# HG changeset patch # User Kevin Walker # Date 1275007151 25200 # Node ID a746cd526cdd797f13d65bc32d734090698629cd # Parent 418919afd07731ce005f8df0f08d1bc52e8c5848# Parent 7afacaa87bdb106b101241d1ca0f42add6cfb3ce merging -- not sure what I'm doing, or why I need to 'commit' again diff -r 418919afd077 -r a746cd526cdd preamble.tex --- a/preamble.tex Thu May 27 17:35:56 2010 -0700 +++ b/preamble.tex Thu May 27 17:39:11 2010 -0700 @@ -13,6 +13,7 @@ \usepackage[section]{placeins} \usepackage{leftidx} +\usepackage{stmaryrd} % additional math symbols, e.g. \mapsfrom \SelectTips{cm}{} % This may speed up compilation of complex documents with many xymatrices. diff -r 418919afd077 -r a746cd526cdd text/appendixes/explicit.tex --- a/text/appendixes/explicit.tex Thu May 27 17:35:56 2010 -0700 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,97 +0,0 @@ -%!TEX root = ../../blob1.tex - -Here's an alternative proof of the special case in which $P$, the parameter space for the family of diffeomorphisms, is a cube. It is much more explicit, for better or worse. - -\begin{proof}[Alternative, more explicit proof of Lemma \ref{extension_lemma}] - - -Fix a finite open cover of $X$, say $(U_l)_{l=1}^L$, along with an -associated partition of unity $(r_l)$. - -We'll define the homotopy $H:I \times P \times X \to X$ via a function -$u:I \times P \times X \to P$, with -\begin{equation*} -H(t,p,x) = F(u(t,p,x),x). -\end{equation*} - -To begin, we'll define a function $u'' : I \times P \times X \to P$, and -a corresponding homotopy $H''$. This homotopy will just be a homotopy of -$F$ through families of maps, not through families of diffeomorphisms. On -the other hand, it will be quite simple to describe, and we'll later -explain how to build the desired function $u$ out of it. - -For each $l = 1, \ldots, L$, pick some $C^\infty$ function $f_l : I \to -I$ which is identically $0$ on a neighborhood of the closed interval $[0,\frac{l-1}{L}]$ -and identically $1$ on a neighborhood of the closed interval $[\frac{l}{L},1]$. (Monotonic? -Fix a bound for the derivative?) We'll extend it to a function on -$k$-tuples $f_l : I^k \to I^k$ pointwise. - -Define $$u''(t,p,x) = \sum_{l=1}^L r_l(x) u_l(t,p),$$ with -$$u_l(t,p) = t f_l(p) + (1-t)p.$$ Notice that the $i$-th component of $u''(t,p,x)$ depends only on the $i$-th component of $p$. - -Let's now establish some properties of $u''$ and $H''$. First, -\begin{align*} -H''(0,p,x) & = F(u''(0,p,x),x) \\ - & = F(\sum_{l=1}^L r_l(x) p, x) \\ - & = F(p,x). -\end{align*} -Next, calculate the derivatives -\begin{align*} -\partial_{p_i} H''(1,p,x) & = \partial_{p_i}u''(1,p,x) \partial_1 F(u(1,p,x),x) \\ -\intertext{and} -\partial_{p_i}u''(1,p,x) & = \sum_{l=1}^L r_l(x) \partial_{p_i} f_l(p). -\end{align*} -Now $\partial_{p_i} f_l(p) = 0$ unless $\frac{l-1}{L} < p_i < \frac{l}{L}$, and $r_l(x) = 0$ unless $x \in U_l$, -so we conclude that for a fixed $p$, $\partial_p H''(1,p,x) = 0$ for all $x$ outside the union of $k$ open sets from the open cover, namely -$\bigcup_{i=1}^k U_{l_i}$ where for each $i$, we choose $l_i$ so $\frac{l_i -1}{L} \leq p_i \leq \frac{l_i}{L}$. It may be helpful to refer to Figure \ref{fig:supports}. - -\begin{figure}[!ht] -\begin{equation*} -\mathfig{0.5}{explicit/supports} -\end{equation*} -\caption{The supports of the derivatives {\color{green}$\partial_p f_1$}, {\color{blue}$\partial_p f_2$} and {\color{red}$\partial_p f_3$}, illustrating the case $k=2$, $L=3$. Notice that any -point $p$ lies in the intersection of at most $k$ supports. The support of $\partial_p u''(1,p,x)$ is contained in the union of these supports.