# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1282196037 25200
# Node ID a9ac20b0a0c231e8e64de9308ad1a86d9427acee
# Parent  050dba5e7bdd909ae5c2f18dbbabe4d7d12dbd83
intro to evmap

diff -r 050dba5e7bdd -r a9ac20b0a0c2 preamble.tex
--- a/preamble.tex	Wed Aug 18 21:05:50 2010 -0700
+++ b/preamble.tex	Wed Aug 18 22:33:57 2010 -0700
@@ -63,6 +63,7 @@
 \newtheorem{module-axiom}{Module Axiom}
 \newenvironment{rem}{\noindent\textsl{Remark.}}{}  % perhaps looks better than rem above?
 \newtheorem{rem*}[prop]{Remark}
+\newtheorem{remark}[prop]{Remark}
 \numberwithin{equation}{section}
 %% example is defined in article_preamble.tex, for compatibility with beamer.
 
@@ -120,6 +121,7 @@
 
 
 \def\bc{{\mathcal B}}
+\def\btc{{\mathcal{BT}}}
 
 \newcommand{\into}{\hookrightarrow}
 \newcommand{\onto}{\twoheadrightarrow}
diff -r 050dba5e7bdd -r a9ac20b0a0c2 text/blobdef.tex
--- a/text/blobdef.tex	Wed Aug 18 21:05:50 2010 -0700
+++ b/text/blobdef.tex	Wed Aug 18 22:33:57 2010 -0700
@@ -236,6 +236,7 @@
 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram),
 we define $\supp(y) \deq \bigcup_i \supp(b_i)$.
 
+\begin{remark} \label{blobsset-remark} \rm
 We note that blob diagrams in $X$ have a structure similar to that of a simplicial set,
 but with simplices replaced by a more general class of combinatorial shapes.
 Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products
@@ -254,5 +255,5 @@
 (When the fields come from an $n$-category, this correspondence works best if we think of each twig label $u_i$ as having the form
 $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cF(B_i) \to C$ is the evaluation map, 
 and $s:C \to \cF(B_i)$ is some fixed section of $e$.)
+\end{remark}
 
-
diff -r 050dba5e7bdd -r a9ac20b0a0c2 text/evmap.tex
--- a/text/evmap.tex	Wed Aug 18 21:05:50 2010 -0700
+++ b/text/evmap.tex	Wed Aug 18 22:33:57 2010 -0700
@@ -11,16 +11,48 @@
 maybe salvage some of the original version of this section as a subsection outlining
 how one might proceed directly.}
 
+In this section we extend the action of homeomorphisms on $\bc_*(X)$
+to an action of {\it families} of homeomorphisms.
+That is, for each pair of homeomorphic manifolds $X$ and $Y$
+we define a chain map
+\[
+    e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) ,
+\]
+where $CH_*(X, Y) = C_*(\Homeo(X, Y))$, the singular chains on the space
+of homeomorphisms from $X$ to $Y$.
+(If $X$ and $Y$ have non-empty boundary, these families of homeomorphisms
+are required to be fixed on the boundaries.)
+See \S \ref{ss:emap-def} for a more precise statement.
+
+The most convenient way to prove that maps $e_{XY}$ with the desired properties exist is to 
+introduce a homotopy equivalent alternate version of the blob complex $\btc_*(X)$
+which is more amenable to this sort of action.
+Recall from Remark \ref{blobsset-remark} that blob diagrams
+have the structure of a sort-of-simplicial set.
+Blob diagrams can also be equipped with a natural topology, which converts this
+sort-of-simplicial set into a sort-of-simplicial space.
+Taking singular chains of this space we get $\btc_*(X)$.
+The details are in \S \ref{ss:alt-def}.
+For technical reasons we also show that requiring the blobs to be
+embedded yields a homotopy equivalent complex.
+
+Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct
+the $CH_*$ actions directly in terms of $\bc_*(X)$.
+This was our original approach, but working out the details created a nearly unreadable mess.
+We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}.
+
 
 \subsection{Alternative definitions of the blob complex}
+\label{ss:alt-def}
 
 
 \subsection{Action of \texorpdfstring{$\CH{X}$}{C_*(Homeo(M))}}
-
+\label{ss:emap-def}
 
 
 
 \subsection{[older version still hanging around]}
+\label{ss:old-evmap-remnants}
 
 \nn{should comment at the start about any assumptions about smooth, PL etc.}