
diff -r e9fac1a6954b -r 62cd552bf8c9 text/evmap.tex
--- a/text/evmap.tex	Tue Jun 30 22:08:56 2009 +0000
+++ b/text/evmap.tex	Sat Jul 04 06:48:22 2009 +0000
@@ -128,7 +128,7 @@
 Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ 
 (by \ref{disjunion} and \ref{bcontract}).
 Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$,
-there is, up to homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$
+there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$
 such that 
 \[
 	\bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) .
@@ -139,13 +139,99 @@
 union of balls.
 On the other hand, Lemma \ref{extension_lemma} allows us to homotope 
 \nn{is this commonly used as a verb?} arbitrary generators to sums of generators with this property.
-\nn{should give a name to this property}
+\nn{should give a name to this property; also forward reference}
 This (roughly) establishes the uniqueness part of the proposition.
 To show existence, we must show that the various choices involved in constructing
 evaluation maps in this way affect the final answer only by a homotopy.
 
-\nn{now for a more detailed outline of the proof...}
+\nn{maybe put a little more into the outline before diving into the details.}
+
+\nn{Note: At the moment this section is very inconsistent with respect to PL versus smooth,
+homeomorphism versus diffeomorphism, etc.
+We expect that everything is true in the PL category, but at the moment our proof
+avails itself to smooth techniques.
+Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$
+rather than $C_*(\Homeo(X))$.}
+
+\medskip
+
+Now for the details.
+
+Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$.
+
+Choose a metric on $X$.
+Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero
+(e.g.\ $\ep_i = 2^{-i}$).
+Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$
+converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$).
+Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $k$
+define
+\[
+	N_{i,k}(p\ot b) \deq \Nbd_{k\ep_i}(|b|) \cup \Nbd_{k\delta_i}(|p|).
+\]
+In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized
+by $k$), with $\ep_i$ controlling the size of the buffer around $|b|$ and $\delta_i$ controlling
+the size of the buffer around $|p|$.
+
+Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$.
+Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b)
+= \deg(p) + \deg(b)$.
+$p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b)
+there exist codimension-zero submanifolds $V_1,\ldots,V_m \sub X$ such that each $V_j$
+is homeomorphic to a disjoint union of balls and
+\[
+	N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b)
+			\subeq V_2 \subeq \cdots \subeq V_m \subeq N_{i,k+m}(p\ot b) .
+\]
+Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$.
+We also require that $b$ is splitable (transverse) along the boundary of each $V_l$.
+
+Note that $G_*^{i,m+1} \subeq G_*^{i,m}$.
 
+As sketched above and explained in detail below, 
+$G_*^{i,m}$ is a subcomplex where it is easy to define
+the evaluation map.
+The parameter $m$ controls the number of iterated homotopies we are able to construct.
+The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of
+$CD_*(X)\ot \bc_*(X)$.
+
+Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$.
+Let $p\ot b \in G_*^{i,m}$.
+If $\deg(p) = 0$, define
+\[
+	e(p\ot b) = p(b) ,
+\]
+where $p(b)$ denotes the obvious action of the diffeomorphism(s) $p$ on the blob diagram $b$.
+For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined
+$e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$.
+Choose $V_1$ as above so that 
+\[
+	N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b) .
+\]
+Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V_1^j$ be the choice of neighborhood
+of $|p_j|\cup |b_j|$ made at the preceding stage of the induction.
+For all $j$, 
+\[
+	V_1^j \subeq N_{i,(k-1)+1}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V_1 .
+\]
+(The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.)
+We therefore have splittings
+\[
+	p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' ,
+\]
+where $p' \in CD_*(V_1)$, $p'' \in CD_*(X\setmin V_1)$, 
+$b' \in \bc_*(V_1)$, $b'' \in \bc_*(X\setmin V_1)$, 
+$e' \in \bc_*(p(V_1))$, and $e'' \in \bc_*(p(X\setmin V_1))$.
+(Note that since the family of diffeomorphisms $p$ is constant (independent of parameters)
+near $\bd V_1)$, the expressions $p(V_1) \sub X$ and $p(X\setmin V_1) \sub X$ are
+unambiguous.)
+We also have that $\deg(b'') = 0 = \deg(p'')$.
+Choose $x' \in \bc_*(p(V_1))$ such that $\bd x' = f'$.
+This is possible by \nn{...}.
+Finally, define
+\[
+	e(p\ot b) \deq x' \bullet p''(b'') .
+\]
 
 
 \medskip




