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