[Torus Algebra: many typo corrections, smargins lipshitz@math.columbia.edu**20111228235225 Ignore-this: 92fc3d2a27a7339f69db434cd0a62b36 ] hunk ./Bminus.tex 17 - \usepackage[pdftex,margin=1in]{geometry} + \usepackage[pdftex,lmargin=1.5in, rmargin=1.5in, tmargin=1in, bmargin=1in]{geometry} hunk ./Bminus.tex 32 - \usepackage[dvips,margin=1in]{geometry} + \usepackage[dvips,lmargin=1.5in, rmargin=1.5in, tmargin=1in, bmargin=1in]{geometry} hunk ./TorusAlgebra.tex 3 +The algebra associated to the torus will be defined in three +steps. First, we define an obvious extension $\ClassicAlg(T^2)$ of the +algebra associated to the torus in~\cite{LOT1}. Then we define an +$\Ainf$-deformation $\Alg_0(\Torus)$ of $\ClassicAlg(T^2)$. Finally +we introduce an additional variable $\Orbit$ (and some relations +involving it), giving the algebra $\Alg(Torus)$ of interest. + +\subsection{The classical algebra} +Idempotent, Reeb chords, product on chords. +\smargin{Write.} + +\subsection{The orbit-less algebra} hunk ./TorusAlgebra.tex 16 -the torus. This could be given either a holomorphic and a -combinatorial definition; we give first the holomorphic one and show -that it is equivalent to a combinatorial one, and verify that it is, -indeed, and $\Ainfty$ algebra. +the torus. This can be given by either a holomorphic or a +combinatorial definition; we give first the holomorphic one, then show +that it is equivalent to a combinatorial one, and then verify that it is, +indeed, an $\Ainfty$ algebra. hunk ./TorusAlgebra.tex 23 -transversally in a single point. +transversally in a single point $p$. hunk ./TorusAlgebra.tex 27 -We fix a conformal structure on $\Torus$. -Fix generic points $p_i\in \alpha_i$ for $i=1,2$. +Fix a conformal structure on $\Torus$ and generic points $p_i\in +\alpha_i$ for $i=1,2$. hunk ./TorusAlgebra.tex 30 -\begin{definition} +\begin{definition}\label{def:hol-cov-disk} hunk ./TorusAlgebra.tex 39 + \smargin{RL: 1. This notion of $p$ hasn't been + introduced. 2. Horrible notation clash between this $p$ and + $p$ as puncture of $\Torus$.} hunk ./TorusAlgebra.tex 44 + \smargin{RL: submersion is not quite right at the corners.} hunk ./TorusAlgebra.tex 47 + \smargin{RL: I guess these disks can't have ``Reeb orbits''; but + this is not yet explicitly ruled out.} hunk ./TorusAlgebra.tex 72 - The initial algebra for $\Alg_0(\Torus)$ the torus is constructed as follows. + The initial algebra $\Alg_0(\Torus)$ for the torus is constructed as follows. hunk ./TorusAlgebra.tex 74 - There are algebra elements $a(\rho)$, where here $\rho$ is an + There are algebra elements $a(\rho)$, where $\rho$ is an hunk ./TorusAlgebra.tex 93 + \smargin{RL: third term on RHS has some typos in (some of) + $\rho_n$, $\rho_0$, $\rho_n$, $\rho_n$.} hunk ./TorusAlgebra.tex 110 -Although one could give a holomorphic proof +Although one could give a holomorphic proof of hunk ./TorusAlgebra.tex 119 -\subsection{Constructing covering disks} +\subsubsection{Constructing covering disks} hunk ./TorusAlgebra.tex 128 -are the complements of $\alpha_1\cup\alpha_2\cap \partial D_)$) +are the complements of $\alpha_1\cup\alpha_2\cap \partial D_0$) hunk ./TorusAlgebra.tex 142 - Two covering disks are $(\phi',D',p')$ and $(\phi,D,p)$ are {\em + Two covering disks $(\phi',D',p')$ and $(\phi,D,p)$ are {\em hunk ./TorusAlgebra.tex 145 - $\phi$ and $\phi'$. + $\phi$ and $\phi'$.% + \smargin{RL: same comment about corners as in Def.~\ref{def:hol-cov-disk}} hunk ./TorusAlgebra.tex 152 - disks and - equivalence classes of - topological covering disks. + disks and equivalence classes of topological covering disks.% + \smargin{RL: this should be ``equivalence classes of holomorphic + covering disks'' (not yet defined above)} hunk ./TorusAlgebra.tex 184 -Before giving the construction, we will also need the following simple observation: +Before giving the construction, we will also need a few simple observations: + +If $f\colon D \to \Torus\setminus D_0$ +is a topological covering disk, then the domain +$D$ is tiled by preimages of $T_0=\Torus\setminus +D_0\cup\alpha_1\cup\alpha_2$. +The number of tiles is the total length of the boundary instructions +divided by four. Moreover, the covering disk can then be specified by gluing +tiles to each other (it is a graph whose vertices correspond to the +tiles and whose edges correspond to gluings). +\smargin{RL: ``it is a graph'': what is ``it''?