--- a/blob1.tex	Mon Jun 02 22:32:54 2008 +0000
+++ b/blob1.tex	Sun Jun 08 21:34:46 2008 +0000
@@ -577,287 +577,9 @@
 
 \nn{what else?}
 
-
-
-
-\section{$n=1$ and Hochschild homology}
-
-In this section we analyze the blob complex in dimension $n=1$
-and find that for $S^1$ the homology of the blob complex is the
-Hochschild homology of the category (algebroid) that we started with.
-\nn{or maybe say here that the complexes are quasi-isomorphic?  in general,
-should perhaps put more emphasis on the complexes and less on the homology.}
-
-Notation: $HB_i(X) = H_i(\bc_*(X))$.
-
-Let us first note that there is no loss of generality in assuming that our system of
-fields comes from a category.
-(Or maybe (???) there {\it is} a loss of generality.
-Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be
-thought of as the morphisms of a 1-category $C$.
-More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$
-are $A(I; a, b)$, and composition is given by gluing.
-If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change
-and neither does $A(I; a, b) = HB_0(I; a, b)$.
-But what about $HB_i(I; a, b)$ for $i > 0$?
-Might these higher blob homology groups be different?
-Seems unlikely, but I don't feel like trying to prove it at the moment.
-In any case, we'll concentrate on the case of fields based on 1-category
-pictures for the rest of this section.)
-
-(Another question: $\bc_*(I)$ is an $A_\infty$-category.
-How general of an $A_\infty$-category is it?
-Given an arbitrary $A_\infty$-category can one find fields and local relations so
-that $\bc_*(I)$ is in some sense equivalent to the original $A_\infty$-category?
-Probably not, unless we generalize to the case where $n$-morphisms are complexes.)
-
-Continuing...
-
-Let $C$ be a *-1-category.
-Then specializing the definitions from above to the case $n=1$ we have:
-\begin{itemize}
-\item $\cC(pt) = \ob(C)$ .
-\item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$.
-Then an element of $\cC(R; c)$ is a collection of (transversely oriented)
-points in the interior
-of $R$, each labeled by a morphism of $C$.
-The intervals between the points are labeled by objects of $C$, consistent with
-the boundary condition $c$ and the domains and ranges of the point labels.
-\item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by
-composing the morphism labels of the points.
-Note that we also need the * of *-1-category here in order to make all the morphisms point
-the same way.
-\item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single
-point (at some standard location) labeled by $x$.
-Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the
-form $y - \chi(e(y))$.
-Thus we can, if we choose, restrict the blob twig labels to things of this form.
-\end{itemize}
-
-We want to show that $HB_*(S^1)$ is naturally isomorphic to the
-Hochschild homology of $C$.
-\nn{Or better that the complexes are homotopic
-or quasi-isomorphic.}
-In order to prove this we will need to extend the blob complex to allow points to also
-be labeled by elements of $C$-$C$-bimodules.
-%Given an interval (1-ball) so labeled, there is an evaluation map to some tensor product
-%(over $C$) of $C$-$C$-bimodules.
-%Define the local relations $U(I; a, b)$ to be the direct sum of the kernels of these maps.
-%Now we can define the blob complex for $S^1$.
-%This complex is the sum of complexes with a fixed cyclic tuple of bimodules present.
-%If $M$ is a $C$-$C$-bimodule, let $G_*(M)$ denote the summand of $\bc_*(S^1)$ corresponding
-%to the cyclic 1-tuple $(M)$.
-%In other words, $G_*(M)$ is a blob-like complex where exactly one point is labeled
-%by an element of $M$ and the remaining points are labeled by morphisms of $C$.
-%It's clear that $G_*(C)$ is isomorphic to the original bimodule-less
-%blob complex for $S^1$.
-%\nn{Is it really so clear?  Should say more.}
-
-%\nn{alternative to the above paragraph:}
-Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$.
-We define a blob-like complex $F_*(S^1, (p_i), (M_i))$.
-The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling
-other points.
-The blob twig labels lie in kernels of evaluation maps.
-(The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.)
-Let $F_*(M) = F_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point.
-In other words, fields for $F_*(M)$ have an element of $M$ at the fixed point $*$
-and elements of $C$ at variable other points.
-
-We claim that the homology of $F_*(M)$ is isomorphic to the Hochschild
-homology of $M$.
-\nn{Or maybe we should claim that $M \to F_*(M)$ is the/a derived coend.
-Or maybe that $F_*(M)$ is quasi-isomorphic (or perhaps homotopic) to the Hochschild
-complex of $M$.}
-This follows from the following lemmas:
-\begin{itemize}
-\item $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$.
-\item An exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$
-gives rise to an exact sequence $0 \to F_*(M_1) \to F_*(M_2) \to F_*(M_3) \to 0$.
-(See below for proof.)
-\item $F_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is
-quasi-isomorphic to the 0-step complex $C$.
-(See below for proof.)
-\item $F_*(C)$ (here $C$ is wearing its $C$-$C$-bimodule hat) is quasi-isomorphic to $\bc_*(S^1)$.
-(See below for proof.)
-\end{itemize}
-
-First we show that $F_*(C\otimes C)$ is
-quasi-isomorphic to the 0-step complex $C$.
-
-Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of
-the point $*$ is $1 \otimes 1 \in C\otimes C$.
-We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism.
-
-Fix a small $\ep > 0$.
-Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$.
-Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex
-generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from
-or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$.
-For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$
-labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$.
-(See Figure xxxx.)
-Note that $y - s_\ep(y) \in U(B_\ep)$.
-\nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.}
-
-Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows.
-Let $x \in F^\ep_*$ be a blob diagram.
-If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to
-$x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$.
-If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows.
-Let $y_i$ be the restriction of $z_i$ to $B_\ep$.
-Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$,
-and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$.
-Define $j_\ep(x) = \sum x_i$.
