# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1282931901 25200
# Node ID 352389c6ddcf218fad8c3920c863a6201fda88a8
# Parent  a60c035e53bdbf762f9d677be219fc584ccd99d5
more on evmap

diff -r a60c035e53bd -r 352389c6ddcf text/evmap.tex
--- a/text/evmap.tex	Thu Aug 26 13:20:13 2010 -0700
+++ b/text/evmap.tex	Fri Aug 27 10:58:21 2010 -0700
@@ -56,7 +56,7 @@
 
 If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted
 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$.
-For a general $k-chain$ $a\in \bc_k(X)$, define the support of $a$ to be the union
+For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union
 of the supports of the blob diagrams which appear in it.
 
 If $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is 
@@ -75,15 +75,15 @@
 and moreover each field labeling a region cut out by the blobs is splittable 
 into fields on smaller regions, each of which is contained in some open set of $\cU$.
 
-\begin{thm}[Small blobs] \label{thm:small-blobs-xx}
+\begin{lemma}[Small blobs] \label{small-blobs-b}
 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence.
-\end{thm}
+\end{lemma}
 
 \begin{proof}
 It suffices to show that for any finitely generated pair of subcomplexes 
 $(C_*, D_*) \sub (\bc_*(X), \sbc_*(X))$
 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$
-and $x + h\bd(x) + \bd h(X) \in \sbc_*(X)_*(X)$ for all $x\in C_*$.
+and $x + h\bd(x) + \bd h(X) \in \sbc_*(X)$ for all $x\in C_*$.
 
 For simplicity we will assume that all fields are splittable into small pieces, so that
 $\sbc_0(X) = \bc_0$.
@@ -297,6 +297,73 @@
 \end{align*}
 \end{proof}
 
+\begin{lemma}
+For manifolds $X$ and $Y$, we have $\btc_*(X\du Y) \simeq \btc_*(X)\otimes\btc_*(Y)$.
+\end{lemma}
+\begin{proof}
+This follows from the Eilenber-Zilber theorem and the fact that
+\[
+	\BD_k(X\du Y) \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) .
+\]
+\end{proof}
+
+For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$}
+if there exists $S' \subeq S$, $a'\in \btc_k(S')$
+and $r\in \btc_0(X\setmin S')$ such that $a = a'\bullet r$.
+
+\newcommand\sbtc{\btc^{\cU}}
+Let $\cU$ be an open cover of $X$.
+Let $\sbtc_*(X)\sub\btc_*(X)$ be the subcomplex generated by
+$a\in \btc_*(X)$ such that there is a decomposition $X = \cup_i D_i$
+such that each $D_i$ is a ball contained in some open set of $\cU$ and
+$a$ is splittable along this decomposition.
+In other words, $a$ can be obtained by gluing together pieces, each of which
+is small with respect to $\cU$.
+
+\begin{lemma} \label{small-top-blobs}
+For any open cover $\cU$ of $X$, the inclusion $\sbtc_*(X)\sub\btc_*(X)$
+is a homotopy equivalence.
+\end{lemma}
+\begin{proof}
+This follows from a combination of Lemma \ref{extension_lemma_c} and the techniques of
+the proof of Lemma \ref{small-blobs-b}.
+
+It suffices to show that we can deform a finite subcomplex $C_*$ of $\btc_*(X)$ into $\sbtc_*(X)$
+(relative to any designated subcomplex of $C_*$ already in $\sbtc_*(X)$).
+The first step is to replace families of general blob diagrams with families that are 
+small with respect to $\cU$.
+This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families.
+Each such family is homotopic to a sum families which can be a ``lifted" to $\Homeo(X)$.
+That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$.
+(We are ignoring a complication related to twig blob labels, which might vary
+independently of $g$, but this complication does not affect the conclusion we draw here.)
+We now apply Lemma \ref{extension_lemma_c} to get families which are supported 
+on balls $D_i$ contained in open sets of $\cU$.
+\end{proof}
+
+
+\begin{proof}[Proof of \ref{lem:bt-btc}]
+Armed with the above lemmas, we can now proceed similarly to the proof of \ref{small-blobs-b}.
+
+It suffices to show that for any finitely generated pair of subcomplexes 
+$(C_*, D_*) \sub (\btc_*(X), \bc_*(X))$
+we can find a homotopy $h:C_*\to \btc_*(X)$ such that $h(D_*) \sub \bc_*(X)$
+and $x + h\bd(x) + \bd h(X) \in \bc_*(X)$ for all $x\in C_*$.
+
+By Lemma \ref{small-top-blobs}, we may assume that $C_* \sub \btc_*^\cU(X)$ for some
+cover $\cU$ of our choosing.
+We choose $\cU$ fine enough so that each generator of $C_*$ is supported on a disjoint union of balls.
+(This is possible since the original $C_*$ was finite and therefore had bounded dimension.)
+
+Since $\bc_0(X) = \btc_0(X)$, we can take $h_0 = 0$.
+
+
+
+
+\nn{...}
+\end{proof}
+
+