--- a/text/appendixes/moam.tex	Fri Jul 30 14:19:11 2010 -0700
+++ b/text/appendixes/moam.tex	Fri Jul 30 14:19:23 2010 -0700
@@ -1,4 +1,54 @@
 %!TEX root = ../../blob1.tex
 
 \section{The method of acyclic models}  \label{sec:moam}
-\todo{...}
\ No newline at end of file
+
+Let $F_*$ and $G_*$ be chain complexes.
+Assume $F_k$ has a basis $\{x_{kj}\}$
+(that is, $F_*$ is free and we have specified a basis).
+(In our applications, $\{x_{kj}\}$ will typically be singular $k$-simplices or 
+$k$-blob diagrams.)
+For each basis element $x_{kj}$ assume we have specified a ``target" $D^{kj}_*\sub G_*$.
+
+We say that a chain map $f:F_*\to G_*$ is {\it compatible} with the above data (basis and targets)
+if $f(x_{kj})\in D^{kj}_*$ for all $k$ and $j$.
+Let $\Compat(D^\bullet_*)$ denote the subcomplex of maps from $F_*$ to $G_*$
+such that the image of each higher homotopy applied to $x_{kj}$ lies in $D^{kj}_*$.
+
+\begin{thm}[Acyclic models]  \label{moam-thm}
+Suppose 
+\begin{itemize}
+\item $D^{k-1,l}_* \sub D^{kj}_*$ whenever $x_{k-1,l}$ occurs in $\bd x_{kj}$
+with non-zero coefficient;
+\item $D^{0j}_0$ is non-empty for all $j$; and
+\item $D^{kj}_*$ is $(k{-}1)$-acyclic (i.e.\ $H_{k-1}(D^{kj}_*) = 0$) for all $k,j$ .
+\end{itemize}
+Then $\Compat(D^\bullet_*)$ is non-empty.
+If, in addition,
+\begin{itemize}
+\item $D^{kj}_*$ is $m$-acyclic for $k\le m \le k+i$ and for all $k,j$,
+\end{itemize}
+then $\Compat(D^\bullet_*)$ is $i$-connected.
+\end{thm}
+
+\begin{proof}
+(Sketch)
+This is a standard result; see, for example, \nn{need citations}.
+
+We will build a chain map $f\in \Compat(D^\bullet_*)$ inductively.
+Choose $f(x_{0j})\in D^{0j}_0$ for all $j$
+(possible since $D^{0j}_0$ is non-empty).
+Choose $f(x_{1j})\in D^{1j}_1$ such that $\bd f(x_{1j}) = f(\bd x_{1j})$
+(possible since $D^{0l}_* \sub D^{1j}_*$ for each $x_{0l}$ in $\bd x_{1j}$
+and $D^{1j}_*$ is 0-acyclic).
+Continue in this way, choosing $f(x_{kj})\in D^{kj}_k$ such that $\bd f(x_{kj}) = f(\bd x_{kj})$
+We have now constructed $f\in \Compat(D^\bullet_*)$, proving the first claim of the theorem.
+
+Now suppose that $D^{kj}_*$ is $k$-acyclic for all $k$ and $j$.
+Let $f$ and $f'$ be two chain maps (0-chains) in $\Compat(D^\bullet_*)$.
+Using a technique similar to above we can construct a homotopy (1-chain) in $\Compat(D^\bullet_*)$
+between $f$ and $f'$.
+Thus $\Compat(D^\bullet_*)$ is 0-connected.
+Similarly, if $D^{kj}_*$ is $(k{+}i)$-acyclic then we can show that $\Compat(D^\bullet_*)$ is $i$-connected.
+\end{proof}
+
+\nn{do we also need some version of ``backwards" acyclic models?  probably}

--- a/text/comm_alg.tex	Fri Jul 30 14:19:11 2010 -0700
+++ b/text/comm_alg.tex	Fri Jul 30 14:19:23 2010 -0700
@@ -31,24 +31,9 @@
 \end{prop}
 
