\documentclass[11pt,notitlepage]{article} \def\bare{n} \usepackage[all]{xy} \usepackage[english,greek]{babel} \usepackage{amsmath, graphicx, amssymb, datetime, multicol, stmaryrd, pifont, amscd, colortbl, mathtools, wasysym, needspace, import, longtable, overpic, enumitem, bbm, pdfpages, ../picins, array, setspace} \usepackage{tensor} \usepackage{txfonts} % for the likes of \coloneqq. \usepackage[usenames,dvipsnames]{xcolor} \usepackage[landscape,textwidth=11in,textheight=8.5in,centering]{geometry} \parindent 0in \usepackage[makeroom]{cancel} % Following http://tex.stackexchange.com/a/847/22475: \usepackage[setpagesize=false]{hyperref} \hypersetup{colorlinks, linkcolor={blue!50!black}, citecolor={blue!50!black}, urlcolor={blue!50!black} } % Following http://tex.stackexchange.com/questions/59340/how-to-highlight-an-entire-paragraph \usepackage[framemethod=tikz]{mdframed} \usepackage[T1]{fontenc} \def\myurl{http://www.math.toronto.edu/~drorbn} \def\thistalk{BonnByWeb-1805} \def\title{Braids and the Grothendieck-Teichm\"uller Group} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\thanks{{Thanks for the opportunity to speak in Bonn!}} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/b18}{http://drorbn.net/b18/}}} \def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}} \def\titleA{{\title}} \def\titleB{{\title}} \def\titleC{{\title}} \definecolor{mgray}{HTML}{B0B0B0} \definecolor{morange}{HTML}{FFA50A} \def\blue{\color{blue}} \def\gray{\color{gray}} \def\mgray{\color{mgray}} \def\morange{\color{morange}} \def\red{\color{red}} \def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}} \def\cbox#1#2{{\setlength{\fboxsep}{0pt}\colorbox{#1}{#2}}} \def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}} \def\qed{{\linebreak[1]\null\hfill\text{$\Box$}}} \def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}} \def\ad{\operatorname{ad}} \def\Ad{\operatorname{Ad}} \def\aft{$\overrightarrow{\text{4T}}$} \def\AS{\mathit{AS}} \def\bbZZ{{\mathbb Z\mathbb Z}} \def\CW{\text{\it CW}} \def\diag{\operatorname{diag}} \def\FL{\text{\it FL}} \def\Hom{\operatorname{Hom}} \def\IHX{\mathit{IHX}} \def\mor{\operatorname{mor}} \def\PvT{{\mathit P\!v\!T}} \def\remove{\!\setminus\!} \def\STU{\mathit{STU}} \def\SW{\text{\it SW}} \def\TC{\mathit{TC}} \def\tr{\operatorname{tr}} \def\vT{{\mathit v\!T}} \def\barT{{\bar T}} \def\bbe{\mathbbm{e}} \def\bbE{{\mathbb E}} \def\bbO{{\mathbb O}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbZ{{\mathbb Z}} \def\bcA{{\bar{\mathcal A}}} \def\calA{{\mathcal A}} \def\calD{{\mathcal D}} \def\calF{{\mathcal F}} \def\calG{{\mathcal G}} \def\calI{{\mathcal I}} \def\calK{{\mathcal K}} \def\calO{{\mathcal O}} \def\calP{{\mathcal P}} \def\calR{{\mathcal R}} \def\calS{{\mathcal S}} \def\calT{{\mathcal T}} \def\calU{{\mathcal U}} \def\fraka{{\mathfrak a}} \def\frakb{{\mathfrak b}} \def\frakg{{\mathfrak g}} \def\frakh{{\mathfrak h}} \def\tilE{\tilde{E}} \def\tilq{\tilde{q}} \def\tDelta{\tensor[^t]{\Delta}{}} \def\tf{\tensor[^t]{f}{}} \def\tF{\tensor[^t]{F}{}} \def\tg{\tensor[^t]{g}{}} \def\tI{\tensor[^t]{I}{}} \def\tm{\tensor[^t]{m}{}} \def\tR{\tensor[^t]{R}{}} \def\tsigma{\tensor[^t]{\sigma}{}} \def\tS{\tensor[^t]{S}{}} \def\tSW{\tensor[^t]{\SW}{}} \def\ASSOm{\text{{\bf ASSO}}^{(m)}} \def\ASSOmm{\text{{\bf ASSO}}^{(m-1)}} \def\GT{\text{{\bf GT}}} \def\GRT{\text{{\bf GRT}}} \def\GRTm{\text{{\bf GRT}}^{(m)}} \def\GRTmm{\text{{\bf GRT}}^{(m-1)}} \def\grtm{{\mathfrak{grt}}^{(m)}} \def\grtmm{{\mathfrak{grt}}^{(m-1)}} \def\PBm{\text{{\bf PaB}}^{(m)}} \def\PCDm{\text{{\bf PaCD}}^{(m)}} \def\PP{\text{{\bf PaP}}} \def\Abstract{{\raisebox{2mm}{\parbox[t]{4.68in}{ {\color{red}Abstract.