Formality of the little Ndisks operad
 Author/Creator
 Lambrechts, Pascal, 1964 author.
 Language
 English.
 Publication
 Providence, Rhode Island : American Mathematical Society, [2014]
 Copyright notice
 ©2013
 Physical description
 vii, 116 pages : illustrations ; 26 cm.
 Series
 Memoirs of the American Mathematical Society ; no. 1079.
Access
Available online

Stacks

Unknown
QA3 .A57 NO.1079

Unknown
QA3 .A57 NO.1079
More options
Contributors
 Contributor
 Volić, Ismar, 1973 author.
Contents/Summary
 Bibliography
 Includes bibliographical references and index.
 Contents

 Notation, linear orders, weak partitions, and operads
 CDGA models for operads
 Real homotopy theory of semialgebraic sets
 The FultonMacPherson operad
 The CDGAs of admissible diagrams
 Cooperad structure on the spaces of (admissible) diagrams
 Equivalence of the cooperads D and H(C)
 The Kontsevich configuration space integrals
 Proofs of the formality theorems.
 Summary
 "The little Ndisks operad, B, along with its variants, is an important tool in homotopy theory. It is defined in terms of configurations of disjoint Ndimensional disks inside the standard unit disk in R [superscript]N and it was initially conceived for detecting and understanding Nfold 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 Ndisks operad over the field of real numbers. More precisely, one can consider the singular chains C [subscript]*(B; R) on B as well as the singular homology H [subscript]*(B; R) of 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. We additionally prove a relative version of the formality for the inclusion of the little mdisks operad in the little Ndisks operad when N ≥ 2m + 1. The formality of the little Ndisks 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 R [superscript]N, N ≥ 4"Page v.
Subjects
Bibliographic information
 Publication date
 2014
 Copyright date
 2013
 Responsibility
 Pascal Lambrechts, Ismar Volić.
 Series
 Memoirs of the American Mathematical Society, 00659266 ; number 1079
 Note
 "Volume 230, number 1079 (first of 5 numbers), July 2014."
 ISBN
 9780821892121
 0821892126