--- a/.hgignore Tue Mar 30 16:31:29 2010 -0700
+++ b/.hgignore Tue Mar 30 16:48:26 2010 -0700
@@ -14,3 +14,13 @@
*.bbl
*.blg
bibliography
+
+# generic OSX crap
+.DS_Store
+
+# latex2pdf temporary files
+diagrams/latex2pdf/defontify.dvi
+diagrams/latex2pdf/defontify.pdf
+diagrams/latex2pdf/defontify.ps
+diagrams/latex2pdf/nofonts.ps
+
--- a/diagrams/latex2pdf/defontify.tex Tue Mar 30 16:31:29 2010 -0700
+++ b/diagrams/latex2pdf/defontify.tex Tue Mar 30 16:48:26 2010 -0700
@@ -15,7 +15,12 @@
$n$-category composition
\newcommand{\cN}{\mathcal{N}}
-$\cN_1 \cN_2 \cN_3 B M B \times \bdy W M \times W$
+$N_0 N_1 N_2 N_3 M_0 M_1 M_2 M_3 I$
+
+$n-\abs{\Gamma}$ edges
+
+$v$
+$\Gamma$
\begin{align*}
abmab & \\
Binary file diagrams/pdf/deligne/intervals.pdf has changed
Binary file diagrams/pdf/deligne/manifolds.pdf has changed
Binary file diagrams/pdf/tempkw/delfig1.pdf has changed
Binary file diagrams/pdf/tempkw/delfig2.pdf has changed
--- a/text/comm_alg.tex Tue Mar 30 16:31:29 2010 -0700
+++ b/text/comm_alg.tex Tue Mar 30 16:48:26 2010 -0700
@@ -95,13 +95,13 @@
\end{proof}
-\begin{prop} \label{ktcdprop}
+\begin{prop} \label{ktchprop}
The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$.
\end{prop}
\begin{proof}
The actions agree in degree 0, and both are compatible with gluing.
-(cf. uniqueness statement in \ref{CDprop}.)
+(cf. uniqueness statement in \ref{CHprop}.)
\end{proof}
\medskip
@@ -128,7 +128,7 @@
and is zero for $i\ge 2$.
\nn{say something about $t$-degrees also matching up?}
-By xxxx and \ref{ktcdprop},
+By xxxx and \ref{ktchprop},
the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$.
Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$.
If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree
--- a/text/deligne.tex Tue Mar 30 16:31:29 2010 -0700
+++ b/text/deligne.tex Tue Mar 30 16:48:26 2010 -0700
@@ -32,7 +32,7 @@
\end{eqnarray*}
See Figure \ref{delfig1}.
\begin{figure}[!ht]
-$$\mathfig{.9}{tempkw/delfig1}$$
+$$\mathfig{.9}{deligne/intervals}$$
\caption{A fat graph}\label{delfig1}\end{figure}
We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval
@@ -53,7 +53,7 @@
It should now be clear how to generalize this to higher dimensions.
In the sequence-of-surgeries description above, we never used the fact that the manifolds
involved were 1-dimensional.
-Thus we can define a $n$-dimensional fat graph to sequence of general surgeries
+Thus we can define a $n$-dimensional fat graph to be a sequence of general surgeries
on an $n$-manifold.
More specifically,
the $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries
@@ -61,7 +61,7 @@
$f_i: R_i\cup N_i \to R_{i+1}\cup M_{i+1}$.
(See Figure \ref{delfig2}.)
\begin{figure}[!ht]
-$$\mathfig{.9}{tempkw/delfig2}$$
+$$\mathfig{.9}{deligne/manifolds}$$
\caption{A fat graph}\label{delfig2}\end{figure}
The components of the $n$-dimensional fat graph operad are indexed by tuples
$(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$.
