# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1318457454 25200
# Node ID 7afa2ffbbac8fb57a9775a5526714b1b2a7c48a4
# Parent  fab3d057beebb55e57e52cdfe6113ee6e5d43462
operad action only up to homotopy; still need to think about this a bit

diff -r fab3d057beeb -r 7afa2ffbbac8 blob_changes_v3
--- a/blob_changes_v3	Sat Oct 08 17:35:05 2011 -0700
+++ b/blob_changes_v3	Wed Oct 12 15:10:54 2011 -0700
@@ -33,4 +33,5 @@
 - rewrote definition of colimit (in "From Balls to Manifolds" subsection) to allow for more general decompositions; also added more details
 - added remark about families of collar maps acting on the blob complex
 - small corrections to proof of product theorem (7.1.1)
-- 
+- added remarks that various homotopy equivalences we construct are well-defined up to a contractible set of choices
+
diff -r fab3d057beeb -r 7afa2ffbbac8 text/deligne.tex
--- a/text/deligne.tex	Sat Oct 08 17:35:05 2011 -0700
+++ b/text/deligne.tex	Wed Oct 12 15:10:54 2011 -0700
@@ -205,7 +205,7 @@
 	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.
+which satisfy the operad compatibility conditions, up to coherent homotopy.
 On $C_0(SC^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above.
 When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}.
 \end{thm}
@@ -228,7 +228,8 @@
 It suffices to show that the above maps are compatible with the relations whereby
 $SC^n_{\ol{M}\ol{N}}$ 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.
+the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative
+(up to coherent homotopy).
 %\nn{should add some detail to above}
 \end{proof}
 
diff -r fab3d057beeb -r 7afa2ffbbac8 text/evmap.tex
--- a/text/evmap.tex	Sat Oct 08 17:35:05 2011 -0700
+++ b/text/evmap.tex	Wed Oct 12 15:10:54 2011 -0700
@@ -391,14 +391,21 @@
 $h_2(b) \in \btc_3(X)$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$
 
 The general case, $h_k$, is similar.
+
+Note that it is possible to make the various choices above so that the homotopies we construct
+are fixed on $\bc_* \sub \btc_*$.
+It follows that we may assume that
+the homotopy inverse to the inclusion constructed above is the identity on $\bc_*$.
+Note that the complex of all homotopy inverses with this property is contractible, 
+so the homotopy inverse is well-defined up to a contractible set of choices.
 \end{proof}
 
-The proof of Lemma \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion
-$\bc_*(X)\sub \btc_*(X)$.
-One might ask for more: a contractible set of possible homotopy inverses, or at least an
-$m$-connected set for arbitrarily large $m$.
-The latter can be achieved with finer control over the various
-choices of disjoint unions of balls in the above proofs, but we will not pursue this here.
+%The proof of Lemma \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion
+%$\bc_*(X)\sub \btc_*(X)$.
+%One might ask for more: a contractible set of possible homotopy inverses, or at least an
+%$m$-connected set for arbitrarily large $m$.
+%The latter can be achieved with finer control over the various
+%choices of disjoint unions of balls in the above proofs, but we will not pursue this here.
 
 
 
@@ -419,7 +426,7 @@
 \eq{
     e_{XY} : \CH{X \to Y} \otimes \bc_*(X) \to \bc_*(Y) ,
 }
-well-defined up to homotopy,
+well-defined up to (coherent) homotopy,
 such that
 \begin{enumerate}
 \item on $C_0(\Homeo(X \to Y)) \otimes \bc_*(X)$ it agrees with the obvious action of 
@@ -459,7 +466,7 @@
 \begin{thm}
 \label{thm:CH-associativity}
 The $\CH{X \to Y}$ actions defined above are associative.
-That is, the following diagram commutes up to homotopy:
+That is, the following diagram commutes up to coherent homotopy:
 \[ \xymatrix@C=5pt{
 & \CH{Y\to Z} \ot \bc_*(Y) \ar[drr]^{e_{YZ}} & &\\
 \CH{X \to Y} \ot \CH{Y \to Z} \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & & \bc_*(Z) \\
diff -r fab3d057beeb -r 7afa2ffbbac8 text/intro.tex
--- a/text/intro.tex	Sat Oct 08 17:35:05 2011 -0700
+++ b/text/intro.tex	Wed Oct 12 15:10:54 2011 -0700
@@ -460,7 +460,8 @@
 \newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}}
 
 \begin{thm:deligne}[Higher dimensional Deligne conjecture]
-The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains.
+The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains
+(up to coherent homotopy).
 Since the little $n{+}1$-balls operad is a suboperad of the $n$-dimensional surgery cylinder operad,
 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball.
 \end{thm:deligne}