

Acknowledgments

1. Introduction

2. Notation, linear orders, weak partitions, and operads

3. CDGA models for operads

4. Real homotopy theory of semialgebraic sets

5. The FultonMacPherson operad

6. The CDGAs of admissible diagrams

7. Cooperad structure on the spaces of (admissible) diagrams

8. Equivalence of the cooperads $\mathcal {D}$ and $\mathrm {H}^*(\mathrm {C}[\bullet ])$

9. The Kontsevich configuration space integrals

10. Proofs of the formality theorems

Index of notation