OPERADS GROTHENDIECK TEICHMULLER GROUPS

Benoit Fresse

Description

?x3 x4x1 x2 x3 x4x1 x2 OPERADS & GROTHENDIECK-TEICHMULLER GROUPS – DRAFT DOCUMENT by Benoit Fresse

• discs


• main topics


• foreword ?


• grothendieck-teichmuller group


• mathematics related


• ?x3 x4x1


• operad theory

Sujets

• Fresse
• Northwestern University
• Operad theory

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 14
• Langue: English
• Poids de l'ouvrage: 7 Mo

Extrait

¨OPERADS & GROTHENDIECK-TEICHMULLER GROUPS
–
DRAFT DOCUMENT
by
Benoit Fresse
x x x x1 2 3 4
x x x x1 2 3 4
∼OPERADS & GROTHENDIECK-TEICHMULLER GROUPS
{
DRAFT DOCUMENT
BENOIT FRESSE
Abstract
This preprint is an extract from a research monograph in preparation on the
homotopy of operads and Grothendieck-Teichmuller groups. The ultimate objective
of this book is to prove that the Grothendieck-Teichmuller group is the group of
homotopy automorphisms of a rational completion of the little 2-discs operad.
The present excerpts include a comprehensive account of the fundamental
concepts of operad theory, a survey chapter on little discs operads as well as a detailed
account on the connections between little 2-discs, braid groups, and
GrothendieckTeichmuller groups, until the formulation of the main result of the monograph.
Most concepts are carefully reviewed in order to make this account accessible to
a broad readership, which should include graduate students as well as researchers
coming from the various elds of mathematics related to our main topics. This
preprint will serve as reference material for a master degree course \Operads 2012",
given by the author at universite Lille 1, from January until April 2012. See:
http://math.univ-lille1.fr/~operads/2012courses.html#Lille
This working draft will not be updated, and the given excerpts should signi -
cantly di er from the nal version of the monograph in preparation. Nevertheless,
a copy with annotated corrections will be made available on the above web-page.
This work has mostly been written during stays at the Ecole Normale Superieure
de Paris, at Northwestern University, and at the Max-Planck-Institut fur
Mathematik in Bonn. The author is grateful to these institutions for outstanding working
conditions, and to numerous colleagues for their warm welcome which has greatly
eased this writing task.
Preprint contents
* Foreword
Overall introduction, pp. v{vii.
Mathematical objectives, pp. ix{xvi.
General conventions, pp. xvii{xxvi.
Synopsis, pp. xxvii{xxxiii.
* Part 0. Background
Chapter I. Operads and algebras over operads, pp. 3{43.
Date: Bonn, 4 January 2012.
1991 Mathematics Subject Classi cation. Primary: 55P48; Secondary: 18G55, 55P10, 55P62,
57T05, 20B27, 20F36.
Research supported in part by ANR grant \OBTH" and ANR grant \HOGT".
12 BENOIT FRESSE
Chapter II. Operads in symmetric monoidal categories, pp. 45{65.
* Part 1. Models of E -operads and Grothendieck-Teichmuller groupsn
Chapter 1. Introduction to E -operad, pp. 105{125.n
2. Braids and the recognition of E -operads, pp. 127{161.2
Chapter 3. Malcev completion of E -operads and Grothendieck-Teichmuller2
groups, pp. 163{193.
* BibliographyForewordOverall Introduction
The rst aim of this book is to give an overall reference, starting from scratch,
on the application of ne algebraic topology methods to operads. Most de nitions,
notably fundamental concepts of operad and homotopy theory, are carefully
reviewed in order to make our account accessible to a broad readership, including
graduate students, as well as researchers coming from the various elds of
mathematics related to our core subject.
Ultimately, our objective is to explain, from a homotopical viewpoint, a deep
relationship between operads and Grothendieck-Teichmuller groups. This
connection, which has arisen from researches on the deformation quantization process in
mathematical physics, gives a new approach to understand internal symmetries of
structures occurring in various constructions of algebra and topology.
The de nition of an operad is reviewed in the rst part of the book. For
the moment, simply recall that an operad is a structure, formed by collections of
abstract operations, which is used to de ne a category of algebras. In our study,
we mainly consider the example of E -operads, n = 1; 2;:::;1, used to model an
hierarchy of homotopy commutative structures, starting with E , fully homotopy1
associative but not commutative, and ending with E , fully homotopy associative1
and commutative. The intermediate E -operads represent structures, which aren
more and more homotopy commutative when n increases, but not fully homotopy
commutative until n =1. For the reader, we should mention that the notion of
anE -operad is synonymous to that of an A -operad, used in the literature when1 1