} -\label{fig:supports} -\end{figure} - -Unfortunately, $H''$ does not have the desired property that it's a homotopy through diffeomorphisms. To achieve this, we'll paste together several copies -of the map $u''$. First, glue together $2^k$ copies, defining $u':I \times P \times X$ by -\begin{align*} -u'(t,p,x)_i & = -\begin{cases} -\frac{1}{2} u''(t, 2p_i, x)_i & \text{if $0 \leq p_i \leq \frac{1}{2}$} \\ -1-\frac{1}{2} u''(t, 2-2p_i, x)_i & \text{if $\frac{1}{2} \leq p_i \leq 1$}. -\end{cases} -\end{align*} -(Note that we're abusing notation somewhat, using the fact that $u''(t,p,x)_i$ depends on $p$ only through $p_i$.) -To see what's going on here, it may be helpful to look at Figure \ref{fig:supports_4}, which shows the support of $\partial_p u'(1,p,x)$. -\begin{figure}[!ht] -\begin{equation*} -\mathfig{0.4}{explicit/supports_4} \qquad \qquad \mathfig{0.4}{explicit/supports_36} -\end{equation*} -\caption{The supports of $\partial_p u'(1,p,x)$ and of $\partial_p u(1,p,x)$ (with $K=3$) are subsets of the indicated region.} -\label{fig:supports_4} -\end{figure} - -Second, pick some $K$, and define -\begin{align*} -u(t,p,x) & = \frac{\floor{K p}}{K} + \frac{1}{K} u'\left(t, K \left(p - \frac{\floor{K p}}{K}\right), x\right). -\end{align*} - -\todo{Explain that the localisation property survives for $u'$ and $u$.} - -We now check that by making $K$ large enough, $H$ becomes a homotopy through diffeomorphisms. We start with -$$\partial_x H(t,p,x) = \partial_x u(t,p,x) \partial_1 F(u(t,p,x), x) + \partial_2 F(u(t,p,x), x)$$ -and observe that since $F(p, -)$ is a diffeomorphism, the second term $\partial_2 F(u(t,p,x), x)$ is bounded away from $0$. Thus if we can control the -size of the first term $\partial_x u(t,p,x) \partial_1 F(u(t,p,x), x)$ we're done. The factor $\partial_1 F(u(t,p,x), x)$ is bounded, and we -calculate \todo{err... this is a mess, and probably wrong.} -\begin{align*} -\partial_x u(t,p,x)_i & = \partial_x \frac{1}{K} u'\left(t, K\left(p - \frac{\floor{K p}}{K}\right), x\right)_i \\ - & = \pm \frac{1}{2 K} \partial_x u''\left(t, (1\mp1)\pm 2K\left(p_i-\frac{\floor{K p}}{K}\right), x\right)_i \\ - & = \pm \frac{1}{2 K} \sum_{l=1}^L (\partial_x r_l(x)) u_l\left(t, (1\mp1)\pm 2K\left(p_i-\frac{\floor{K p}}{K}\right)\right)_i. \\ -\intertext{Since the target of $u_l$ is just the unit cube $I^k$, we can make the estimate} -\norm{\partial_x u(t,p,x)_i} & \leq \frac{1}{2 K} \sum_{l=1}^L \norm{\partial_x r_l(x)}. -\end{align*} -The sum here is bounded, so for large enough $K$ this is small enough that $\partial_x H(t,p,x)$ is never zero. - -\end{proof} \ No newline at end of file diff -r 418919afd077 -r a746cd526cdd text/appendixes/famodiff.tex --- a/text/appendixes/famodiff.tex Thu May 27 17:35:56 2010 -0700 +++ b/text/appendixes/famodiff.tex Thu May 27 17:39:11 2010 -0700 @@ -9,14 +9,10 @@ (That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$; for fixed $x$, $r_\alpha(x) \ne 0$ for only finitely many $\alpha$; and $\sum_\alpha r_\alpha = 1$.) Since $X$ is compact, we will further assume that $r_\alpha = 0$ (globally) -for all but finitely many $\alpha$. +for all but finitely many $\alpha$. \nn{Can't we just assume $\cU$ is finite? Is something subtler happening? -S} -Let -\[ - CM_*(X, T) \deq C_*(\Maps(X\to T)) , -\] -the singular chains on the space of continuous maps from $X$ to $T$. -$CM_k(X, T)$ is generated by continuous maps +Consider $C_*(\Maps(X\to T))$, the singular chains on the space of continuous