blob: comparison blob1.tex

equal deleted inserted replaced

-:17348691adc7
+:4ef2f77a4652
 \def\sup{\supset}
 %\def\setmin{\smallsetminus}
 \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}}
 \def\nn#1{{{\it \small [#1]}}}
 \item $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$.
 \item An exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$
 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 $1 \otimes 1 \in C\otimes C$.
 We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism.
 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
+Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex
-or contained in all blobs, and the two boundary points of $B_\ep$ are not labeled points.
+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.
 Let $x \in F^\ep_*$ be a blob diagram.
 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.
 The key property of $j_\ep$ is
 \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
+where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field
-mentioned in $x \in F^\ep_*$ (call the restriction $y$) with $s_\ep(y)$.
+mentioned in $x \in F^\ep_*$ with $s_\ep(y)$.
-Note that $\sigma_\ep(x) \in F'$.
+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 $F'_* \to F_*(C\otimes C)$.
 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller
 $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$.
 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 $F_*(C\otimes C)$.
 \medskip
 Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob.
 We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence.
 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
 in ``basic properties" section above} away from $*$.
 Thus any cycle lies in the image of the normal blob complex of a disjoint union
 This completes the proof that $F_*(C\otimes C)$ is
 homotopic to the 0-step complex $C$.
 \medskip
-Next we show that $F_*(C)$ is homotopic \nn{q-isom?} to $\bc_*(S^1)$
+Next we show that $F_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$.
-\nn{...}
+$F_*(C)$ 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
 \nn{still need to prove exactness claim}
 \nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex?
 Do we need a map from hoch to blob?
 Does the above exactness and contractibility guarantee such a map without writing it
 down explicitly?
 Probably it's worth writing down an explicit map even if we don't need to.}
 \section{Action of $C_*(\Diff(X))$}  \label{diffsect}
 \begin{itemize}
 \item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$;
 \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 $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$
+\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 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}
 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 $\cU$.
 This insures that the resulting homotopy $h_i$ is compatible.
 Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$.
 \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}
 Let $J$ denote the handle decomposition of $P$ corresponding to $L$.
 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 $p_c \in c \sub P$.
+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
+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 $P$.
 For $p \in D$ we define
 Next we verify that $u$ has the desired properties.
 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
 \eq{
 %	\pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) .
 \nn{bounded away from zero, or something like that}.
 (Recall that $X$ and $P$ are compact.)
 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.
 \medskip
 This is similar to the previous case, except that the normal bundle is 2-dimensional instead of
 1-dimensional.
 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
 on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$.
 If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and
 \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
+\nn{Need to let the input $n$-category $C$ be a graded thing
-(e.g.~DGA or $A_\infty$ $n$-category).)
+(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?...}
 \begin{itemize}
 \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
 \item something about higher derived coend things (derived 2-coend, e.g.)
 \end{itemize}
 \end{document}
 %Recall that for $n$-category picture fields there is an evaluation map

changeset 7	4ef2f77a4652
parent 5	61751866cf69
child 8	15e6335ff1d4