
diff -r d363611b1f59 -r 5ab0e6f0d89e text/evmap.tex
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+++ b/text/evmap.tex	Fri Jun 05 16:10:37 2009 +0000
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+%!TEX root = ../blob1.tex
+
+Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of
+the space of diffeomorphisms
+\nn{or homeomorphisms}
+between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$).
+For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general
+than simplices --- they can be based on any linear polyhedron.
+\nn{be more restrictive here?  does more need to be said?}
+We also will use the abbreviated notation $CD_*(X) \deq CD_*(X, X)$.
+
+\begin{prop}  \label{CDprop}
+For $n$-manifolds $X$ and $Y$ there is a chain map
+\eq{
+    e_{XY} : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) .
+}
+On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$
+(Proposition (\ref{diff0prop})).
+For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, 
+the following diagram commutes up to homotopy
+\eq{ \xymatrix{
+     CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}}    & \bc_*(Y) \\
+     CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
+        \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}}  \ar[u]^{\gl \otimes \gl}  &
+            \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl}
+} }
+Any other map satisfying the above two properties is homotopic to $e_X$.
+\end{prop}
+
+\nn{need to rewrite for self-gluing instead of gluing two pieces together}
+
+\nn{Should say something stronger about uniqueness.
+Something like: there is
+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
+any two homotopies and second order homotopic, and so on.}
+
+\nn{Also need to say something about associativity.
+Put it in the above prop or make it a separate prop?
+I lean toward the latter.}
+\medskip
+
+The proof will occupy the remainder of this section.
+\nn{unless we put associativity prop at end}
+
+Without loss of generality, we will assume $X = Y$.
+
+\medskip
+
+Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$.
+We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all
+$x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if there is a family of diffeomorphisms $f' : P \times S \to S$ and a `background'
+diffeomorphism $f_0 : X \to X$ so that
+\begin{align}
+	f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\
+\intertext{and}
+	f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}.
+\end{align}
+Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$.
+
+Let $\cU = \{U_\alpha\}$ be an open cover of $X$.
+A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is
+{\it adapted to $\cU$} if there is a factorization
+\eq{
+    P = P_1 \times \cdots \times P_m
+}
+(for some $m \le k$)
+and families of diffeomorphisms
+\eq{
+    f_i :  P_i \times X \to X
+}
+such that
+\begin{itemize}
+\item each $f_i$ is supported on some connected $V_i \sub X$;
+\item the sets $V_i$ are mutually disjoint;
+\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s,
+where $k_i = \dim(P_i)$; and
+\item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$
+for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$.
+\end{itemize}
+A chain $x \in CD_k(X)$ is (by definition) adapted to $\cU$ if it is the sum
+of singular cells, each of which is adapted to $\cU$.
+
+(Actually, in this section we will only need families of diffeomorphisms to be 
+{\it weakly adapted} to $\cU$, meaning that the support of $f$ is contained in the union
+of at most $k$ of the $U_\alpha$'s.)
+
+\begin{lemma}  \label{extension_lemma}
+Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.
+Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
+Furthermore, one can choose the homotopy so that its support is equal to the support of $x$.
+\end{lemma}
+
+The proof will be given in Section \ref{sec:localising}.
+
+\medskip
+
+Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}.
+
+Let $p$ be a singular cell in $CD_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$.
+Suppose that there exists $V \sub X$ such that
+\begin{enumerate}
+\item $V$ is homeomorphic to a disjoint union of balls, and
+\item $\supp(p) \cup \supp(b) \sub V$.
+\end{enumerate}
+Let $W = X \setmin V$, and let $V' = p(V)$ and $W' = p(W)$.
+We then have a factorization 
+\[
+	p = \gl(q, r),
+\]
+where $q \in CD_k(V, V')$ and $r' \in CD_0(W, W')$.
+According to the commutative diagram of the proposition, we must have
+\[
+	e_X(p) = e_X(\gl(q, r)) = gl(e_{VV'}(q), e_{WW'}(r)) .
+\]
+\nn{need to add blob parts to above}
+Since $r$ is a plain, 0-parameter family of diffeomorphisms, 
+\medskip
+
+\nn{to be continued....}
+
+
+%\nn{say something about associativity here}
+
+
+
+
+