one only deals with purely homotopy associative structures.
The notion of E -operad formally refers to a class of operads, rather than ton
a singled out object. This class consists, in the initial de nition, of topological
operads which are homotopically equivalent to a reference model, the
BoardmanVogt operad of little n-discs D . The operad of little n-cubes, which is a simplen
variant of the little n-discs operad, is also used in the literature to provide an
equivalent de nition of the class of E -operads. The second part of the book isn
devoted to detailed recollections on these notions. Nonetheless, as we explain soon,
the ultimate objective of the book is not to study E -operads themselves, butn
homotopy automorphisms groups attached to these structures.
Before explaining this goal, we survey some motivating applications of E -n
operads, which are not our main matter (we only give short introductions to these
topics in the book), but illustrate our approach of the subject.
The operads of little n-discs D were initially introduced to collect operationsn
acting on iterated loop spaces. The rst main application, which has motivated the
de nition of these operads, was the Boardman-Vogt and May recognition theorems
asserting, in the most basic outcome, that any connected space equipped with an
vvi OVERALL INTRODUCTION
naction of D is homotopy equivalent to an n-fold loop space
X (see [16, 17]n
and [72]).
nRecall that the set of connected components of an n-fold loop space
X is
identi ed with the nth homotopy group (X) of the space X, which is abeliann
nfor n> 1. The action of D on
X includes, for any n> 0, a product operationn
n n n :
X
X!
X which, at the level of connected components, gives the
composition operation of the group (X). The operad D carries the homotopyn n
making this product associative, as well as commutative for n > 1, and includes
further operations, representing ne homotopy constraints, which we need to form
na faithful picture of the structure of the n-fold loop space
X.
Since the initial topological de nition, new applications of E -operads haven
been discovered in the elds of algebra and mathematical physics, mostly after
the proof of the Deligne conjecture asserting that the Hochschild cochain
complex C (A;A) of an associative algebra A inherits an action of an E -operad. In2
this context, we use a chain version of the previously considered topological little
2-discs operad D .2
The cohomology of the Hochschild cochain complexC (A;A) is identi ed in
degree 0 with the centerZ(A) of the associative algebraA. In a sense, the Hochschild
cochain complex represents a derived version of this ordinary center Z(A). From
this point of view, the construction of an E -structure on C (A;A) determines,2
again, a ne level of homotopical commutativity of the derived center, beyond an
apparent commutativity at the cohomology level. The rst proofs of the Deligne
conjecture have been given by Kontsevich-Soibelman [58] and McClure-Smith [73]. The
interpretation in terms of derived centers has been advertised by Kontsevich [56] in
order to formulate a natural extension of the conjecture in the context of algebras
associated to E -operads, for any n 1.n
The veri cation of the Deligne conjecture has yielded a second generation of
proofs, promoted by Tamarkin [89] and Kontsevich [56], of the Kontsevich formality
theorem giving the existence of deformation quantizations. The new approach of
this problem also involves Drinfeld’s theory of associators, which are used to
transport the E -structure yielded by the Deligne conjecture on the Hochschild cochain2
complex to the cohomology. In the nal outcome, one obtains that each associator
gives rise to a deformation quantization functor. This result has hinted the
existence of a deep connection between the deformation quantization problem and the
program, initiated in Grothendieck’s famous \esquisse" [49], aiming to understand
Galois groups through geometric actions on curves. The Grothendieck-Teichmuller
groups are devices, introduced in this program, encoding the information which can
be captured through the actions considered by Grothendieck. The correspondence
between associators and deformation quantizations imply that a rational
prounipo1
tent version of the Grothendieck-Teichmuller group GT (Q) acts on the moduli
space of deformation quantizations. The initial motivation for our work was the
desire to understand this connection from a homotopical viewpoint, in terms of
homotopical structures associated to E -operads. The homotopy automorphisms2
of operads come in at this point.
Recall again that an operad is a structure encoding a category of algebras. The
homotopy automorphisms of an operad P are transformations, de ned at the operad
level, encoding natural homotopy equivalences on the category of algebras