@@ -82,9 +82,9 @@
\label{prop:deligne}
There is a collection of maps
\begin{eqnarray*}
- C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_0), \bc_*(N_0))\otimes\cdots\otimes
-\mapinf(\bc_*(M_{k-1}), \bc_*(N_{k-1})) & \\
- & \hspace{-11em}\to \mapinf(\bc_*(M_k), \bc_*(N_k))
+ C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes
+\mapinf(\bc_*(M_{k}), \bc_*(N_{k})) & \\
+ & \hspace{-11em}\to \mapinf(\bc_*(M_0), \bc_*(N_0))
\end{eqnarray*}
which satisfy an operad type compatibility condition. \nn{spell this out}
\end{prop}
--- a/text/evmap.tex Tue Mar 30 16:31:29 2010 -0700
+++ b/text/evmap.tex Tue Mar 30 16:48:26 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, $\ep_i$, $\delta_i$:
--- a/text/ncat.tex Tue Mar 30 16:31:29 2010 -0700
+++ b/text/ncat.tex Tue Mar 30 16:48:26 2010 -0700
@@ -443,7 +443,7 @@
which fix $\bd X$.
These action maps are required to be associative up to homotopy
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that
-a diagram like the one in Proposition \ref{CDprop} commutes.
+a diagram like the one in Proposition \ref{CHprop} commutes.
\nn{repeat diagram here?}
\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}
\end{axiom-numbered}
@@ -967,7 +967,7 @@
which fix $\bd M$.
These action maps are required to be associative up to homotopy
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that
-a diagram like the one in Proposition \ref{CDprop} commutes.
+a diagram like the one in Proposition \ref{CHprop} commutes.
\nn{repeat diagram here?}
\nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}}
--- a/text/smallblobs.tex Tue Mar 30 16:31:29 2010 -0700
+++ b/text/smallblobs.tex Tue Mar 30 16:48:26 2010 -0700
@@ -4,15 +4,16 @@
Fix $\cU$, an open cover of $M$. Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$. Say that an open cover $\cV$ is strictly subordinate to $\cU$ if every open set of $\cV$ is contained in some closed set which is contained in some open set of $\cU$.
\begin{lem}
-For any open cover $\cU$ of $M$ and strictly subordinate open cover $\cV$, we can choose an up-to-homotopy representative $\ev_{X,\cU,\cV}$ of the chain map $\ev_X$ of Property ?? which gives the action of families of homeomorphisms, so that the restriction of $\ev_{X,\cU,\cV} : \CH{X} \tensor \bc_*(X) \to \bc_*(X)$ to the subcomplex $\CH{X} \tensor \bc^{\cV}_*(X)$ has image contained in the small blob complex $\bc^{\cU}_*(X)$.
+\label{lem:CH-small-blobs}
+For any open cover $\cU$ of $M$ and strictly subordinate open cover $\cV$, and for any $k \in \Natural$, we can choose an up-to-homotopy representative $\ev_{M,\cU,\cV,k}$ of the chain map $\ev_M$ of Property \ref{property:evaluation} which gives the action of families of homeomorphisms, which restricts to give a map $$\ev_{M,\cU,\cV,k} : C_{*\leq k}(\Homeo(M)) \tensor \bc^{\cV}_*(M) \to \bc^{\cU}_*(M).$$
\end{lem}
\begin{rem}
-This says that while we can't quite get a map $\CH{X} \tensor \bc^{\cU}_*(X) \to \bc^{\cU}_*(X)$, we can get by if we give ourselves arbitrarily little room to maneuver, by making the blobs we act on slightly smaller.
+This says that while we can't quite get a map $\CH{M} \tensor \bc^{\cU}_*(M) \to \bc^{\cU}_*(M)$, we can get by if we give ourselves arbitrarily little room to maneuver, by making the blobs we act on slightly smaller.
