--- a/text/deligne.tex Thu Aug 11 13:54:38 2011 -0700
+++ b/text/deligne.tex Thu Aug 11 22:14:11 2011 -0600
@@ -178,7 +178,8 @@
p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k))
\to \hom(\bc_*(M_0), \bc_*(N_0)) .
\]
-Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$) to be the composition
+Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define
+$p(\ol{f})(\alpha_1\ot\cdots\ot\alpha_k)$ to be the composition
\[
\bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1)
\stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1)
@@ -201,7 +202,7 @@
\label{thm:deligne}
There is a collection of chain maps
\[
- C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes
+ C_*(SC^n_{\ol{M}\ol{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes
\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0))
\]
which satisfy the operad compatibility conditions.
@@ -216,7 +217,7 @@
a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad.
\begin{proof}
-As described above, $SC^n_{\overline{M}, \overline{N}}$ is equal to the disjoint
+As described above, $SC^n_{\ol{M}\ol{N}}$ is equal to the disjoint
union of products of homeomorphism spaces, modulo some relations.
By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$
a chain map
@@ -225,7 +226,7 @@
\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) .
\]
It suffices to show that the above maps are compatible with the relations whereby
-$SC^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s.
+$SC^n_{\ol{M}\ol{N}}$ is constructed from the various $P$'s.
This in turn follows easily from the fact that
the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative.
%\nn{should add some detail to above}