--- a/text/hochschild.tex Mon Apr 26 10:43:42 2010 -0700
+++ b/text/hochschild.tex Mon Apr 26 21:54:41 2010 -0700
@@ -310,8 +310,34 @@
\end{align*}
where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$.
-The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly surjective ($\ev$ surjects onto $M$); we now show that it's injective. \todo{}
+The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly surjective ($\ev$ surjects onto $M$); we now show that it's injective.
+This is equivalent to showing that
+\[
+ \ev\inv(\ker(\pi)) \sub \bd K_1(M) .
+\]
+The above inclusion follows from
+\[
+ \ker(\ev) \sub \bd K_1(M)
+\]
+and
+\[
+ \ker(\pi) \sub \ev(\bd K_1(M)) .
+\]
+Let $x = \sum x_i$ be in the kernel of $\ev$, where each $x_i$ is a configuration of
+labeled points in $S^1$.
+Since the sum is finite, we can find an interval (blob) $B$ in $S^1$
+such that for each $i$ the $C$-labeled points of $x_i$ all lie to the right of the
+base point *.
+Let $y_i$ be the restriction of $x_i$ to $B$ and $y = \sum y_i$.
+Let $r$ be the ``empty" field on $S^1 \setmin B$.
+It follows that $y \in U(B)$ and
+\[
+ \bd(B, y, r) = x .
+\]
+$\ker(\pi)$ is generated by elements of the form $cm - mc$.
+As shown in Figure \ref{fig:hochschild-1-chains}, $cm - mc$ lies in $\ev(\bd K_1(M))$.
\end{proof}
+
\begin{proof}[Proof of Lemma \ref{lem:hochschild-free}]
We show that $K_*(C\otimes C)$ is
quasi-isomorphic to the 0-step complex $C$. We'll do this in steps, establishing quasi-isomorphisms and homotopy equivalences
@@ -334,7 +360,7 @@
given by replacing the restriction $y$ to $N_\ep$ of each field
appearing in an element of $K_*^\ep$ with $s_\ep(y)$.
Note that $\sigma_\ep(x) \in K'_*$.
-\begin{figure}[!ht]
+\begin{figure}[t]
\begin{align*}
y & = \mathfig{0.2}{hochschild/y} &
s_\ep(y) & = \mathfig{0.2}{hochschild/sy}
@@ -413,7 +439,8 @@
Let $x \in K'_k$.
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 $B$.
+We can decompose $x = x' \bullet p$,
+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 $\bd y = x' - x'' - h_{l-1}\bd x'$.
@@ -456,7 +483,7 @@
In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams
as shown in Figure \ref{fig:hochschild-1-chains}.
-\begin{figure}[!ht]
+\begin{figure}[t]
\begin{equation*}
\mathfig{0.4}{hochschild/1-chains}
\end{equation*}
@@ -467,7 +494,7 @@
\label{fig:hochschild-1-chains}
\end{figure}
-\begin{figure}[!ht]
+\begin{figure}[t]
\begin{equation*}
\mathfig{0.6}{hochschild/2-chains-0}
\end{equation*}
@@ -478,7 +505,7 @@
\label{fig:hochschild-2-chains}
\end{figure}
-\begin{figure}[!ht]
+\begin{figure}[t]
\begin{equation*}
A = \mathfig{0.1}{hochschild/v_1} + \mathfig{0.1}{hochschild/v_2} + \mathfig{0.1}{hochschild/v_3} + \mathfig{0.1}{hochschild/v_4}
\end{equation*}