Formality of the little N-disks operad / Pascal Lambrechts, Ismar Volić.

Format:

Book

Author/Creator:

Lambrechts, Pascal, 1964- author.

Volić, Ismar, 1973- author.

Series:

Memoirs of the American Mathematical Society ; Volume 230, Number 1079.

Memoirs of the American Mathematical Society, 1947-6221 ; Volume 230, Number 1079

Language:

English

Subjects (All):

Homotopy theory.

Operads.

Loop spaces.

Physical Description:

1 online resource (130 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2013.

Language Note:

English

Summary:

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 $\mathbb{R} DEGREESN$ 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\ge

Contents:

""Contents""; ""Acknowledgments""; ""Chapter 1. Introduction""; ""1. Plan of the paper""; ""Chapter 2. Notation, linear orders, weak partitions, and operads""; ""2.1. Notation""; ""2.2. Linear orders""; ""2.3. Weak ordered partitions""; ""2.4. Operads and cooperads""; ""Chapter 3. CDGA models for operads""; ""Chapter 4. Real homotopy theory of semi-algebraic sets""; ""Chapter 5. The Fulton-MacPherson operad""; ""5.1. Compactification of configuration spaces in â??^{â??}""; ""5.2. The operad structure""; ""5.3. The canonical projections""

""5.4. Decomposition of the boundary of [ ] into codimension 0 faces""""5.5. Spaces of singular configurations""; ""5.6. Pullback of a canonical projection along an operad structure map""; ""5.7. Decomposition of the fiberwise boundary along a canonical projection""; ""5.8. Orientation of [ ]""; ""5.9. Proof of the local triviality of the canonical projections""; ""Chapter 6. The CDGAs of admissible diagrams""; ""6.1. Diagrams""; ""6.2. The module ( ) of diagrams""; ""6.3. Product of diagrams""; ""6.4. A differential on the space of diagrams""

""6.5. The CDGA ( ) of admissible diagrams""""Chapter 7. Cooperad structure on the spaces of (admissible) diagrams""; ""7.1. Construction of the cooperad structure maps Î?_{ } and Î?_{ }""; ""7.2. Î?_{ } and Î?_{ } are morphisms of algebras""; ""7.3. Î?_{ } is a chain map""; ""7.4. Proof that the cooperad structure is well-defined""; ""Chapter 8. Equivalence of the cooperads and â??*( [â??])""; ""Chapter 9. The Kontsevich configuration space integrals""; ""9.1. Construction of the Kontsevich configuration space integral ""; ""9.2. is a morphism of algebras""

""9.3. Vanishing of on non-admissible diagrams""""9.4. and are chain maps""; ""9.5. and are almost morphisms of cooperads""; ""Chapter 10. Proofs of the formality theorems""; ""Index of notation""; ""Bibliography""

Notes:

"Volume 230, Number 1079 (first of 5 numbers)."

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-1669-7