 \begin{proof}
-%To define the chain maps between the two complexes we will use the following lemma:
-%
-%\begin{lemma}
-%Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with
-%a basis (e.g.\ blob diagrams or singular simplices).
-%For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$
-%such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$.
-%Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that
-%$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty).
-%\end{lemma}
-%
-%\begin{proof}
-%\nn{easy, but should probably write the details eventually}
-%\nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that}
-%\end{proof}
-We will use acyclic models \nn{need ref}.
+We will use acyclic models (\S \ref{sec:moam}).
 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$
-satisfying the conditions of \nn{need ref}.
+satisfying the conditions of Theorem \ref{moam-thm}.
 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a 
 finite unordered collection of points of $M$ with multiplicities, which is
 a point in $\Sigma^\infty(M)$.
@@ -63,15 +48,29 @@
 Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a 
 subspace of $\Sigma^\infty(M)$.
 It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from 
-\nn{need ref, or state condition}.
+Theorem \ref{moam-thm}.
 Thus we have defined (up to homotopy) a map from 
-$\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$.
+$\bc_*(M, k[t])$ to $C_*(\Sigma^\infty(M))$.
 
-Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace
-$R(c)_* \sub \bc_*(M^n, k[t])$.
+Next we define a map going the other direction.
+First we replace $C_*(\Sigma^\infty(M))$ with a homotopy equivalent 
+subcomplex $S_*$ of small simplices.
+Roughly, we define $c\in C_*(\Sigma^\infty(M))$ to be small if the 
+corresponding track of points in $M$
+is contained in a disjoint union of balls.
+Because there could be different, inequivalent choices of such balls, we must a bit more careful.
+\nn{this runs into the same issues as in defining evmap.
+either refer there for details, or use the simp-space-ish version of the blob complex,
+which makes things easier here.}
+
+\nn{...}
+
+
+We will define, for each simplex $c$ of $S_*$, a contractible subspace
+$R(c)_* \sub \bc_*(M, k[t])$.
 If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and 
 $\Sigma^\infty(M)$ described above.
-Now let $c$ be an $i$-simplex of $\Sigma^j(M)$.
+Now let $c$ be an $i$-simplex of $S_*$.
 Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$.
 We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$
 is homotopy equivalent to the subcomplex of small simplices.
@@ -80,7 +79,7 @@
 \nn{do we need to define this precisely?}
 Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter.
 \nn{need to say more precisely how small}
-Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$.
+Define $R(c)_*$ to be $\bc_*(D; k[t]) \sub \bc_*(M; k[t])$.
 This is contractible by Proposition \ref{bcontract}.
 We can arrange that the boundary/inclusion condition is satisfied if we start with
 low-dimensional simplices and work our way up.
@@ -92,12 +91,13 @@
 
 
 \begin{prop} \label{ktchprop}
-The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$.
+The above maps are compatible with the evaluation map actions of $C_*(\Homeo(M))$.
 \end{prop}
 
 \begin{proof}
 The actions agree in degree 0, and both are compatible with gluing.
 (cf. uniqueness statement in Theorem \ref{thm:CH}.)
+\nn{if Theorem \ref{thm:CH} is rewritten/rearranged, make sure uniqueness discussion is properly referenced from here}
 \end{proof}
 
 \medskip
@@ -108,16 +108,16 @@
 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section}
 Let us check this directly.
 
-The algebra $k[t]$ has Koszul resolution 
+The algebra $k[t]$ has a resolution 
 $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, 
 which has coinvariants $k[t] \xrightarrow{0} k[t]$. 
-This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$.
+So we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$.
 (See also  \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: 
 $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one.
 
 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other.
 The fixed points of this flow are the equally spaced configurations.
-This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation).
+This defines a deformation retraction from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation).
 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, 
 and the holonomy of the $\Delta^{j-1}$ bundle
 over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices.
@@ -128,7 +128,7 @@
 and is zero for $i\ge 2$.
 Note that the $j$-grading here matches with the $t$-grading on the algebraic side.
 
-By xxxx and Proposition \ref{ktchprop}, 
+By Proposition \ref{ktchprop}, 
 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$.
 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$.
 If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree
@@ -163,7 +163,7 @@
 corresponding to $X$.
 The homology calculation we desire follows easily from this.
 
-\nn{say something about cyclic homology in this case?  probably not necessary.}
+%\nn{say something about cyclic homology in this case?  probably not necessary.}
 
 \medskip
 

--- a/text/kw_macros.tex	Fri Jul 30 14:19:11 2010 -0700
+++ b/text/kw_macros.tex	Fri Jul 30 14:19:23 2010 -0700
@@ -60,7 +60,7 @@
 
 % \DeclareMathOperator{\pr}{pr} etc.
 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
-\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad};
+\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat};
 
 \DeclareMathOperator*{\colim}{colim}
 \DeclareMathOperator*{\hocolim}{hocolim}