--- a/blob1.tex Mon Apr 21 20:27:52 2008 +0000
+++ b/blob1.tex Tue Apr 22 05:13:02 2008 +0000
@@ -23,6 +23,7 @@
\def\setmin{\setminus}
\def\ep{\epsilon}
\def\sgl{_\mathrm{gl}}
+\def\op{^\mathrm{op}}
\def\deq{\stackrel{\mathrm{def}}{=}}
\def\pd#1#2{\frac{\partial #1}{\partial #2}}
@@ -681,14 +682,14 @@
gives rise to an exact sequence $0 \to F_*(M_1) \to F_*(M_2) \to F_*(M_3) \to 0$.
(See below for proof.)
\item $F_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is
-homotopic to the 0-step complex $C$.
+quasi-isomorphic to the 0-step complex $C$.
(See below for proof.)
-\item $F_*(C)$ (here $C$ is wearing its $C$-$C$-bimodule hat) is homotopic to $\bc_*(S^1)$.
+\item $F_*(C)$ (here $C$ is wearing its $C$-$C$-bimodule hat) is quasi-isomorphic to $\bc_*(S^1)$.
(See below for proof.)
\end{itemize}
First we show that $F_*(C\otimes C)$ is
-homotopic to the 0-step complex $C$.
+quasi-isomorphic to the 0-step complex $C$.
Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of
the point $*$ is $1 \otimes 1 \in C\otimes C$.
@@ -696,11 +697,13 @@
Fix a small $\ep > 0$.
Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$.
-Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex where $B_\ep$ is either disjoint from
-or contained in all blobs, and the two boundary points of $B_\ep$ are not labeled points.
+Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex
+generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from
+or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$.
For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$
labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$.
(See Figure xxxx.)
+Note that $y - s_\ep(y) \in U(B_\ep)$.
\nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.}
Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows.
@@ -708,10 +711,11 @@
If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to
$x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$.
If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows.
-Let $y_i$ be the restriction of $z_i$ to $*$.
+Let $y_i$ be the restriction of $z_i$ to $B_\ep$.
Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$,
and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$.
Define $j_\ep(x) = \sum x_i$.
+\nn{need to check signs coming from blob complex differential}
Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also.
@@ -719,9 +723,9 @@
\eq{
\bd j_\ep + j_\ep \bd = \id - \sigma_\ep ,
}
-where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction of each field
-mentioned in $x \in F^\ep_*$ (call the restriction $y$) with $s_\ep(y)$.
-Note that $\sigma_\ep(x) \in F'$.
+where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field
+mentioned in $x \in F^\ep_*$ with $s_\ep(y)$.
+Note that $\sigma_\ep(x) \in F'_*$.
If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$
is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$.
@@ -730,16 +734,16 @@
Instead, we'll be less ambitious and just show that
$F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
-If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$
+If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have
$x \in F_*^\ep$.
(This is true for any chain in $F_*(C\otimes C)$, since chains are sums of
finitely many blob diagrams.)
Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map
-is surjective on homology.
+$F'_* \sub F_*(C\otimes C)$ is surjective on homology.
If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$
and
\eq{
- \bd x = \bd (\sigma_\ep(y) + j_\ep(x)) .
+ \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) .
}
Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology.
This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
@@ -751,7 +755,8 @@
First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with
$S^1$ replaced some (any) neighborhood of $* \in S^1$.
-Then $G''_*$ and $G'_*$ are both contractible.
+Then $G''_*$ and $G'_*$ are both contractible
+and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence.
For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting
$G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$.
For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe
@@ -798,8 +803,24 @@
\medskip
-Next we show that $F_*(C)$ is homotopic \nn{q-isom?} to $\bc_*(S^1)$
-\nn{...}
+Next we show that $F_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$.
+$F_*(C)$ differs from $\bc_*(S^1)$ only in that the base point *
+is always a labeled point in $F_*(C)$, while in $\bc_*(S^1)$ it may or may not be.
+In other words, there is an inclusion map $i: F_*(C) \to \bc_*(S^1)$.
+
+We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows.
+If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if
+* is a labeled point in $y$.
+Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *.
+Let $x \in \bc_*(S^1)$.
+Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in
+$x$ with $y$.
+It is easy to check that $s$ is a chain map and $s \circ i = \id$.
+
+Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points
+in a neighborhood $B_\ep$ of *, except perhaps *.
+Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$.
+\nn{rest of argument goes similarly to above}
\bigskip
@@ -813,6 +834,9 @@
+
+
+
\section{Action of $C_*(\Diff(X))$} \label{diffsect}
Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of
@@ -865,10 +889,10 @@
\item the $V_i$'s 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 $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$
-for all $p = (p_1, \ldots, p_m)$.