maps from $X$ to $T$. +$C_k(\Maps(X \to T))$ is generated by continuous maps \[ f: P\times X \to T , \] @@ -24,7 +20,7 @@ Recall that $f$ is {\it supported} on $S\sub X$ if $f(p, x)$ does not depend on $p$ when $x \notin S$, and that $f$ is {\it adapted} to $\cU$ if $f$ is supported on the union of at most $k$ of the $U_\alpha$'s. -A chain $c \in CM_*(X, T)$ is adapted to $\cU$ if it is a linear combination of +A chain $c \in C_*(\Maps(X \to T))$ is adapted to $\cU$ if it is a linear combination of generators which are adapted. \begin{lemma} \label{basic_adaptation_lemma} @@ -40,14 +36,12 @@ the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. \item If $f$ has support $S\sub X$, then $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$. -Furthermore, if $Q\sub\bd P$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$ -has support $S'$, then +Furthermore, if $Q$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$ +has support $S' \subset X$, then $F: (I\times Q)\times X\to T$ also has support $S'$. -\item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a -[immersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism] -then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$. -(Of course we must assume that $X$ and $T$ are the appropriate -sort of manifolds for this to make sense.) +\item Suppose both $X$ and $T$ are smooth manifolds, metric spaces, or PL manifolds, and let $\cX$ denote the subspace of $\Maps(X \to T)$ consisting of immersions or of diffeomorphisms (in the smooth case), bi-Lipschitz homeomorphisms (in the metric case), or PL homeomorphisms (in the PL case). + If $f$ is smooth, Lipschitz or PL, as appropriate, and $f(p, \cdot):X\to T$ is in $\cX$ for all $p \in P$ +then $F(t, p, \cdot)$ is also in $\cX$ for all $t\in I$ and $p\in P$. \end{enumerate} \end{lemma} @@ -80,7 +74,7 @@ For each (top-dimensional) $k$-cell $C$ of each $K_\alpha$, choose a point $p(C) \in C \sub P$. If $C$ meets a subpolyhedron $Q$ of $\bd P$, we require that $p(C)\in Q$. (It follows that if $C$ meets both $Q$ and $Q'$, then $p(C)\in Q\cap Q'$. -This puts some mild constraints on the choice of $K_\alpha$.) +Ensuring this is possible corresponds to some mild constraints on the choice of the $K_\alpha$.) Let $D$ be a $k$-handle of $\jj$. For each $\alpha$ let $C(D, \alpha)$ be the $k$-cell of $K_\alpha$ which contains $D$ @@ -134,7 +128,7 @@ \right) . \end{equation} -This completes the definition of $u: I \times P \times X \to P$. +This completes the definition of $u: I \times P \times X \to P$. The formulas above are consistent: for $p$ at the boundary between a $k-j$-handle and a $k-(j+1)$-handle the corresponding expressions in Equation \eqref{eq:u} agree, since one of the normal coordinates becomes $0$ or $1$. Note that if $Q\sub \bd P$ is a convex linear subpolyhedron, then $u(I\times Q\times X) \sub Q$. \medskip @@ -150,7 +144,7 @@ Next we show that for each handle $D$ of $J$, $F(1, \cdot, \cdot) : D\times X \to X$ is a singular cell adapted to $\cU$. Let $k-j$ be the index of $D$. -Referring to Equation \ref{eq:u}, we see that $F(1, p, x)$ depends on $p$ only if +Referring to Equation \eqref{eq:u}, we see that $F(1, p, x)$ depends on $p$ only if $r_\beta(x) \ne 0$ for some $\beta\in\cN$, i.e.