} hunk ./TorusAlgebra.tex 199 - instructions, or each tile, the four Reeb arcs are encountered in + instructions, for each tile, the four Reeb arcs are encountered in hunk ./TorusAlgebra.tex 214 -If $f\colon D \to \Torus\setminus D_0$ -is a topological covering disk, then the domain -$D$ is tiled by preimages of $T_0=\Torus\setminus -D_0\cup\alpha_1\cup\alpha_2$. -The number of tiles is the total length of the boundary instructions -divided by four. Moreover, the covering disk can then be specified by gluing -tiles to each other (it is a graph whose vertices correspond to the -tiles and whose edges correspond to gluings). The following -algorithm reconstructs $D$ from its boundary instructions; -indeed, it also specifies which tile -each arc belongs to. + +The following algorithm reconstructs $D$ from its boundary +instructions; indeed, it also specifies which tile each arc belongs +to. hunk ./TorusAlgebra.tex 228 + \smargin{RL: $j\leq i$? Seems not entirely consistent later.} hunk ./TorusAlgebra.tex 233 - and $a_i$ is {\em not} the last arc on the boundary of $T$, + and $a_i$ is {\em not} the last arc on the boundary of $T$,% + \smargin{RL: what's the ordering on arcs on $\bdy T$? i.e., which is + the first (or last) one?}% hunk ./TorusAlgebra.tex 239 - along an $\alpha$-edge, so that the arc $a_i$ thought of as an arc in $T$ is + along an $\alpha$-edge, so that the arc $a_i$, thought of as an arc in $T$, is hunk ./TorusAlgebra.tex 244 - $a_{i+1}$ is associated to the predecessor to $T$. + $a_{i+1}$ is associated to the predecessor to $T$. % + \smargin{RL: what does ``$a_{i+1}$ is associated to the predecessor + to $T$'' mean?}% hunk ./TorusAlgebra.tex 255 -which is tiled by copies of $T_0$, one of which is initial. Each -non-initial tile has a predecessor. It also produces -a way to associate to each arc in the sequence of prospective boundary -instructions -a corresponding tile. +which is tiled by copies of $T_0$, one of which is initial. % +\smargin{RL: I don't understand what the surface is in the + non-admissible case.}% +Each non-initial tile has a predecessor. It also produces a way to +associate to each arc in the sequence of prospective boundary +instructions a corresponding tile. hunk ./TorusAlgebra.tex 264 - A sequence of reeb chords $(r_1,\dots,r_n)$ is called {\em strongly + A sequence of Reeb chords $(r_1,\dots,r_n)$ is called {\em strongly hunk ./TorusAlgebra.tex 308 + \smargin{RL hasn't read this proof yet.} + hunk ./TorusAlgebra.tex 404 - if $r_n=r_n'\uplus r_n''$ so that $(r_1,\dots,r_{n-1},\rho_n')$ + if $r_n=r_n'\uplus r_n''$ so that $(r_1,\dots,r_{n-1},r_n')$ hunk ./TorusAlgebra.tex 423 - Moreover, if it is left- resp. right- extended by some $r$, then - $r$ - is unique. + Moreover, if it is left-extended (respectively right-extended) by + some $r$, then $r$ is unique. hunk ./TorusAlgebra.tex 429 - to a left-extended sequence, we find that the first, say, $i$ arcs + to a left-extended sequence, we find that the first $i$ arcs, say, hunk ./TorusAlgebra.tex 436 + \smargin{RL didn't follow this proof.} hunk ./TorusAlgebra.tex 448 - $\mu_n(\rho_1,\dots,r_n)=\iota$, the left idempotent of $r_1$. + $\mu_n(r_1,\dots,r_n)=\iota$, the left idempotent of $r_1$. hunk ./TorusAlgebra.tex 452 - $\mu_n(\rho_1,\dots,r_n)=a(r)$. + $\mu_n(r_1,\dots,r_n)=a(r)$. hunk ./TorusAlgebra.tex 456 - $\mu_n(\rho_1,\dots,r_n)=a(r)$. + $\mu_n(r_1,\dots,r_n)=a(r)$. hunk ./TorusAlgebra.tex 467 - Definition~\ref{eq:DefMuN}, + Equation~\eqref{eq:DefMuN}, hunk ./TorusAlgebra.tex 471 -\subsection{From covering disks to $\Alg_0(\Torus)$} +\subsubsection{From covering disks to $\Alg_0(\Torus)$} hunk ./TorusAlgebra.tex 475 -subsequence, indexed by $1\leq i< j \leq n$, write +subsequence, indexed by $1\leq i< j \leq n$, write % +\smargin{RL: ``$1\leq i