\end{rem}
\begin{proof}
-\todo{We have to choose the open cover differently for each $k$...}
We choose yet another open cover, $\cW$, which so fine that the union (disjoint or not) of any one open set $V \in \cV$ with $k$ open sets $W_i \in \cW$ is contained in a disjoint union of open sets of $\cU$.
-\todo{explain why we can do this, and then why it works.}
+Now, in the proof of Proposition \ref{CHprop}
+\todo{I think I need to understand better that proof before I can write this!}
\end{proof}
\begin{thm}[Small blobs]
@@ -24,12 +25,12 @@
On $0$-blobs, $s$ is just the identity; a blob diagram without any blobs is compatible with any open cover. Nevertheless, we'll begin introducing nomenclature at this point: for configuration $\beta$ of disjoint embedded balls in $M$ we'll associate a one parameter family of homeomorphisms $\phi_\beta : \Delta^1 \to \Homeo(M)$ (here $\Delta^m$ is the standard simplex $\setc{\mathbf{x} \in \Real^{m+1}}{\sum_{i=0}^m x_i = 1}$). For $0$-blobs, where $\beta = \eset$, all these homeomorphisms are just the identity.
-\todo{have to decide which open cover we're going to use in the action of homeomorphisms, and then ensure that we make $\beta$ sufficiently small to apply the lemma above.}
+When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ `makes $\beta$ small' if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$. Further, we'll say a homeomorphism `makes $\beta$ $\epsilon$-small' if the image of each ball is contained in some open ball of radius $\epsilon$.
-On a $1$-blob $b$, with ball $\beta$, $s$ is defined as the sum of two terms. Essentially, the first term `makes $\beta$ small', while the other term `gets the boundary right'. First, pick a one-parameter family $\phi_\beta : \Delta^1 \to \Homeo(M)$ of homeomorphisms, so $\phi_\beta(1,0)$ is the identity and $\phi_\beta(0,1)$ makes the ball $\beta$ small. Next, pick a two-parameter family $\phi_{\eset \prec \beta} : \Delta^2 \to \Homeo(M)$ so that $\phi_{\eset \prec \beta}(0,x_1,x_2)$ makes the ball $\beta$ small for all $x_1+x_2=1$, while $\phi_{\eset \prec \beta}(x_0,0,x_2) = \phi_\eset(x_0,x_2)$ and $\phi_{\eset \prec \beta}(x_0,x_1,0) = \phi_\beta(x_0,x_1)$. (It's perhaps not obvious that this is even possible --- see Lemma \ref{lem:extend-small-homeomorphisms} below.) We now define $s$ by
+On a $1$-blob $b$, with ball $\beta$, $s$ is defined as the sum of two terms. Essentially, the first term `makes $\beta$ small', while the other term `gets the boundary right'. First, pick a one-parameter family $\phi_\beta : \Delta^1 \to \Homeo(M)$ of homeomorphisms, so $\phi_\beta(1,0)$ is the identity and $\phi_\beta(0,1)$ makes the ball $\beta$ small --- in fact, not just small with respect to $\cU$, but $\epsilon/2$-small, where $\epsilon > 0$ is such that every $\epsilon$ ball is contained in some open set of $\cU$. Next, pick a two-parameter family $\phi_{\eset \prec \beta} : \Delta^2 \to \Homeo(M)$ so that $\phi_{\eset \prec \beta}(0,x_1,x_2)$ makes the ball $\beta$ $\frac{3\epsilon}{4}$-small for all $x_1+x_2=1$, while $\phi_{\eset \prec \beta}(x_0,0,x_2) = \phi_\eset(x_0,x_2)$ and $\phi_{\eset \prec \beta}(x_0,x_1,0) = \phi_\beta(x_0,x_1)$. (It's perhaps not obvious that this is even possible --- see Lemma \ref{lem:extend-small-homeomorphisms} below.) We now define $s$ by
$$s(b) = \restrict{\phi_\beta}{x_0=0}(b) + \restrict{\phi_{\eset \prec \beta}}{x_0=0}(\bdy b).$$
-Here, $\restrict{\phi_\beta}{x_0=0} = \phi_\beta(0,1)$ is just a homeomorphism, which we apply to $b$, while $\restrict{\phi_{\eset \prec \beta}}{x_0=0}$ is a one parameter family of homeomorphisms which acts on the $0$-blob $\bdy b$ to give a $1$-blob.
