Homotopy of Operads and Grothendieck-Teichmuller Groups

Part 2: the Applications of (Rational) Homotopy Theory Methods

AvBenoit Fresse

Inbunden, Engelska, 2017

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Beskrivning

The 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.

Produktinformation

Utgivningsdatum:2017-06-30
Mått:178 x 254 x undefined mm
Vikt:1 460 g
Format:Inbunden
Språk:Engelska
Serie:Mathematical Surveys and Monographs
Antal sidor:715
Förlag:American Mathematical Society
ISBN:9781470434823

Utforska kategorier

Matematik inom Naturvetenskap och teknik
Algebra inom Naturvetenskap och teknik

Mer om författaren

Benoit Fresse, Universite de Lille 1, Villeneuve d'Ascq, France.

Innehållsförteckning

Homotopy theory and its applications to operads. General methods of homotopy theory: Model categories and homotopy theoryMapping spaces and simplicial model categoriesSimplicial structures and mapping spaces in general model categoriesCofibrantly generated model categoriesModules, algebras, and the rational homotopy of spaces: Differential graded modules, simplicial modules, and cosimplicial modulesDifferential graded algebras, simplicial algebras, and cosimplicial algebrasModels for the rational homotopy of spacesThe (rational) homotopy of operads: The model category of operads in simplicial setsThe homotopy theory of (Hopf) cooperadsModels for the rational homotopy of (non-unitary) operadsThe homotopy theory of (Hopf) $\Lambda$-cooperadsModels for the rational homotopy of unitary operadsApplications of the rational homotopy to $E_n$-operads: Complete Lie algebras and rational models of classifying spacesFormality and rational models of $E_n$-operadsThe computation of homotopy automorphism spaces of operads: Introduction to the results of the computations for the $E_n$-operadsThe applications of homotopy spectral sequences: Homotopy spsectral sequences and mapping spaces of operadsApplications of the cotriple cohomology of operadsApplications of the Koszul duality of operadsThe case of $E_n$-operads: The applications of the Koszul duality for $E_n$-operadsThe interpretation of the result of the spectral sequence in the case of $E_2$-operadsConclusion: A survey of further research on operadic mapping spaces and their applications: Graph complexes and $E_n$-operadsFrom $E_n$-operads to embedding spacesAppendices: Cofree cooperads and the bar duality of operadsGlossary of notationBibliographyIndex