author | Kevin Walker <kevin@canyon23.net> |
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%!TEX root = ../blob1.tex |
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\section{The blob complex for $A_\infty$ $n$-categories} |
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\label{sec:ainfblob} |
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Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob |
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complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of Section \ref{ss:ncat_fields}. |
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We will show below |
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in Corollary \ref{cor:new-old} |
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that when $\cC$ is obtained from a topological $n$-category $\cD$ as the blob complex of a point, this agrees (up to homotopy) with our original definition of the blob complex |
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for $\cD$. |
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An important technical tool in the proofs of this section is provided by the idea of `small blobs'. |
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Fix $\cU$, an open cover of $M$. |
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Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set. |
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\begin{thm}[Small blobs] \label{thm:small-blobs} |
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The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
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\end{thm} |
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The proof appears in \S \ref{appendix:small-blobs}. |
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\subsection{A product formula} |
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\label{ss:product-formula} |
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\noop{ |
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Let $Y$ be a $k$-manifold, $F$ be an $n{-}k$-manifold, and |
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\[ |
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E = Y\times F . |
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\] |
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Let $\cC$ be an $n$-category. |
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Let $\cF$ be the $k$-category of Example \ref{ex:blob-complexes-of-balls}, |
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\[ |
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\cF(X) = \cC(X\times F) |
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\] |
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for $X$ an $m$-ball with $m\le k$. |
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} |
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\nn{need to settle on notation; proof and statement are inconsistent} |
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\begin{thm} \label{thm:product} |
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Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from |
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Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\bc_*(F; C)$ defined by |
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\begin{equation*} |
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\bc_*(F; C) = \cB_*(B \times F, C). |
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\end{equation*} |
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Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' |
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blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' |
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(i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$: |
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\begin{align*} |
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\cB_*(Y \times F; C) & \htpy \cl{\bc_*(F; C)}(Y) |
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\end{align*} |
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\end{thm} |
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\begin{proof} |
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We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
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First we define a map |
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\[ |
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\psi: \cl{\bc_*(F; C)}(Y) \to \bc_*(Y\times F;C) . |
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\] |
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In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$ |
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(where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
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$Y\times F$. |
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In filtration degrees 1 and higher we define the map to be zero. |
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It is easy to check that this is a chain map. |
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In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ |
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and a map |
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\[ |
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\phi: G_* \to \cl{\bc_*(F; C)}(Y) . |
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\] |
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Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
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decomposition of $Y\times F$ into the pieces $X_i\times F$. |
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Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there |
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exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. |
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It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ is homotopic to a subcomplex of $G_*$. |
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(If the blobs of $a$ are small with respect to a sufficiently fine cover then their |
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projections to $Y$ are contained in some disjoint union of balls.) |
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Note that the image of $\psi$ is equal to $G_*$. |
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We will define $\phi: G_* \to \cl{\bc_*(F; C)}(Y)$ using the method of acyclic models. |
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Let $a$ be a generator of $G_*$. |
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Let $D(a)$ denote the subcomplex of $\cl{\bc_*(F; C)}(Y)$ generated by all $(b, \ol{K})$ |
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such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing |
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in an iterated boundary of $a$ (this includes $a$ itself). |
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(Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
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see Subsection \ref{ss:ncat_fields}.) |
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By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
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$b$ split according to $K_0\times F$. |
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To simplify notation we will just write plain $b$ instead of $b^\sharp$. |
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Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
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$a$ (or one of its iterated boundaries), filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
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filtration degree 2 stuff which kills the homology created by the |
