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OverviewThe ultimate goal of this book is to explain that the Grothendieck-Teichmuller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads. The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck-Teichmuller group in the case of the little 2-disc operad. This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory. Full Product DetailsAuthor: Benoit FressePublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 1.460kg ISBN: 9781470434823ISBN 10: 1470434822 Pages: 715 Publication Date: 30 June 2017 Audience: Professional and scholarly , Professional & Vocational Format: Hardback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsHomotopy theory and its applications to operads. General methods of homotopy theory: Model categories and homotopy theory Mapping spaces and simplicial model categories Simplicial structures and mapping spaces in general model categories Cofibrantly generated model categories Modules, algebras, and the rational homotopy of spaces: Differential graded modules, simplicial modules, and cosimplicial modules Differential graded algebras, simplicial algebras, and cosimplicial algebras Models for the rational homotopy of spaces The (rational) homotopy of operads: The model category of operads in simplicial sets The homotopy theory of (Hopf) cooperads Models for the rational homotopy of (non-unitary) operads The homotopy theory of (Hopf) $\Lambda$-cooperads Models for the rational homotopy of unitary operads Applications of the rational homotopy to $E_n$-operads: Complete Lie algebras and rational models of classifying spaces Formality and rational models of $E_n$-operads The computation of homotopy automorphism spaces of operads: Introduction to the results of the computations for the $E_n$-operads The applications of homotopy spectral sequences: Homotopy spsectral sequences and mapping spaces of operads Applications of the cotriple cohomology of operads Applications of the Koszul duality of operads The case of $E_n$-operads: The applications of the Koszul duality for $E_n$-operads The interpretation of the result of the spectral sequence in the case of $E_2$-operads Conclusion: A survey of further research on operadic mapping spaces and their applications: Graph complexes and $E_n$-operads From $E_n$-operads to embedding spaces Appendices: Cofree cooperads and the bar duality of operads Glossary of notation Bibliography IndexReviewsAuthor InformationBenoit Fresse, Universite de Lille 1, Villeneuve d'Ascq, France. Tab Content 6Author Website:Countries AvailableAll regions |
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