**Mathematical Surveys and Monographs**

Volume: 217;
2017;
704 pp;
Hardcover

MSC: Primary 55;
Secondary 18; 57; 20

Print ISBN: 978-1-4704-3482-3

Product Code: SURV/217.2

List Price: $135.00

Individual Member Price: $108.00

**Electronic ISBN: 978-1-4704-3757-2
Product Code: SURV/217.2.E**

List Price: $135.00

Individual Member Price: $108.00

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# Homotopy of Operads and Grothendieck–Teichmüller Groups: Part 2: The Applications of (Rational) Homotopy Theory Methods

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*Benoit Fresse*

The ultimate goal of this 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.

#### Table of Contents

# Table of Contents

## Homotopy of Operads and Grothendieck-Teichmuller Groups: Part 2: The Applications of (Rational) Homotopy Theory Methods

- Cover Cover11
- Title page iii4
- Contents v6
- Preliminaries ix10
- Part II . Homotopy Theory and its Applications to Operads 138
- Part II(a) . General Methods of Homotopy Theory 340
- Part II(b) . Modules, Algebras, and the Rational Homotopy of Spaces 125162
- Chapter 5. Differential Graded Modules, Simplicial Modules, and Cosimplicial Modules 127164
- 5.0. Background: dg-modules and simplicial modules 128165
- 5.1. The model category of cochain graded dg-modules 139176
- 5.2. Monoidal structures and the Eilenberg–Zilber equivalence 146183
- 5.3. Hom-objects on dg-modules and simplicial modules 152189
- 5.4. Appendix: Contracting chain-homotopies and extra-degeneracies 159196

- Chapter 6. Differential Graded Algebras, Simplicial Algebras, and Cosimplicial Algebras 163200
- Chapter 7. Models for the Rational Homotopy of Spaces 183220

- Part II(c) . The (Rational) Homotopy of Operads 211248
- Chapter 8. The Model Category of Operads in Simplicial Sets 213250
- 8.0. The category of operads in simplicial sets 215252
- 8.1. The model category of symmetric sequences 216253
- 8.2. The model category of non-unitary operads 226263
- 8.3. The model category of -sequences 240277
- 8.4. The model category of augmented non-unitary Λ-operads 255292
- 8.5. Simplicial structures and the cotriple resolution of operads 265302

- Chapter 9. The Homotopy Theory of (Hopf) Cooperads 273310
- Chapter 10. Models for the Rational Homotopy of (Non-unitary) Operads 319356
- Chapter 11. The Homotopy Theory of (Hopf) Λ-cooperads 333370
- Chapter 12. Models for the Rational Homotopy of Unitary Operads 367404

- Part II(d) . Applications of the Rational Homotopy to 𝐸_{𝑛}-operads 377414
- Chapter 13. Complete Lie Algebras and Rational Models of Classifying Spaces 379416
- Chapter 14. Formality and Rational Models of 𝐸_{𝑛}-operads 411448
- 14.0. Preliminaries on additive operads and additive cooperads 414451
- 14.1. The graded Drinfeld–Kohno Lie algebra operads and the applications of Chevalley–Eilenberg cochain complexes 418455
- 14.2. The chord diagram operad and the rational model of 𝐸₂-operads 439476
- 14.3. Appendix: Reminders on the Drinfeld–Kohno Lie algebra operad 446483

- Part III . The Computation of Homotopy Automorphism Spaces of Operads 449486
- Introduction to the Results of the Computations for 𝐸₂-operads 451488
- Part III(a) . The Applications of Homotopy Spectral Sequences 461498
- Part III(b) . The Case of 𝐸_{𝑛}-operads 541578
- Conclusion: A Survey of Further Research on Operadic Mapping Spaces and their Applications 589626
- Appendices 615652
- Back Cover Back Cover1743