
diff -r aac9fd8d6bc6 -r ea489bbccfbf blob1.tex
--- a/blob1.tex	Thu Jun 26 17:56:20 2008 +0000
+++ b/blob1.tex	Fri Jun 27 04:24:25 2008 +0000
@@ -611,11 +611,11 @@
 
 \nn{Should say something stronger about uniqueness.
 Something like: there is
-a contractible subcomplex of the complex of chain maps 
+a contractible subcomplex of the complex of chain maps
 $CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.),
 and all choices in the construction lie in the 0-cells of this
 contractible subcomplex.
-Or maybe better to say any two choices are homotopic, and 
+Or maybe better to say any two choices are homotopic, and
 any two homotopies and second order homotopic, and so on.}
 
 \nn{Also need to say something about associativity.
@@ -669,7 +669,7 @@
 The strategy for the proof of Proposition \ref{CDprop} is as follows.
 We will identify a subcomplex
 \[
-	G_* \sub CD_*(X) \otimes \bc_*(X)
+    G_* \sub CD_*(X) \otimes \bc_*(X)
 \]
 on which the evaluation map is uniquely determined (up to homotopy) by the conditions
 in \ref{CDprop}.
@@ -678,12 +678,12 @@
 \nn{need to be more specific here}
 
 Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$.
-Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$ 
+Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$
 intersects at most one blob $B$ of $b$.
 Since $V \cup B$ might not itself be a ball, we need a more careful and complicated definition.
 Choose a metric for $X$.
-We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that 
-$N_\epsilon(b) \cup \supp(p)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon
+We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that
+$\supp(p) \cup N_\epsilon(b)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon
 neighborhood of the support of $b$.
 \nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.}
 
@@ -704,15 +704,16 @@
 let $p\otimes b$ be a generator of $G_k$.
 Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$.
 There is a factorization $p = p' \circ g$, where $g\in \Diff(X)$ and $p'$ is a family of diffeomorphisms which is the identity outside of $C$.
+\scott{Shouldn't this be $p = g\circ p'$?}
 Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$.
-We may assume inductively that $e_X(\bd(p\otimes b))$ has the form $x\bullet g(b'')$, where
+We may assume inductively \scott{why? I don't get this.} that $e_X(\bd(p\otimes b))$ has the form $x\bullet g(b'')$, where
 $x \in \bc_*(g(C))$.
 Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$.
 \nn{need to say more if degree of $x$ is 0}
 Define $e_X(p\otimes b) = y\bullet g(b'')$.
 
 We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made.
-If we make a different series of choice of the chain $y$ in the previous paragraph, 
+If we make a different series of choice of the chain $y$ in the previous paragraph,
 we can inductively construct a homotopy between the two sets of choices,
 again relying on the contractibility of $\bc_*(g(G))$.
 A similar argument shows that this homotopy is unique up to second order homotopy, and so on.
@@ -726,25 +727,25 @@
 a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and
 this homotopy are unique up to iterated homotopy.
 
-Given $k>0$ and a blob diagram $b$, we say that a cover of $X$ $\cU$ is $k$-compatible with
-$b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union 
+Given $k>0$ and a blob diagram $b$, we say that a cover $\cU$ of $X$ is $k$-compatible with
+$b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union
 $U_1\cup\cdots\cup U_k$ is a union of balls which satisfies the condition used to define $G_*$ above.
-Note that if a family of diffeomorphisms $p$ is adapted to 
-$\cU$ and $b$ is a blob diagram occurring in $x$, then $p\otimes b \in G_*$.
+Note that if a family of diffeomorphisms $p$ is adapted to
+$\cU$ and $b$ is a blob diagram occurring in $x$ \scott{huh, what's $x$ here?}, then $p\otimes b \in G_*$.
 \nn{maybe emphasize this more; it's one of the main ideas in the proof}
 
 Let $k$ be the degree of $x$ and choose a cover $\cU$ of $X$ such that $\cU$ is
 $k$-compatible with each of the (finitely many) blob diagrams occurring in $x$.
-We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to 
+We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to
 construct the homotopy to $G_*$.
 First we will construct a homotopy $h \in G_*$ from $\bd x$ to a cycle $z$ such that
 each family of diffeomorphisms $p$ occurring in $z$ is adapted to $\cU$.
 Then we will construct a homotopy (rel boundary) $r$ from $x + h$ to $y$ such that
 each family of diffeomorphisms $p$ occurring in $y$ is adapted to $\cU$.
 This implies that $y \in G_*$.
-$r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek.
+The homotopy $r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek.
 
-We will define $h$ inductive on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$.
+We will define $h$ inductively on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$.
 Define $h$ to be zero on bidegree $(0, k-1)$.
 Let $p\otimes b$ be a generator occurring in $\bd x$ with bidegree $(1, k-2)$.
 Using Lemma \ref{extension_lemma}, construct a homotopy $q$ from $p$ to $p'$ which is adapted to $\cU$.
@@ -761,7 +762,7 @@
 
 The homotopy $r$ is constructed similarly.
 
-\nn{need to say something about uniqueness of $r$, $h$ etc.  
+\nn{need to say something about uniqueness of $r$, $h$ etc.
 postpone this until second draft.}
 
 At this point, we have finished defining the evaluation map.




