Book: Homotopy of Operads & Grothendieck-Teichmüller Groups

By Benoit Fresse


The first aim of this book is to give an overall reference, starting from scratch, on the applications of methods of algebraic topology to operads. To be more specific, one of the main objectives is the development of a rational homotopy theory for operads. Most definitions, notably fundamental concepts of operad and homotopy theory, are carefully reviewed in order to make this book accessible to a broad readership of graduate students and researchers interested in the applications of operads.
The second purpose of the book is to explain, from a homotopical viewpoint, a deep relationship between operads and Grothendieck-Teichmüller groups. This connection, which has been foreseen by M. Kontsevich (from researches on the deformation quantization process in mathematical physics), gives a new approach to understanding internal symmetries of structures occurring in various constructions of algebra and topology. In the book, the background required by an in-depth study of this subject is set up, and the interpretation of the Grothendieck-Teichmüller group in terms of the homotopy of operads is made precise. The book is actually organized for this ultimate objective, which readers can take either as a main motivation or as a leading example to learn about general theories.


Reading guide and overview of the content of the book

This monograph comprises three main parts which form a progression up to our ultimate mathematical goal. Part I “From operads to Grothendieck-Teichmüller groups” is mainly devoted to the algebraic foundations of our subject. In Part II “Homotopy theory and its applications to operads”, we develop our rational homotopy theory of operads after a comprehensive review of the applications of methods of homotopy theory. In Part III “The computation of homotopy automorphism spaces of operads”, we work out our problem of giving a homotopy interpretation of the Grothendieck-Teichmüller group. Three appendices A-B-C give detailed surveys on the applications of trees in operad theory, on the definition of free objects and of free resolutions in the category of operads, and on the Koszul duality of operads.
These parts are widely independent from each others. Each part of this book is also divided into subparts which, by themselves, form self-contained groupings of chapters, devoted to specific topics, and organized according to an internal progression of the level of the chapters each. There is a progression in the level of the parts of the book too, but the chapters are written so that a reader with a minimal background could tackle any of these subparts straight away in order to get a self-contained reference and an overview of the literature on each of the subjects addressed in this monograph.

Contents of the first volume, published as "Homotopy of Operads and Grothendieck–Teichmüller Groups: Part 1: The Algebraic Theory and its Topological Background"

Contents of the second volume, published as "Homotopy of Operads and Grothendieck–Teichmüller Groups: Part 2: The Applications of (Rational) Homotopy Theory Methods"

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