
diff -r fe9295fcf31d -r dfefae16073c text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Mon Jun 07 06:01:39 2010 +0200
+++ b/text/a_inf_blob.tex	Mon Jun 07 13:43:38 2010 +0200
@@ -106,7 +106,7 @@
 filtration degree 1 stuff, and so on.
 More formally,
  
-\begin{lemma}
+\begin{lemma} \label{lem:d-a-acyclic}
 $D(a)$ is acyclic.
 \end{lemma}
 
@@ -372,9 +372,66 @@
 Ricardo Andrade also told us about a similar result.
 \end{rem}
 
-\nn{proof is again similar to that of Theorem \ref{product_thm}.  should probably say that explicitly}
+\begin{proof}
+The proof is again similar to that of Theorem \ref{product_thm}.
+
+We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
+
+Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of
+$j$-fold mapping cylinders, $j \ge 0$.
+So, as an abelian group (but not as a chain complex), 
+\[
+	\cB^\cT(M) = \bigoplus_{j\ge 0} C^j,
+\]
+where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders.
+
+Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by
+decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms
+of $\cT$.
+Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs
+$(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous
+maps from the $n{-}1$-skeleton of $K$ to $T$.
+The summand indexed by $(K, \vphi)$ is
+\[
+	\bigotimes_b D_*(b, \vphi),
+\]
+where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes
+chains of maps from $b$ to $T$ compatible with $\vphi$.
+We can take the product of these chains of maps to get a chains of maps from
+all of $M$ to $K$.
+This defines $\psi$ on $C^0$.
 
-\begin{proof}
+We define $\psi(C^j) = 0$ for $j > 0$.
+It is not hard to see that this defines a chain map from 
+$\cB^\cT(M)$ to $C_*(\Maps(M\to T))$.
+
+The image of $\psi$ is the subcomplex $G_*\sub C_*(\Maps(M\to T))$ generated by 
+families of maps whose support is contained in a disjoint union of balls.
+It follows from Lemma \ref{extension_lemma_c} 
+that $C_*(\Maps(M\to T))$ is homotopic to a subcomplex of $G_*$.
+
+We will define a map $\phi:G_*\to \cB^\cT(M)$ via acyclic models.
+Let $a$ be a generator of $G_*$.
+Define $D(a)$ to be the subcomplex of $\cB^\cT(M)$ generated by all 
+pairs $(b, \ol{K})$, where $b$ is a generator appearing in an iterated boundary of $a$
+and $\ol{K}$ is an index of the homotopy colimit $\cB^\cT(M)$.
+(See the proof of Theorem \ref{product_thm} for more details.)
+The same proof as of Lemma \ref{lem:d-a-acyclic} shows that $D(a)$ is acyclic.
+By the usual acyclic models nonsense, there is a (unique up to homotopy)
+map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$.
+Furthermore, we may choose $\phi$ such that for all $a$ 
+\[
+	\phi(a) = (a, K) + r
+\]
+where $(a, K) \in C^0$ and $r\in \bigoplus_{j\ge 1} C^j$.
+
+It is now easy to see that $\psi\circ\phi$ is the identity on the nose.
+Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity.
+(See the proof of Theorem \ref{product_thm} for more details.)
+\end{proof}
+
+\noop{
+% old proof (just start):
 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
 We then use Lemma \ref{extension_lemma_c} to show that $g$ induces isomorphisms on homology.
 
@@ -407,8 +464,7 @@
 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$.
 
 \nn{...}
-
-\end{proof}
+}
 
 \nn{maybe should also mention version where we enrich over
 spaces rather than chain complexes;}