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filtration degree 1 stuff, and so on. |
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More formally, |
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\begin{lemma} \label{lem:d-a-acyclic} |
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$D(a)$ is acyclic. |
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\end{lemma} |
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\begin{proof} |
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We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least} |
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leave the general case to the reader. |
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Let $K$ and $K'$ be two decompositions of $Y$ compatible with $a$. |
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We want to show that $(a, K)$ and $(a, K')$ are homologous via filtration degree 1 stuff. |
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\nn{need to say this better; these two chains don't have the same boundary.} |
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We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
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the case. |
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(Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) |
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However, we {\it can} find another decomposition $L$ such that $L$ shares common |
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refinements with both $K$ and $K'$. |
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Let $KL$ and $K'L$ denote these two refinements. |
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Then filtration degree 1 chains associated to the four anti-refinemnts |
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$KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
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give the desired chain connecting $(a, K)$ and $(a, K')$ |
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(see Figure \ref{zzz4}). |
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\begin{figure}[!ht] |
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\begin{equation*} |
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\begin{tikzpicture} |
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\foreach \x/\label in {-3/K, 0/L, 3/K'} { |
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\node(\label) at (\x,0) {$\label$}; |
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} |
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\foreach \x/\la/\lb in {-1.5/K/L, 1.5/K'/L} { |
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\node(\la \lb) at (\x,-1.5) {$\la \lb$}; |
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\draw[->] (\la \lb) -- (\la); |
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\draw[->] (\la \lb) -- (\lb); |
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} |
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\end{tikzpicture} |
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\end{equation*} |
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\caption{Connecting $K$ and $K'$ via $L$} |
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\label{zzz4} |
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\end{figure} |
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Consider a different choice of decomposition $L'$ in place of $L$ above. |
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This leads to a cycle consisting of filtration degree 1 stuff. |
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We want to show that this cycle bounds a chain of filtration degree 2 stuff. |
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Choose a decomposition $M$ which has common refinements with each of |
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$K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
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(We also also require that $KLM$ antirefines to $KM$, etc.) |
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Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. |
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(Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) |
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\begin{figure}[!ht] |
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%\begin{equation*} |
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%\mathfig{1.0}{tempkw/zz5} |
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%\end{equation*} |
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\begin{equation*} |
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\begin{tikzpicture} |
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\node(M) at (0,0) {$M$}; |
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\foreach \angle/\label in {0/K', 45/K'L, 90/L, 135/KL, 180/K, 225/KL', 270/L', 315/K'L'} { |
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\node(\label) at (\angle:4) {$\label$}; |
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} |
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\foreach \label in {K', L, K, L'} { |
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\node(\label M) at ($(M)!0.6!(\label)$) {$\label M$}; |
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\draw[->] (\label M)--(M); |
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\draw[->] (\label M)--(\label); |
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} |
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\foreach \k in {K, K'} { |
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\foreach \l in {L, L'} { |
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\node(\k \l M) at (intersection cs: first line={(\k M)--(\l)}, second line={(\l M)--(\k)}) {$\k \l M$}; |
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\draw[->] (\k \l M)--(M); |
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\draw[->] (\k \l M)--(\k \l ); |
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\draw[->] (\k \l M)--(\k M); |
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\draw[->] (\k \l M)--(\l); |
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\draw[->] (\k \l M)--(\l M); |
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\draw[->] (\k \l M)--(\k); |
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} |
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} |
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\draw[->] (K'L') to[bend right=10] (K'); |
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\draw[->] (K'L') to[bend left=10] (L'); |
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\draw[->] (KL') to[bend left=10] (K); |
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\draw[->] (KL') to[bend right=10] (L'); |
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\draw[->] (K'L) to[bend left=10] (K'); |
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\draw[->] (K'L) to[bend right=10] (L); |
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\draw[->] (KL) to[bend right=10] (K); |
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\draw[->] (KL) to[bend left=10] (L); |
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\end{tikzpicture} |
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\end{equation*} |
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\caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
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\label{zzz5} |
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\end{figure} |
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Continuing in this way we see that $D(a)$ is acyclic. |
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\end{proof} |
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We are now in a position to apply the method of acyclic models to get a map |
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$\phi:G_* \to \cl{\bc_*(F; C)}(Y)$. |