-\nn{need to check signs coming from blob complex differential}
-
-Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also.
-
-The key property of $j_\ep$ is
-\eq{
-    \bd j_\ep + j_\ep \bd = \id - \sigma_\ep ,
-}
-where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field
-mentioned in $x \in F^\ep_*$ with $s_\ep(y)$.
-Note that $\sigma_\ep(x) \in F'_*$.
-
-If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$
-is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$.
-One strategy would be to try to stitch together various $j_\ep$ for progressively smaller
-$\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$.
-Instead, we'll be less ambitious and just show that
-$F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
-
-If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have
-$x \in F_*^\ep$.
-(This is true for any chain in $F_*(C\otimes C)$, since chains are sums of
-finitely many blob diagrams.)
-Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map
-$F'_* \sub F_*(C\otimes C)$ is surjective on homology.
-If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$
-and
-\eq{
-    \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) .
-}
-Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology.
-This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
-
-\medskip
-
-Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob.
-We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence.
-
-First, a lemma:  Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with
-$S^1$ replaced some (any) neighborhood of $* \in S^1$.
-Then $G''_*$ and $G'_*$ are both contractible
-and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence.
-For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting
-$G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$.
-For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe
-in ``basic properties" section above} away from $*$.
-Thus any cycle lies in the image of the normal blob complex of a disjoint union
-of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}).
-Actually, we need the further (easy) result that the inclusion
-$G''_* \to G'_*$ induces an isomorphism on $H_0$.
-
-Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that
-for all $x \in F'_*$ we have
-\eq{
-    x - \bd h(x) - h(\bd x) \in F''_* .
-}
-Since $F'_0 = F''_0$, we can take $h_0 = 0$.
-Let $x \in F'_1$, with single blob $B \sub S^1$.
-If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$.
-If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$).
-Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$.
-Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$.
-Define $h_1(x) = y$.
-The general case is similar, except that we have to take lower order homotopies into account.
-Let $x \in F'_k$.
-If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$.
-Otherwise, let $B$ be the outermost blob of $x$ containing $*$.
-By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$.
-So $x' \in G'_l$ for some $l \le k$.
-Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$.
-Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$.
-Define $h_k(x) = y \bullet p$.
-This completes the proof that $i: F''_* \to F'_*$ is a homotopy equivalence.
-\nn{need to say above more clearly and settle on notation/terminology}
-
-Finally, we show that $F''_*$ is contractible.
-\nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now}
-Let $x$ be a cycle in $F''_*$.
-The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a
-ball $B \subset S^1$ containing the union of the supports and not containing $*$.
-Adding $B$ as a blob to $x$ gives a contraction.
-\nn{need to say something else in degree zero}
-
-This completes the proof that $F_*(C\otimes C)$ is
-homotopic to the 0-step complex $C$.
-
-\medskip
-
-Next we show that $F_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$.
-$F_*(C)$ differs from $\bc_*(S^1)$ only in that the base point *
-is always a labeled point in $F_*(C)$, while in $\bc_*(S^1)$ it may or may not be.
-In other words, there is an inclusion map $i: F_*(C) \to \bc_*(S^1)$.
-
-We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows.
-If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if
-* is a labeled point in $y$.
-Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *.
-Let $x \in \bc_*(S^1)$.
-Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in
-$x$ with $y$.
-It is easy to check that $s$ is a chain map and $s \circ i = \id$.
-
-Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points
-in a neighborhood $B_\ep$ of *, except perhaps *.
-Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$.
-\nn{rest of argument goes similarly to above}
-
-\bigskip
-
-\nn{still need to prove exactness claim}
-
-\nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex?
-Do we need a map from hoch to blob?
-Does the above exactness and contractibility guarantee such a map without writing it
-down explicitly?
-Probably it's worth writing down an explicit map even if we don't need to.}
-
-
-We can also describe explicitly a map from the standard Hochschild
-complex to the blob complex on the circle. \nn{What properties does this
-map have?}
-
-\begin{figure}%
-$$\mathfig{0.6}{barycentric/barycentric}$$
-\caption{The Hochschild chain $a \tensor b \tensor c$ is sent to
-the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.}
-\label{fig:Hochschild-example}%
-\end{figure}
-
-As an example, Figure \ref{fig:Hochschild-example} shows the image of the Hochschild chain $a \tensor b \tensor c$. Only the $0$-cells are shown explicitly.
-The edges marked $x, y$ and $z$ carry the $1$-chains
-\begin{align*}
-x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\
-y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\
-z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab}
-\end{align*}
-and the $2$-chain labelled $A$ is
-\begin{equation*}
-A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}.
-\end{equation*}
-Note that we then have
-\begin{equation*}
-\bdy A = x+y+z.
-\end{equation*}
-
-In general, the Hochschild chain $\Tensor_{i=1}^n a_i$ is sent to the sum of $n!$ blob $(n-1)$-chains, indexed by permutations,
-$$\phi\left(\Tensor_{i=1}^n a_i\right) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$
-with ... (hmmm, problems making this precise; you need to decide where to put the labels, but then it's hard to make an honest chain map!)
-
+\section{Hochschild homology when $n=1$}
+\label{sec:hochschild}
+\input{text/hochschild}
 
 \section{Action of $C_*(\Diff(X))$}  \label{diffsect}