+\item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot) \circ g$
+for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$.
\end{itemize}
-A chain $x \in C_k(\Diff(M))$ is (by definition) adapted to $\cU$ if is is the sum
+A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum
of singular cells, each of which is adapted to $\cU$.
\begin{lemma} \label{extension_lemma}
@@ -910,7 +934,14 @@
\nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this}
\end{proof}
+\medskip
+((argument continues roughly as follows: up to homotopy, there is only one way to define $e_X$
+on compatible $x\otimes y \in CD_*(X)\otimes \bc_*(X)$.
+This is because $x$ is the gluing of $x'$ and $x''$, where $x'$ has degree zero and is defined on
+the complement of the $D_i$'s, and $x''$ is defined on the $D_i$'s.
+We have no choice on $x'$, since we already know the map on 0-parameter families of diffeomorphisms.
+We have no choice, up to homotopy, on $x''$, since the target chain complex is contractible.))
\section{Families of Diffeomorphisms} \label{fam_diff_sect}
@@ -968,8 +999,8 @@
Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and,
more importantly, a $k{-}i$-dimensional normal coordinate.
-For each $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$.
-Let $D$ be a $k$-handle of $J$, and let $d$ also denote the corresponding
+For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$.
+Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding
$k$-cell of $L$.
To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s
which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$.
@@ -1026,7 +1057,7 @@
Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$.
Therefore $F$ is a homotopy from $f$ to something.
-Next we show that the $K_\alpha$'s are sufficiently fine cell decompositions,
+Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions,
then $F$ is a homotopy through diffeomorphisms.
We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.
We have
@@ -1040,6 +1071,7 @@
Also, $\pd{f}{p}$ is bounded.
So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done.
It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$
+(which is bounded)
and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s.
These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine.
This completes the proof that $F$ is a homotopy through diffeomorphisms.
@@ -1083,7 +1115,7 @@
We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell
and a 2-cell with support $U_\beta$.
-Next case: $j=2$, $m=2$, $j_1 = j_2 = 2$.
+Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$.
In this case the codimension 2 cell $D$ is the intersection of two
codimension 1 cells, from $K_\beta$ and $K_\gamma$.
We can write $D = D' \times I \times I$, where the normal coordinates are constant
@@ -1103,13 +1135,77 @@
\section{$A_\infty$ action on the boundary}
+Let $Y$ be an $n{-}1$-manifold.
+The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary
+conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure
+of an $A_\infty$ category.
+
+Composition of morphisms (multiplication) depends of a choice of homeomorphism
+$I\cup I \cong I$. Given this choice, gluing gives a map
+\eq{
+ \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c)
+ \cong \bc_*(Y\times I; a, c)
+}
+Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various
+higher associators of the $A_\infty$ structure, more or less canonically.
+
+\nn{is this obvious? does more need to be said?}
+
+Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$.
+
+Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism
+$(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$
+(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the
+$A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$.
+Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood
+of $Y$ in $X$.
+
+In the next section we use the above $A_\infty$ actions to state and prove
+a gluing theorem for the blob complexes of $n$-manifolds.
+
+
+
+
+
+
\section{Gluing} \label{gluesect}
+Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy
+of $Y \du -Y$ contained in its boundary.
+Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$.
+We wish to describe the blob complex of $X\sgl$ in terms of the blob complex
+of $X$.
+More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$,
+where $c\sgl \in \cC(\bd X\sgl)$,
+in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation
+of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$.
+
+\begin{thm}
+$\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product
+of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$.
+\end{thm}
+
+The proof will occupy the remainder of this section.
+
+\nn{...}
+
+\bigskip
+
+\nn{need to define/recall def of (self) tensor product over an $A_\infty$ category}
+
+
+
+
+
\section{Extension to ...}
-(Need to let the input $n$-category $C$ be a graded thing
-(e.g.~DGA or $A_\infty$ $n$-category).)
+\nn{Need to let the input $n$-category $C$ be a graded thing
+(e.g.~DGA or $A_\infty$ $n$-category).}
+
+\nn{maybe this should be done earlier in the exposition?
+if we can plausibly claim that the various proofs work almost
+the same with the extended def, then maybe it's better to extend late (here)}
\section{What else?...}
@@ -1118,7 +1214,7 @@
\item Derive Hochschild standard results from blob point of view?
\item $n=2$ examples
\item Kh
-\item dimension $n+1$
+\item dimension $n+1$ (generalized Deligne conjecture?)
\item should be clear about PL vs Diff; probably PL is better
(or maybe not)
\item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category
@@ -1127,6 +1223,8 @@
+
+
\end{document}