\ only if $x\in \bigcup_{\beta\in\cN} U_\beta$. Since the cardinality of $\cN$ is $j$ which is less than or equal to $k$, @@ -176,7 +170,7 @@ \medskip Now for claim 4 of the lemma. -Assume that $X$ and $T$ are smooth manifolds and that $f$ is a family of diffeomorphisms. +Assume that $X$ and $T$ are smooth manifolds and that $f$ is a smooth family of diffeomorphisms. We must show that we can choose the $K_\alpha$'s and $u$ so that $F(t, p, \cdot)$ is a diffeomorphism for all $t$ and $p$. It suffices to @@ -188,8 +182,8 @@ } Since $f$ is a family of diffeomorphisms and $X$ and $P$ are compact, $\pd{f}{x}$ is non-singular and bounded away from zero. -Also, $\pd{f}{p}$ is bounded. -So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. +Also, since $f$ is smooth $\pd{f}{p}$ is bounded. +Thus if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. It follows from Equation \eqref{eq:u} above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ (which is bounded) and the differences amongst the various $p(D_0,\alpha)$'s and $q_{\beta i}$'s. @@ -200,7 +194,7 @@ through essentially unchanged. Next we consider the case where $f$ is a family of bi-Lipschitz homeomorphisms. -We assume that $f$ is Lipschitz in $P$ direction as well. +Recall that we assume that $f$ is Lipschitz in the $P$ direction as well. The argument in this case is similar to the one above for diffeomorphisms, with bounded partial derivatives replaced by Lipschitz constants. Since $X$ and $P$ are compact, there is a universal bi-Lipschitz constant that works for @@ -214,15 +208,14 @@ \end{proof} \begin{lemma} -Let $CM_*(X\to T)$ be the singular chains on the space of continuous maps -[resp. immersions, diffeomorphisms, PL homeomorphisms, bi-Lipschitz homeomorphisms] -from $X$ to $T$, and let $G_*\sub CM_*(X\to T)$ denote the chains adapted to an open cover $\cU$ +Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, bi-Lipschitz homeomorphisms or PL homeomorphisms. +Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$ of $X$. -Then $G_*$ is a strong deformation retract of $CM_*(X\to T)$. +Then $G_*$ is a strong deformation retract of $\cX_*$. \end{lemma} \begin{proof} -If suffices to show that given a generator $f:P\times X\to T$ of $CM_k(X\to T)$ with -$\bd f \in G_{k-1}$ there exists $h\in CM_{k+1}(X\to T)$ with $\bd h = f + g$ and $g \in G_k$. +If suffices to show that given a generator $f:P\times X\to T$ of $\cX_k$ with +$\bd f \in G_{k-1}$ there exists $h\in \cX_{k+1}$ with $\bd h = f + g$ and $g \in G_k$. This is exactly what Lemma \ref{basic_adaptation_lemma} gives us. More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$. @@ -276,11 +269,3 @@ } % end \noop -\medskip -\hrule -\medskip - -\nn{do we want to keep this alternative construction?} - -\input{text/appendixes/explicit.tex} - diff -r 418919afd077 -r a746cd526cdd text/intro.tex --- a/text/intro.tex Thu May 27 17:35:56 2010 -0700 +++ b/text/intro.tex Thu May 27 17:39:11 2010 -0700 @@ -8,7 +8,7 @@ \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.) \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains -on the configurations space of unlabeled points in $M$. +on the configuration space of unlabeled points in $M$. %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ \end{itemize} The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of resolution), diff -r 418919afd077 -r a746cd526cdd text/ncat.tex --- a/text/ncat.tex Thu May 27 17:35:56 2010 -0700 +++ b/text/ncat.tex Thu May 27 17:39:11 2010 -0700 @@ -581,7 +581,16 @@ See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. -\begin{example}[Blob complexes of balls (with a fiber)] \rm \label{ex:blob-complexes-of-balls} Fix an $m$-dimensional manifold $F$ and system of fields $\cE$. We will define an $A_\infty$ $(n-m)$-category $\cC$. When $X$ is a $k$-ball or $k$-sphere, with $k