--- a/text/evmap.tex Tue Mar 30 10:03:48 2010 -0700
+++ b/text/evmap.tex Tue Mar 30 15:12:27 2010 -0700
@@ -1,37 +1,37 @@
%!TEX root = ../blob1.tex
-\section{Action of \texorpdfstring{$\CD{X}$}{$C_*(Diff(M))$}}
+\section{Action of \texorpdfstring{$\CH{X}$}{$C_*(Homeo(M))$}}
\label{sec:evaluation}
-Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of
-the space of diffeomorphisms
-\nn{or homeomorphisms; need to fix the diff vs homeo inconsistency}
-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
+Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of
+the space of homeomorphisms
+\nn{need to fix the diff vs homeo inconsistency}
+between the $n$-manifolds $X$ and $Y$ (extending a fixed homeomorphism $\bd X \to \bd Y$).
+For convenience, we will permit the singular cells generating $CH_*(X, Y)$ 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 $CD_*(X) \deq CD_*(X, X)$.
+We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$.
-\begin{prop} \label{CDprop}
+\begin{prop} \label{CHprop}
For $n$-manifolds $X$ and $Y$ there is a chain map
\eq{
- e_{XY} : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) .
+ e_{XY} : CH_*(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 $\bc_*(X)$
+On $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Homeo(X, Y)$ on $\bc_*(X)$
(Proposition (\ref{diff0prop})).
For any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$,
the following diagram commutes up to homotopy
\eq{ \xymatrix{
- CD_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) \\
- CD_*(X, Y) \otimes \bc_*(X)
+ CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]^(.7){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) \\
+ CH_*(X, Y) \otimes \bc_*(X)
\ar@/_4ex/[r]_{e_{XY}} \ar[u]^{\gl \otimes \gl} &
\bc_*(Y) \ar[u]_{\gl}
} }
%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)
+% CH_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\
+% CH_*(X_1, Y_1) \otimes CH_*(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}
%} }
@@ -43,7 +43,7 @@
\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.),
+$CH_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (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
@@ -61,10 +61,10 @@
\medskip
-Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$.
+Let $f: P \times X \to X$ be a family of homeomorphisms 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
+$x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if there is a family of homeomorphisms $f' : P \times S \to S$ and a `background'
+homeomorphism $f_0 : X \to X$ so that
\begin{align}
f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\
\intertext{and}
@@ -73,13 +73,13 @@
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
+A $k$-parameter family of homeomorphisms $f: P \times X \to X$ 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
+and families of homeomorphisms
\eq{
f_i : P_i \times X \to X
}
@@ -90,18 +90,18 @@
\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 $g \in \Diff(X)$.
+for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$.
\end{itemize}
-A chain $x \in CD_k(X)$ is (by definition) adapted to $\cU$ if it is the sum
+A chain $x \in CH_k(X)$ 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
+(Actually, in this section we will only need families of homeomorphisms 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 $\cU$.
+Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.
+Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$.
Furthermore, one can choose the homotopy so that its support is equal to the support of $x$.
\end{lemma}
@@ -127,10 +127,10 @@
\medskip
-Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}.
+Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}.
%Suppose for the moment that evaluation maps with the advertised properties exist.
-Let $p$ be a singular cell in $CD_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$.
+Let $p$ be a singular cell in $CH_k(X)$ 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
@@ -141,18 +141,18 @@
\[
p = \gl(q, r),
\]
-where $q \in CD_k(V, V')$ and $r \in CD_0(W, W')$.
+where $q \in CH_k(V, V')$ and $r \in CH_0(W, W')$.
We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$.
According to the commutative diagram of the proposition, we must have
\[
e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) =
gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) .
\]
-Since $r$ is a plain, 0-parameter family of diffeomorphisms, we must have
+Since $r$ is a plain, 0-parameter family of homeomorphisms, we must have
\[
e_{WW'}(r\otimes b_W) = r(b_W),
\]
-where $r(b_W)$ denotes the obvious action of diffeomorphisms on blob diagrams (in
+where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in
this case a 0-blob diagram).
Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$
(by \ref{disjunion} and \ref{bcontract}).
@@ -175,8 +175,7 @@
\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.