} The ``Grothendieck-Teichm\"uller Group'' (\GT) appears as a ``depth certificate'' in many recent works --- ``we do $A$ to $B$, apply the result to $C$, and get something related to \GT, therefore it must be interesting''. Interesting or not, in my talk I will explain how \GT{} arose first, in Drinfel'd's work on associators, and how it can be used to show that ``every bounded-degree associator extends'', that ``rational associators \text{exist'',} and that ``the pentagon implies the hexagon''. In a nutshell: the filtered tower of braid groups (with bells and whistles attached) is isomorphic to its associated graded, but the isomorphism is neither canonical nor unique --- such an isomorphism is precisely the thing called ``an associator''. But the set of isomorphisms between two isomorphic objects {\bf always} has two groups acting simply transitively on it --- the group of automorphisms of the first object acting on the right, and the group of automorphisms of the second object acting on the left. In the case of associators, that first group is what Drinfel'd calls the Grothendieck-Teichm\"uller group \GT, and the second group, isomorphic but not canonically to the first and denoted \GRT, is the one several recent works seem to refer to. \vskip 2mm Almost everything is in my \href{http://www.math.toronto.edu/~drorbn/LOP.html\#Associators}{old paper} ``On Associators and the Grothendieck-Teichm\"uller Group I'', also at \arXiv{q-alg/9606021}. The ``$\pentagon\rightarrow\hexagon$'' material is in \arXiv{math/0702128} by Furusho and \arXiv{math/1010.0754} by B-N and Dancso. }}}} \def\BabyExample{{\raisebox{2mm}{\parbox[t]{2.5in}{ {\color{red}Baby(?) Example.} $PB_n$: pure braids; $I\subset\bbQ PB_n$ the augmentation ideal; $B^{(m)}=\bbQ PB_n/I^{m+1}$ (filtered!); $\hat{B}=\varprojlim B^{(m)}$ (filtered!). Then $\text{gr }B^{(m)}=C^{(m)}$ and then $\text{gr }\hat{B}=\hat{C}$ where $C=\langle t^{ij}=t^{ji}\colon [t^{ij},t^{kl}]=[t^{ij},t^{ik}+t^{jk}]=0\rangle$, so $B^{(m)}$ and $\hat{B}$ are isomorphic to $C^{(m)}$ and $\hat{C}$, but not canonically. Me not know that the groups \GT{} and \GRT{} here have been analyzed. }}}} \def\From{{\raisebox{3mm}{\parbox[t]{2.4in}{ From Drinfel'd's {\em On quasitriangular Quasi-Hopf algebras and a group closely connected with $\text{Gal}(\bar{{\bbQ}}/{\bbQ})$}, Leningrad Math.{} J.{} {\bf 2} (1991) 829--860. }}}} \def\B{{$ B^{(m)} = \left( \PBm,{\mathbf S}:\PBm\to\PP,d_i,s_i,{\mathbf\Box},\sigma \right) $:}} \def\C{{$ C^{(m)} = \left( \PCDm,{\mathbf S}:\PCDm\to\PP,d_i,s_i,{\mathbf\Box},\tilde{R} \right)$:}} \def\tilR{{$\tilde{R}=X\exp\frac{H}{2}$}} \def\GammaEquations{{\raisebox{3mm}{\parbox[t]{3.2in}{ \[ d_4\Gamma\cdot d_2\Gamma\cdot d_0\Gamma = d_1\Gamma\cdot d_3\Gamma \] \[ 1=\Gamma\cdot(\Gamma^{-1})^{132}\cdot\Gamma^{312} \] \[ d_1t^{12}=\Gamma\cdot \left(t^{23}\cdot(\Gamma^{-1})^{132}+(\Gamma^{-1})^{132}\cdot t^{13}\right) \cdot\Gamma^{312} \] \[ e^{\epsilon(t^{13}+t^{23})} = \Gamma\cdot e^{\epsilon t^{23}}\cdot(\Gamma^{-1})^{132}\cdot e^{\epsilon t^{13}}\cdot\Gamma^{312} \] }}}} \def\MainTheorem{{\raisebox{3mm}{\parbox[t]{1.95in}{ {\color{red}Main Theorem.} The projection $\ASSOm\to\ASSOmm$ is surjective. (yey!) \noindent\par {\color{red}Sketch. } Given $\ASSOm\neq\emptyset$ (hard, analytic), sufficient is surjectivity of $\GRTm\to\GRTmm$, enough is surjectivity of $\grtm\to\grtmm$, polyhedron on left use, little homological algebra too. }}}} \begin{document} \latintext %\setlength{\jot}{0ex} \setlength{\abovedisplayskip}{0.5ex} \setlength{\belowdisplayskip}{0.5ex} \setlength{\abovedisplayshortskip}{0ex} \setlength{\belowdisplayshortskip}{0ex} \begin{center} \null\vfill\scalebox{0.95}{\input{GT1.pdftex_t}}\vfill\null \null\vfill\scalebox{0.95}{\input{GT2.pdftex_t}}\vfill\null \end{center} \end{document} \endinput