-\todo{Does $s$ actually land in small blobs?}
+Here, $\restrict{\phi_\beta}{x_0=0} = \phi_\beta(0,1)$ is just a homeomorphism, which we apply to $b$, while $\restrict{\phi_{\eset \prec \beta}}{x_0=0}$ is a one parameter family of homeomorphisms which acts on the $0$-blob $\bdy b$ to give a $1$-blob. To be precise, this action is via the chain map identified in Lemma \ref{lem:CH-small-blobs} as $\ev_{M, \cU, \cV, 1}$, where $\cV$ is the open cover by $\epsilon/2$ balls. From this, it is immediate that $s(b) \in \bc^{\cU}_1(M)$, as desired.
+
We now check that $s$, as defined so far, is a chain map, calculating
\begin{align*}
\bdy (s(b)) & = \restrict{\phi_\beta}{x_0=0}(\bdy b) + (\bdy \restrict{\phi_{\eset \prec \beta}}{x_0=0})(\bdy b) \\
@@ -37,9 +38,12 @@
& = \restrict{\phi_\eset}{x_0=0}(\bdy b) \\
& = s(\bdy b)
\end{align*}
-Next, we compute the compositions $s \circ i$ and $i \circ s$. If we start with a small $1$-blob diagram $b$, first include it up to the full blob complex then apply $s$, we get exactly back to $b$, at least assuming we adopt the convention that for any ball $\beta$ which is already small, we choose the families of homeomorphisms $\phi_\beta$ and $\phi_{\eset \prec \beta}$ to always be the identity. In the other direction, $i \circ s$, we will need to construct a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ for $*=0$ or $1$. This is defined by $h(b) = \phi_\eset(b)$ when $b$ is a $0$-blob (here $\phi_\eset$ is a one parameter family of homeomorphisms, so this is a $1$-blob), and $h(b) = \phi_\beta(b) + \phi_{\eset \prec \beta}(\bdy b)$ when $b$ is a $1$-blob (here $\beta$ is the ball in $b$, and the first term is the action of a one parameter family of homeomorphisms on a $1$-blob, and the second term is the action of a two parameter family of homeomorphisms on a $0$-blob, so both are $2$-blobs).
+Next, we compute the compositions $s \circ i$ and $i \circ s$. If we start with a small $1$-blob diagram $b$, first include it up to the full blob complex then apply $s$, we get exactly back to $b$, at least assuming we adopt the convention that for any ball $\beta$ which is already small, we choose the families of homeomorphisms $\phi_\beta$ and $\phi_{\eset \prec \beta}$ to always be the identity. In the other direction, $i \circ s$, we will need to construct a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ for $*=0$ or $1$.
+In what follows, it will be necessary to use different actions of families of homeomorphisms at different stages. We'll write $\ev_k$ for the chain map $\ev_{M,\cU, \cV_k, k}$ from Lemma \ref{lem:CH-small-blobs}, where $\cV_k$ is the open cover by $\epsilon(1-2^{-k})$ balls.