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We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is in filtration degree zero |
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and $r$ has filtration degree greater than zero. |
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We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. |
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First, $\psi\circ\phi$ is the identity on the nose: |
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\[ |
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\psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0. |
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\] |
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Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and |
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$\psi$ glues those pieces back together, yielding $a$. |
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We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees. |
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Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models. |
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To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above. |
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Both the identity map and $\phi\circ\psi$ are compatible with this |
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collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps |
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are homotopic. |
123 | 211 |
|
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This concludes the proof of Theorem \ref{thm:product}. |
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\end{proof} |
214 |
||
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\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category} |
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|
123 | 217 |
\medskip |
113 | 218 |
|
123 | 219 |
\begin{cor} |
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\label{cor:new-old} |
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The blob complex of a manifold $M$ with coefficients in a topological $n$-category $\cC$ is homotopic to the homotopy colimit invariant of $M$ defined using the $A_\infty$ $n$-category obtained by applying the blob complex to a point: |
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$$\bc_*(M; \cC) \htpy \cl{\bc_*(pt; \cC)}(M).$$ |
123 | 223 |
\end{cor} |
224 |
\begin{proof} |
|
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Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point. |
123 | 226 |
\end{proof} |
113 | 227 |
|
228 |
\medskip |
|
133 | 229 |
|
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Theorem \ref{thm:product} extends to the case of general fiber bundles |
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\[ |
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F \to E \to Y . |
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\] |
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We outline one approach here and a second in Subsection xxxx. |
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We can generalize the definition of a $k$-category by replacing the categories |
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of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ |
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(c.f. \cite{MR2079378}). |
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Call this a $k$-category over $Y$. |
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A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
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assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$. |
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Let $\cF_E$ denote this $k$-category over $Y$. |
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We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
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get a chain complex $\cF_E(Y)$. |
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The proof of Theorem \ref{thm:product} goes through essentially unchanged |
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to show that |
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\[ |
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\bc_*(E) \simeq \cF_E(Y) . |
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\] |
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|
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|
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|
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\nn{put this later} |
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\nn{The second approach: Choose a decomposition $Y = \cup X_i$ |
312 | 256 |
such that the restriction of $E$ to $X_i$ is a product $F\times X_i$. |
257 |
Choose the product structure as well. |
|
258 |
To each codim-1 face $D_i\cap D_j$ we have a bimodule ($S^0$-module). |
|
259 |
And more generally to each codim-$j$ face we have an $S^{j-1}$-module. |
|
260 |
Decorate the decomposition with these modules and do the colimit. |
|
261 |
} |
|
262 |
||
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\nn{There is a version of this last construction for arbitrary maps $E \to Y$ |
312 | 264 |
(not necessarily a fibration).} |
265 |
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267 |
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\subsection{A gluing theorem} |
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\label{sec:gluing} |
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|
133 | 271 |
Next we prove a gluing theorem. |
272 |
Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$. |
|
273 |
We will need an explicit collar on $Y$, so rewrite this as |
|
274 |
$X = X_1\cup (Y\times J) \cup X_2$. |
|
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Given this data we have: |
133 | 276 |
\begin{itemize} |
277 |
\item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball |
|
278 |
$D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$ |
|
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(for $m+k = n$). |
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(See Example \ref{ex:blob-complexes-of-balls}.) |
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%\nn{need to explain $c$}. |
133 | 282 |
\item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly. |
283 |
\item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked |
|
284 |
$m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$) |
|
285 |
or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$). |
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(See Example \nn{need example for this}.) |
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\end{itemize} |
288 |
||
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\begin{thm} |
|
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\label{thm:gluing} |
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$\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
133 | 292 |
\end{thm} |
293 |
||
294 |
\begin{proof} |
|
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\nn{for now, just prove $k=0$ case.} |
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The proof is similar to that of Theorem \ref{thm:product}. |
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We give a short sketch with emphasis on the differences from |
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the proof of Theorem \ref{thm:product}. |
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Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
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Recall that this is a homotopy colimit based on decompositions of the interval $J$. |
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|
342 | 303 |
We define a map $\psi:\cT\to \bc_*(X)$. |
304 |
On filtration degree zero summands it is given |
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by gluing the pieces together to get a blob diagram on $X$. |
