[added a picture and a little about the DD id bimodule petero@math.columbia.edu**20120417142425 Ignore-this: 359606b7dbce3edf9d54be165d193b2c ] hunk ./Bminus.tex 355 - $\MAlg$. Call $i$ \emph{multipliable} if $a_ia_{i+1}\neq 0$.% + $\ClassicAlg$. Call $i$ \emph{multipliable} if $a_ia_{i+1}\neq 0$.% hunk ./Bminus.tex 359 - Let $T$ be a tree with $n$ inputs and $\mu_T\co (\MAlg)^{\otimes n}\to - \MAlg$ the corresponding operation. If $\mu_T(a_1,\dots,a_n)$ then + Let $T$ be a tree with $n$ inputs and $\mu_T\co (\ClassicAlg)^{\otimes n}\to + \ClassicAlg$ the corresponding operation. If $\mu_T(a_1,\dots,a_n)\neq 0$ then hunk ./Bminus.tex 369 - First, suppose that none of the $a_i$'s contain a Reeb orbit. - \smargin{Write.} + Write out the tree as a sequence of operations. We prove the + inequality by induction on the number of vertices. If a vertex + corresponds to a left- or right-extendable sequence, the number of + multipliables drops by at most one, but we also passed through + a vertex. If the vertex is neither, the output is an idempotent, so the + multipliables stays the same. + + Look at the last vertex. The number of multipliables drop by at most + one across each vertex. In the case of equality it must drop by + exactly one at each vertex. It follows that the last vertex has a + multipliable pair in it, and hence it in fact has to have valence + $3$. Apply induction to the two input trees. hunk ./Makefile 3 -SOURCES = Associahedron.tex $(PROJECT).tex defs.tex macros.tex +SOURCES = Associahedron.tex $(PROJECT).tex defs.tex macros.tex TorusAlgebra.tex hunk ./Makefile 32 - LeftwardsSequences.fig + LeftwardsSequences.fig AinftyOperations.fig AinftyRelations.fig hunk ./TorusAlgebra.tex 14 +\begin{figure} + \centering + \input{AinftyOperations} + \caption{\textbf{$A_{\infty}$ operations}. + Pictures of curves representing operations.} +\end{figure} + + +\begin{figure} + \centering + \input{AinftyRelations} + \caption{\textbf{$A_{\infty}$ relations}. + Ends of moduli spaces giving $A_{\infty}$ relations.} +\end{figure} + hunk ./TorusAlgebra.tex 1019 + +We define $\MAlg(\Torus)$, by introducing a new central element $\RO$ +to $\ClassicAlg(\Torus)$, satisfying $\RO^2=0$; i.e. +\[ +\MAlg(\Torus)=\ClassicAlg(\Torus)[\RO]/\RO^2=0. +\] +This gives $\MAlg(\Torus)$ its multiplicative structure ($\mu_2$). +Next, we describe its differential. + +The differential $d$ on $\MAlg(\Torus)$ is uniquely specified by the property that +\[ d\RO=\rho_{1234}+\rho_{2341}+\rho_{3412}+\rho_{2341},\] where here +the right-hand-side is a central element of $\ClassicAlg(\Torus)$. +This property, together with the Leibniz rule, allows one to describe +the differential of any element of $\MAlg(\Torus)$. + +For example, we have that +\begin{align*} + d(\rho_{2}\cdot \RO)& =\rho_{23412} \\ + d(\rho_{23412}\cdot \RO) &= \rho_{234123412}. +\end{align*} + +Having defined $\mu_1$ and $\mu_2$ on $\MAlg(\Torus)$, it remains to +describe $\mu_n$ for $n>2$. + +We add new $\Ainfty$ operations coming from the $\Ainfty$ operations +in $\MAlg(\Torus)$ two ways. + +Suppose that $\mu_n(a_1,\dots,a_n)=a$ is an operation in +$\MAlg(\Torus)$, and there is a subsequence of algebra elements +$\{a_{n_i}\}_{i=1}^k$, where $a_{n_i}$ are algebra elements associated +to length $4$ chords. We will introduce four new families of +$\Ainfty$ algebra +operations. + +The first {\em generic type} of operations are associated to sequences $\{m_i\}_{i=1}^k$ with the properties that +\begin{align*} +n_i