# HG changeset patch # User kevin@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1244162321 0 # Node ID d363611b1f593bfbf620a733a66db846e097f6dc # Parent 4f2ea5eabc8fc006656698f6361d2685ac455d69 ... diff -r 4f2ea5eabc8f -r d363611b1f59 blob1.tex --- a/blob1.tex Thu Jun 04 19:28:55 2009 +0000 +++ b/blob1.tex Fri Jun 05 00:38:41 2009 +0000 @@ -26,7 +26,7 @@ \def\lf{\overline{\cC}} \def\nn#1{{{\it \small [#1]}}} - +\long\def\noop#1{} % equations \newcommand{\eq}[1]{\begin{displaymath}#1\end{displaymath}} @@ -879,26 +879,29 @@ \section{Action of $\CD{X}$} \label{sec:evaluation} -Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of +Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of the space of diffeomorphisms -of the $n$-manifold $X$ (fixed on $\bd X$). -For convenience, we will permit the singular cells generating $CD_*(X)$ to be more general +\nn{or homeomorphisms} +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 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 $CD_*(X) \deq CD_*(X, X)$. \begin{prop} \label{CDprop} For each $n$-manifold $X$ there is a chain map \eq{ - e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) . + e_X : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) . } -On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$ +On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$ (Proposition (\ref{diff0prop})). -For any splitting $X = X_1 \cup X_2$, the following diagram commutes +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) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\ - CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) + CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(Y) \\ + CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & - \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} + \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl} } } Any other map satisfying the above two properties is homotopic to $e_X$. \end{prop} @@ -922,6 +925,8 @@ The proof will occupy the remainder of this section. \nn{unless we put associativity prop at end} +Without loss of generality, we will assume $X = Y$. + \medskip Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. @@ -955,9 +960,13 @@ \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 $g \in \Diff(X)$. \end{itemize} -A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum +A chain $x \in CD_k(X)$ 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 +{\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 $\cU$. @@ -968,122 +977,21 @@ \medskip -The strategy for the proof of Proposition \ref{CDprop} is as follows. -We will identify a subcomplex -\[ - G_* \sub CD_*(X) \otimes \bc_*(X) -\] -on which the evaluation map is uniquely determined (up to homotopy) by the conditions -in \ref{CDprop}. -We then show that the inclusion of $G_*$ into the full complex -is an equivalence in the appropriate sense. -\nn{need to be more specific here} +Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}. Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$. -Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$ -intersects at most one blob $B$ of $b$. -Since $V \cup B$ might not itself be a ball, we need a more careful and complicated definition. -Choose a metric for $X$. -We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that -$\supp(p) \cup N_\epsilon(b)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon -neighborhood of the support of $b$. -\nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.} - -\nn{need to worry about case where the intrinsic support of $p$ is not a union of balls. -probably we can just stipulate that it is (i.e. only consider families of diffeos with this property). -maybe we should build into the definition of ``adapted" that support takes up all of $U_i$.} - -\nn{need to eventually show independence of choice of metric. maybe there's a better way than -choosing a metric. perhaps just choose a nbd of each ball, but I think I see problems -with that as well. -the bottom line is that we need a scheme for choosing unions of balls -which satisfies the $C$, $C'$, $C''$ claim made a few paragraphs below.} - -Next we define the evaluation map $e_X$ on $G_*$. -We'll proceed inductively on $G_i$. -The induction starts on $G_0$, where the evaluation map is determined -by the action of $\Diff(X)$ on $\bc_*(X)$ -because $G_0 \sub CD_0\otimes \bc_0$. -Assume we have defined the evaluation map up to $G_{k-1}$ and -let $p\otimes b$ be a generator of $G_k$. -Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. -There is a factorization $p = g \circ p'$, where $g\in \Diff(X)$ and $p'$ is a family of diffeomorphisms which is the identity outside of $C$. -Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$. -We may assume inductively -(cf the end of this paragraph) -that $e_X(\bd(p\otimes b))$ has a similar factorization $x\bullet g(b'')$, where -$x \in \bc_*(g(C))$ and $\bd x = 0$. -Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$. -Define $e_X(p\otimes b) = y\bullet g(b'')$. - -We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made. -If we make a different series of choice of the chain $y$ in the previous paragraph, -we can inductively construct a homotopy between the two sets of choices, -again relying on the contractibility of $\bc_*(g(G))$. -A similar argument shows that this homotopy is unique up to second order homotopy, and so on. - -Given a different set of choices $\{C'\}$ of the unions of balls $\{C\}$, -we can find a third set of choices $\{C''\}$ such that $C, C' \sub C''$. -The argument now proceeds as in the previous paragraph. -\nn{should maybe say more here; also need to back up claim about third set of choices} -\nn{this definitely needs reworking} - -Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists -a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and -this homotopy are unique up to iterated homotopy. - -Given $k>0$ and a blob diagram $b$, we say that a cover $\cU$ of $X$ is $k$-compatible with -$b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union -$U_1\cup\cdots\cup U_k$ is a union of balls which satisfies the condition used to define $G_*$ above. -It follows from Lemma \ref{extension_lemma} -that if $\cU$ is $k$-compatible with $b$ and -$p$ is a $k$-parameter family of diffeomorphisms which is adapted to $\cU$, then -$p\otimes b \in G_*$. -\nn{maybe emphasize this more; it's one of the main ideas in the proof} - -Let $k$ be the degree of $x$ and choose a cover $\cU$ of $X$ such that $\cU$ is -$k$-compatible with each of the (finitely many) blob diagrams occurring in $x$. -We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to -construct the homotopy to $G_*$. -First we will construct a homotopy $h \in G_*$ from $\bd x$ to a cycle $z$ such that -each family of diffeomorphisms $p$ occurring in $z$ is adapted to $\cU$. -Then we will construct a homotopy (rel boundary) $r$ from $x + h$ to $y$ such that -each family of diffeomorphisms $p$ occurring in $y$ is adapted to $\cU$. -This implies that $y \in G_*$. -The homotopy $r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek. - -We will define $h$ inductively on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$. -Define $h$ to be zero on bidegree $(0, k-1)$. -Let $p\otimes b$ be a generator occurring in $\bd x$ with bidegree $(1, k-2)$. -Using Lemma \ref{extension_lemma}, construct a homotopy (rel $\bd$) $q$ from $p$ to $p'$ which is adapted to $\cU$. -Define $h$ at $p\otimes b$ to be $q\otimes b$. -Let $p'\otimes b'$ be a generator occurring in $\bd x$ with bidegree $(2, k-3)$. -Let $s$ denote the sum of the $q$'s from the previous step for generators -adjacent to $(\bd p')\otimes b'$. -\nn{need to say more here} -Apply Lemma \ref{extension_lemma} to $p'+s$ -yielding a family of diffeos $q'$. -Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$. -Continuing in this way, we define all of $h$. - -The homotopy $r$ is constructed similarly. - -\nn{need to say something about uniqueness of $r$, $h$ etc. -postpone this until second draft.} - -At this point, we have finished defining the evaluation map. -The uniqueness statement in the proposition is clear from the method of proof. -All that remains is to show that the evaluation map gets along well with cutting and gluing, -as claimed in the proposition. -This is in fact not difficult, since the myriad choices involved in defining the -evaluation map can be made in parallel for the top and bottom -arrows in the commutative diagram. - -This completes the proof of Proposition \ref{CDprop}. +Suppose that there exists $V \sub X$ such that +\begin{enumerate} +\item $V$ is homeomorphic to a disjoint union of balls, and +\item $\supp(p) \cup \supp(b) \sub V$. +\end{enumerate} \medskip -\nn{say something about associativity here} +\nn{to be continued....} + + +%\nn{say something about associativity here} \input{text/A-infty.tex} diff -r 4f2ea5eabc8f -r d363611b1f59 text/hochschild.tex --- a/text/hochschild.tex Thu Jun 04 19:28:55 2009 +0000 +++ b/text/hochschild.tex Fri Jun 05 00:38:41 2009 +0000 @@ -218,8 +218,7 @@ Suppose we have a short exact sequence of $C$-$C$ bimodules $$\xymatrix{0 \ar[r] & K \ar@{^{(}->}[r]^f & E \ar@{->>}[r]^g & Q \ar[r] & 0}.$$ We'll write $\hat{f}$ and $\hat{g}$ for the image of $f$ and $g$ under the functor. Most of what we need to check is easy. -If $(a \tensor k \tensor b) \in \ker(C \tensor K \tensor C \to K)$, to have $\hat{f}(a \tensor k \tensor b) = 0 \in \ker(C \tensor E \tensor C \to E)$, we must have $f(k) = 0 \in E$, so -be $k=0$ itself. If $(a \tensor e \tensor b) \in \ker(C \tensor E \tensor C \to E)$ is in the image of $\ker(C \tensor K \tensor C \to K)$ under $\hat{f}$, clearly +If $(a \tensor k \tensor b) \in \ker(C \tensor K \tensor C \to K)$, to have $\hat{f}(a \tensor k \tensor b) = 0 \in \ker(C \tensor E \tensor C \to E)$, we must have $f(k) = 0 \in E$, which implies $k=0$ itself. If $(a \tensor e \tensor b) \in \ker(C \tensor E \tensor C \to E)$ is in the image of $\ker(C \tensor K \tensor C \to K)$ under $\hat{f}$, clearly $e$ is in the image of the original $f$, so is in the kernel of the original $g$, and so $\hat{g}(a \tensor e \tensor b) = 0$. If $\hat{g}(a \tensor e \tensor b) = 0$, then $g(e) = 0$, so $e = f(\widetilde{e})$ for some $\widetilde{e} \in K$, and $a \tensor e \tensor b = \hat{f}(a \tensor \widetilde{e} \tensor b)$. Finally, the interesting step is in checking that any $q = \sum_i a_i \tensor q_i \tensor b_i$ such that $\sum_i a_i q_i b_i = 0$ is in the image of $\ker(C \tensor E \tensor C \to C)$ under $\hat{g}$. @@ -232,7 +231,7 @@ \end{align*} (here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$). -Identical arguments show that the functors +Similar arguments show that the functors \begin{equation} \label{eq:ker-functor}% M \mapsto \ker(C^{\tensor k} \tensor M \tensor C^{\tensor l} \to M)