+
+The homotopy $h$ is defined by $$h(b) = \ev_1(\phi_\eset, b)$$ when $b$ is a $0$-blob (here $\phi_\eset$ is a one parameter family of homeomorphisms, so this is a $1$-blob), and $$h(b) = \ev_1(\phi_\beta,b) + \ev_2(\phi_{\eset \prec \beta},\bdy b)$$ when $b$ is a $1$-blob (here $\beta$ is the ball in $b$, and the first term is the action of a one parameter family of homeomorphisms on a $1$-blob, and the second term is the action of a two parameter family of homeomorphisms on a $0$-blob, so both are $2$-blobs).
\begin{align*}
-(\bdy h+h \bdy)(b) & = \bdy (\phi_{\beta}(b) + \phi_{\eset \prec \beta}{\bdy b}) + \phi_\eset(\bdy b) \\
+(\bdy h+h \bdy)(b) & = \bdy (\phi_{\beta}(b) + \phi_{\eset \prec \beta}(\bdy b)) + \phi_\eset(\bdy b) \\
& = \restrict{\phi_\beta}{x_0=0}(b) - \restrict{\phi_\beta}{x_1=0}(b) - \phi_\beta(\bdy b) + (\bdy \phi_{\eset \prec \beta})(\bdy b) + \phi_\eset(\bdy b) \\
& = \restrict{\phi_\beta}{x_0=0}(b) - b - \phi_\beta(\bdy b) + \restrict{\phi_{\eset \prec \beta}}{x_0=0}(\bdy b) - \phi_\eset(\bdy b) + \phi_\beta(\bdy b) + \phi_\eset(\bdy b) \\
& = \restrict{\phi_\beta}{x_0=0}(b) - b + \restrict{\phi_{\eset \prec \beta}}{x_0=0}(\bdy b) \\
@@ -49,6 +53,8 @@
We now describe the general case. For a $k$-blob diagram $b \in \bc_k(M)$, denote by $b_\cS$ for $\cS \subset \{0, \ldots, k-1\}$ the blob diagram obtained by erasing the corresponding blobs. In particular, $b_\eset = b$, $b_{\{0,\ldots,k-1\}} \in \bc_0(M)$, and $d b_\cS = \sum_{i \notin \cS} \pm b_{\cS \cup \{i\}}$.
Similarly, for a disjoint embedding of $k$ balls $\beta$ (that is, a blob diagram but without the labels on regions), $\beta_\cS$ denotes the result of erasing a subset of blobs. We'll write $\beta' \prec \beta$ if $\beta' = \beta_\cS$ for some $\cS$. Finally, for finite sequences, we'll write $i \prec i'$ if $i$ is subsequence of $i'$, and $i \prec_1 i$ if the lengths differ by exactly 1.
+Now we fix a sequence of strictly subordinate covers for $\cU$. First choose an $\epsilon > 0$ so every $\epsilon$ ball is contained in some open set of $\cU$. Let $\cV_{k \geq 1}$ be the open cover of $M$ by $\epsilon (1-2^{-k})$ balls, and $\cV_0 = \cU$. Certainly $\cV_k$ is strictly subordinate to $\cU$. We will write $\ev_{k \geq 0}$ for the chain map written in Lemma \ref{lem:CH-small-blobs} as $\ev_{M,\cU,\cV,k}$.
+
For a $2$-blob $b$, with balls $\beta$, $s$ is the sum of $5$ terms. Again, there is a term that makes $\beta$ small, while the others `get the boundary right'. It may be useful to look at Figure \ref{fig:erectly-a-tent-badly} to help understand the arrangement.
\begin{figure}[!ht]
\todo{}
@@ -67,17 +73,16 @@
\end{itemize}
It's not immediately obvious that it's possible to make such choices, but it follows readily from the following Lemma.
-When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ `makes $\beta$ small' if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$.
\begin{lem}
\label{lem:extend-small-homeomorphisms}
-Fix a collection of disjoint embedded balls $\beta$ in $M$. Suppose we have a map $f : X \to \Homeo(M)$ on some compact $X$ such that for each $x \in \bdy X$, $f(x)$ makes $\beta$ small. Then we can extend $f$ to a map $\tilde{f} : X \times [0,1] \to \Homeo(M)$ so that $\tilde{f}(x,0) = f(x)$ and for every $x \in \bdy X \times [0,1] \cup X \times \{1\}$, $\tilde{f}(x)$ makes $\beta$ small.
+Fix a collection of disjoint embedded balls $\beta$ in $M$ and some open cover $\cV$. Suppose we have a map $f : X \to \Homeo(M)$ on some compact $X$ such that for each $x \in \bdy X$, $f(x)$ makes $\beta$ $\cV$-small. Then we can extend $f$ to a map $\tilde{f} : X \times [0,1] \to \Homeo(M)$ so that $\tilde{f}(x,0) = f(x)$ and for every $x \in \bdy X \times [0,1] \cup X \times \{1\}$, $\tilde{f}(x)$ makes $\beta$ $\cV$-small.
\end{lem}
\begin{proof}
-Fix a metric on $M$, and pick $\epsilon > 0$ so every $\epsilon$ ball in $M$ is contained in some open set of $\cU$. First construct a family of homeomorphisms $g_s : M \to M$, $s \in [1,\infty)$ so $g_1$ is the identity, and $g_s(\beta_i) \subset \beta_i$ and $\rad g_s(\beta_i) \leq \frac{1}{s} \rad \beta_i$ for each ball $\beta_i$.
+Fix a metric on $M$, and pick $\epsilon > 0$ so every $\epsilon$ ball in $M$ is contained in some open set of $\cV$. First construct a family of homeomorphisms $g_s : M \to M$, $s \in [1,\infty)$ so $g_1$ is the identity, and $g_s(\beta_i) \subset \beta_i$ and $\rad g_s(\beta_i) \leq \frac{1}{s} \rad \beta_i$ for each ball $\beta_i$.
There is some $K$ which uniformly bounds the expansion factors of all the homeomorphisms $f(x)$, that is $d(f(x)(a), f(x)(b)) < K d(a,b)$ for all $x \in X, a,b \in M$. Write $S=\epsilon^{-1} K \max_i \{\rad \beta_i\}$ (note that is $S<1$, we can just take $S=1$, as already $f(x)$ makes $\beta$ small for all $x$). Now define $\tilde{f}(t, x) = f(x) \compose g_{(S-1)t+1}$.
-If $x \in \bdy X$, then $g_{(S-1)t+1}(\beta_i) \subset \beta_i$, and by hypothesis $f(x)$ makes $\beta_i$ small, so $\tilde{f}(t, x)$ makes $\beta$ small for all $t \in [0,1]$. Alternatively, $\rad g_S(\beta_i) \leq \frac{1}{S} \rad \beta_i \leq \frac{\epsilon}{K}$, so $\rad \tilde{f}(1,x)(\beta_i) \leq \epsilon$, and so $\tilde{f}(1,x)$ makes $\beta$ small for all $x \in X$.
+If $x \in \bdy X$, then $g_{(S-1)t+1}(\beta_i) \subset \beta_i$, and by hypothesis $f(x)$ makes $\beta_i$ small, so $\tilde{f}(t, x)$ makes $\beta$ $\cV$-small for all $t \in [0,1]$. Alternatively, $\rad g_S(\beta_i) \leq \frac{1}{S} \rad \beta_i \leq \frac{\epsilon}{K}$, so $\rad \tilde{f}(1,x)(\beta_i) \leq \epsilon$, and so $\tilde{f}(1,x)$ makes $\beta$ $\cV$-small for all $x \in X$.
\end{proof}
We'll need a stronger version of Property \ref{property:evaluation}; while the evaluation map $ev: \CD{M} \tensor \bc_*(M) \to \bc_*(M)$ is not unique, it has an up-to-homotopy representative (satisfying the usual conditions) which restricts to become a chain map $ev: \CD{M} \tensor \bc^{\cU}_*(M) \to \bc^{\cU}_*(M)$. The proof is straightforward: when deforming the family of diffeomorphisms to shrink its supports to a union of open sets, do so such that those open sets are subordinate to the cover.