--- a/text/a_inf_blob.tex Sun Jul 18 08:07:50 2010 -0600
+++ b/text/a_inf_blob.tex Sun Jul 18 11:07:47 2010 -0600
@@ -208,14 +208,18 @@
\medskip
+Taking $F$ above to be a point, we obtain the following corollary.
+
\begin{cor}
\label{cor:new-old}
-The blob complex of a manifold $M$ with coefficients in a topological $n$-category $\cC$ is homotopic to the homotopy colimit invariant of $M$ defined using the $A_\infty$ $n$-category obtained by applying the blob complex to a point:
-$$\bc_*(M; \cC) \htpy \cl{\bc_*(pt; \cC)}(M).$$
+Let $\cE$ be a system of fields (with local relations) and let $\cC_\cE$ be the $A_\infty$
+$n$-category obtained from $\cE$ by taking the blob complex of balls.
+Then for all $n$-manifolds $Y$ the old-fashioned and new-fangled blob complexes are
+homotopy equivalent:
+\[
+ \bc^\cE_*(Y) \htpy \cl{\cC_\cE}(Y) .
+\]
\end{cor}
-\begin{proof}
-Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point.
-\end{proof}
\medskip
@@ -223,7 +227,7 @@
\[
F \to E \to Y .
\]
-We outline one approach here and a second in Subsection xxxx.
+We outline one approach here and a second in \S \ref{xyxyx}.
We can generalize the definition of a $k$-category by replacing the categories
of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$
@@ -233,11 +237,11 @@
assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$.
Let $\cF_E$ denote this $k$-category over $Y$.
We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to
-get a chain complex $\cF_E(Y)$.
+get a chain complex $\cl{\cF_E}(Y)$.
The proof of Theorem \ref{thm:product} goes through essentially unchanged
to show that
\[
- \bc_*(E) \simeq \cF_E(Y) .
+ \bc_*(E) \simeq \cl{\cF_E}(Y) .
\]
\nn{remark further that this still works when the map is not even a fibration?}
@@ -276,16 +280,19 @@
\item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked
$m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$)
or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$).
-(See Example \nn{need example for this}.)
+(See Example \ref{bc-module-example}.)
\end{itemize}
+\nn{statement (and proof) is only for case $k=n$; need to revise either above or below; maybe
+just say that until we define functors we can't do more}
+
\begin{thm}
\label{thm:gluing}
$\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
\end{thm}
\begin{proof}
-\nn{for now, just prove $k=0$ case.}
+We will assume $k=n$; the other cases are similar.
The proof is similar to that of Theorem \ref{thm:product}.
We give a short sketch with emphasis on the differences from
the proof of Theorem \ref{thm:product}.
@@ -294,9 +301,9 @@
Recall that this is a homotopy colimit based on decompositions of the interval $J$.
We define a map $\psi:\cT\to \bc_*(X)$.
-On filtration degree zero summands it is given
+On 0-simplices it is given
by gluing the pieces together to get a blob diagram on $X$.
-On filtration degree 1 and greater $\psi$ is zero.
+On simplices of dimension 1 and greater $\psi$ is zero.
The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split
over some decomposition of $J$.
@@ -313,20 +320,6 @@
Theorem \ref{thm:product}.
\end{proof}
-\noop{
-Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
-Let $D$ be an $n{-}k$-ball.
-There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$.
-To get a map in the other direction, we replace $\bc_*(D\times X)$ with a subcomplex
-$\cS_*$ which is adapted to a fine open cover of $D\times X$.
-For sufficiently small $j$ (depending on the cover), we can find, for each $j$-blob diagram $b$
-on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding
-decomposition of $D\times X$.
-The proof that these two maps are inverse to each other is the same as in
-Theorem \ref{thm:product}.
-}
-
-
\medskip
\subsection{Reconstructing mapping spaces}
@@ -361,31 +354,25 @@
We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
-Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of
-$j$-fold mapping cylinders, $j \ge 0$.
-So, as an abelian group (but not as a chain complex),
-\[
- \cB^\cT(M) = \bigoplus_{j\ge 0} C^j,
-\]
-where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders.
-
-Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by
+Recall that
+the 0-simplices of the homotopy colimit $\cB^\cT(M)$
+are a direct sum of chain complexes with the summands indexed by
decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms
of $\cT$.
Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs
$(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous
-maps from the $n{-}1$-skeleton of $K$ to $T$.
+map from the $n{-}1$-skeleton of $K$ to $T$.
The summand indexed by $(K, \vphi)$ is
\[
\bigotimes_b D_*(b, \vphi),
\]
where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes
chains of maps from $b$ to $T$ compatible with $\vphi$.
-We can take the product of these chains of maps to get a chains of maps from
+We can take the product of these chains of maps to get chains of maps from
all of $M$ to $K$.
-This defines $\psi$ on $C^0$.
+This defines $\psi$ on 0-simplices.
-We define $\psi(C^j) = 0$ for $j > 0$.
+We define $\psi$ to be zero on $(\ge1)$-simplices.
It is not hard to see that this defines a chain map from
$\cB^\cT(M)$ to $C_*(\Maps(M\to T))$.
@@ -407,49 +394,13 @@
\[
\phi(a) = (a, K) + r
\]
-where $(a, K) \in C^0$ and $r\in \bigoplus_{j\ge 1} C^j$.
+where $(a, K)$ is a 0-simplex and $r$ is a sum of simplices of dimension 1 and greater.
It is now easy to see that $\psi\circ\phi$ is the identity on the nose.
Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity.
(See the proof of Theorem \ref{thm:product} for more details.)
\end{proof}
-\noop{
-% old proof (just start):
-We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
-We then use Lemma \ref{extension_lemma_c} to show that $g$ induces isomorphisms on homology.
-
-Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of
-$j$-fold mapping cylinders, $j \ge 0$.
-So, as an abelian group (but not as a chain complex),
-\[
- \cB^\cT(M) = \bigoplus_{j\ge 0} C^j,
-\]
-where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders.
-
-Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by
-decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms
-of $\cT$.
-Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs
-$(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous
-maps from the $n{-}1$-skeleton of $K$ to $T$.
-The summand indexed by $(K, \vphi)$ is
-\[
- \bigotimes_b D_*(b, \vphi),
-\]
-where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes
-chains of maps from $b$ to $T$ compatible with $\vphi$.
-We can take the product of these chains of maps to get a chains of maps from
-all of $M$ to $K$.
-This defines $g$ on $C^0$.
-
-We define $g(C^j) = 0$ for $j > 0$.
-It is not hard to see that this defines a chain map from
-$\cB^\cT(M)$ to $C_*(\Maps(M\to T))$.
-
-\nn{...}
-}
-
\nn{maybe should also mention version where we enrich over
spaces rather than chain complexes;}
@@ -461,13 +412,4 @@
\medskip
\nn{still to do: general maps}
-\todo{}
-Various citations we might want to make:
-\begin{itemize}
-\item \cite{MR2061854} McClure and Smith's review article
-\item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad)
-\item \cite{MR0236922,MR0420609} Boardman and Vogt
-\item \cite{MR1256989} definition of framed little-discs operad
-\end{itemize}
-