diff -r 17348691adc7 -r 4ef2f77a4652 blob1.tex --- 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}