+\nn{Note: At the moment this section is very inconsistent with respect to PL versus smooth, 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 $C_*(\Diff(X))$
@@ -195,7 +194,7 @@
converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$).
Let $\phi_l$ be an increasing sequence of positive numbers
satisfying the inequalities of Lemma \ref{xx2phi}.
-Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$
+Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$
define
\[
N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|).
@@ -204,8 +203,8 @@
by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling
the size of the buffers 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)
+Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$.
+Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ 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 $V_0,\ldots,V_m \sub X$ such that each $V_j$
@@ -225,7 +224,7 @@
The parameter $m$ controls the number of iterated homotopies we are able to construct
(see Lemma \ref{m_order_hty}).
The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of
-$CD_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}).
+$CH_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}).
Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$.
Let $p\ot b \in G_*^{i,m}$.
@@ -233,7 +232,7 @@
\[
e(p\ot b) = p(b) ,
\]
-where $p(b)$ denotes the obvious action of the diffeomorphism(s) $p$ on the blob diagram $b$.
+where $p(b)$ denotes the obvious action of the homeomorphism(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 $V = V_0$ as above so that
@@ -251,10 +250,10 @@
\[
p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet b'' ,
\]
-where $p' \in CD_*(V)$, $p'' \in CD_*(X\setmin V)$,
+where $p' \in CH_*(V)$, $p'' \in CH_*(X\setmin V)$,
$b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$,
$f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$.
-(Note that since the family of diffeomorphisms $p$ is constant (independent of parameters)
+(Note that since the family of homeomorphisms $p$ is constant (independent of parameters)
near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are
unambiguous.)
We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$.
@@ -316,7 +315,7 @@
p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 ,
\;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 ,
\]
-where $p'_1 \in CD_*(V_1)$, $p''_1 \in CD_*(X\setmin V_1)$,
+where $p'_1 \in CH_*(V_1)$, $p''_1 \in CH_*(X\setmin V_1)$,
$b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$,
$f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$.
Inductively, $\bd f'_1 = 0$ and $f_1'' = p_1''(b_1'')$.
@@ -335,11 +334,11 @@
An easy variation on the above lemma shows that $e_{i,m}$ and $e_{i,m+1}$ are $m$-th
order homotopic.
-Next we show how to homotope chains in $CD_*(X)\ot \bc_*(X)$ to one of the
+Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the
$G_*^{i,m}$.
Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero.
Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$.
-Let $h_j: CD_*(X)\to CD_*(X)$ be a chain map homotopic to the identity whose image is spanned by diffeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}.
+Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}.
Recall that $h_j$ and also the homotopy connecting it to the identity do not increase
supports.
Define
@@ -352,9 +351,9 @@
(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.)
\begin{lemma} \label{Gim_approx}
-Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$.
+Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$.
Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$
-there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$
+there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CH_n(X)$
we have $g_j(p)\ot b \in G_*^{i,m}$.
\end{lemma}
@@ -387,7 +386,7 @@
\]
and also so that $\phi_t \gamma_j$ is less than the constant $\rho(M)$ of Lemma \ref{xxzz11}.
-Let $j \ge j_i$ and $p\in CD_n(X)$.
+Let $j \ge j_i$ and $p\in CH_n(X)$.
Let $q$ be a generator appearing in $g_j(p)$.
Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$,
which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$.
@@ -516,9 +515,9 @@
\begin{itemize}
\item We need to assemble the maps for the various $G^{i,m}$ into
-a map for all of $CD_*\ot \bc_*$.
-One idea: Think of the $g_j$ as a sort of homotopy (from $CD_*\ot \bc_*$ to itself)
-parameterized by $[0,\infty)$. For each $p\ot b$ in $CD_*\ot \bc_*$ choose a sufficiently
+a map for all of $CH_*\ot \bc_*$.
+One idea: Think of the $g_j$ as a sort of homotopy (from $CH_*\ot \bc_*$ to itself)
+parameterized by $[0,\infty)$. For each $p\ot b$ in $CH_*\ot \bc_*$ choose a sufficiently
large $j'$. Use these choices to reparameterize $g_\bullet$ so that each
$p\ot b$ gets pushed as far as the corresponding $j'$.
\item Independence of metric, $\ep_i$, $\delta_i$: