**Mathematical Surveys and Monographs**

Volume: 217;
2017;
1236 pp;
Hardcover

MSC: Primary 55; 18; 57; 20;
**Print ISBN: 978-1-4704-3480-9
Product Code: SURV/217**

List Price: $250.00

AMS Member Price: $200.00

MAA Member Price: $225.00

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#### Supplemental Materials

# Homotopy of Operads and Grothendieck–Teichmüller Groups: Parts 1 and 2

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*Benoit Fresse*

The Grothendieck–Teichmüller group was defined by Drinfeld
in quantum group theory with insights coming from the Grothendieck
program in Galois theory. The ultimate goal of this book set is to explain
that this group has a topological interpretation as a group of
homotopy automorphisms associated to the operad of little 2-discs,
which is an object used to model commutative homotopy structures in
topology.

The first part of this two-part set gives a comprehensive survey on
the algebraic aspects of this subject. The book explains the
definition of an operad in a general context, reviews the definition
of the little discs operads, and explains the definition of the
Grothendieck–Teichmüller group from the viewpoint of the theory of
operads. In the course of this study, the relationship between the
little discs operads and the definition of universal operations
associated to braided monoidal category structures is explained. Also
provided is a comprehensive and self-contained survey of the
applications of Hopf algebras to the definition of a rationalization
process, the Malcev completion, for groups and groupoids.

Most definitions are carefully reviewed in the book; it requires
minimal prerequisites to be accessible to a broad readership of
graduate students and researchers interested in the applications of
operads.

The ultimate goal of the second part of the book is to explain that
the Grothendieck–Teichmüller 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–Teichmüller 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.

#### Readership

Graduate students and researchers interested in algebraic topology and algebraic geometry.

#### Reviews & Endorsements

Even if the main goal of the first volume is to provide the reader with the necessary prerequisites to understand the deep theory developed by the author in the second volume, it, nevertheless, provides the literature with an interesting treatise, accessible to graduate students and to researchers working in any field. It should first be mentioned that the mathematical content covered here is absolutely beautiful...In the end, the present first volume of this treatise on the homotopy of operads and Grothendieck-Teichmüller groups represents a huge amount of work and is a valuable addition to the current mathematial literature.

-- Bruno Vallette, Mathematical Reviews

The scope of this book is vast, with a highly interesting result as its culmination. Yet, the book will be valuable for researchers beyond simply those wanting to understand the main theorem. It is a good reference for many topics in homotopy theory and operad theory.

-- Julia Bergner, Mathematical Reviews