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On filtration degree 1 and greater $\psi$ is zero. |
133 | 307 |
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The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split |
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over some decomposition of $J$. |
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It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to |
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a subcomplex of $G_*$. |
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|
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Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models. |
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As in the proof of Theorem \ref{thm:product}, we assign to a generator $a$ of $G_*$ |
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an acyclic subcomplex which is (roughly) $\psi\inv(a)$. |
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The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have |
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a common refinement. |
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|
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The proof that these two maps are inverse to each other is the same as in |
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Theorem \ref{thm:product}. |
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321 |
\end{proof} |
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\noop{ |
133 | 324 |
Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
325 |
Let $D$ be an $n{-}k$-ball. |
|
326 |
There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$. |
|
327 |
To get a map in the other direction, we replace $\bc_*(D\times X)$ with a subcomplex |
|
328 |
$\cS_*$ which is adapted to a fine open cover of $D\times X$. |
|
329 |
For sufficiently small $j$ (depending on the cover), we can find, for each $j$-blob diagram $b$ |
|
330 |
on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding |
|
331 |
decomposition of $D\times X$. |
|
332 |
The proof that these two maps are inverse to each other is the same as in |
|
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Theorem \ref{thm:product}. |
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} |
133 | 335 |
|
336 |
||
337 |
\medskip |
|
211 | 338 |
|
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\subsection{Reconstructing mapping spaces} |
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\label{sec:map-recon} |
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|
211 | 342 |
The next theorem shows how to reconstruct a mapping space from local data. |
343 |
Let $T$ be a topological space, let $M$ be an $n$-manifold, |
|
344 |
and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ |
|
345 |
of Example \ref{ex:chains-of-maps-to-a-space}. |
|
346 |
Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
|
347 |
want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
|
348 |
To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
|
349 |
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\begin{thm} |
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\label{thm:map-recon} |
342 | 352 |
The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ |
353 |
is quasi-isomorphic to singular chains on maps from $M$ to $T$. |
|
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$$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
211 | 355 |
\end{thm} |
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\begin{rem} |
342 | 357 |
Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology |
358 |
of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers |
|
359 |
the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. |
|
360 |
This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} |
|
361 |
that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which |
|
362 |
is trivial at all but the topmost level. |
|
363 |
Ricardo Andrade also told us about a similar result. |
|
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\end{rem} |
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|
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\begin{proof} |
400
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The proof is again similar to that of Theorem \ref{thm:product}. |
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368 |
|
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369 |
We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
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370 |
|
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371 |
Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
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$j$-fold mapping cylinders, $j \ge 0$. |
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373 |
So, as an abelian group (but not as a chain complex), |
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\[ |
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\cB^\cT(M) = \bigoplus_{j\ge 0} C^j, |
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\] |
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377 |
where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders. |
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378 |
|
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Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by |
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380 |
decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms |
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of $\cT$. |
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Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs |
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$(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous |
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384 |
maps from the $n{-}1$-skeleton of $K$ to $T$. |
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385 |
The summand indexed by $(K, \vphi)$ is |
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386 |
\[ |
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387 |
\bigotimes_b D_*(b, \vphi), |
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388 |
\] |
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389 |
where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes |
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390 |
chains of maps from $b$ to $T$ compatible with $\vphi$. |
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391 |
We can take the product of these chains of maps to get a chains of maps from |
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392 |
all of $M$ to $K$. |
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393 |
This defines $\psi$ on $C^0$. |
325 | 394 |
|
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We define $\psi(C^j) = 0$ for $j > 0$. |
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396 |
It is not hard to see that this defines a chain map from |
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397 |
$\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
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398 |
|
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399 |
The image of $\psi$ is the subcomplex $G_*\sub C_*(\Maps(M\to T))$ generated by |
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400 |
families of maps whose support is contained in a disjoint union of balls. |
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401 |
It follows from Lemma \ref{extension_lemma_c} |
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402 |
that $C_*(\Maps(M\to T))$ is homotopic to a subcomplex of $G_*$. |
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403 |
|
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404 |
We will define a map $\phi:G_*\to \cB^\cT(M)$ via acyclic models. |
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405 |
Let $a$ be a generator of $G_*$. |
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406 |
Define $D(a)$ to be the subcomplex of $\cB^\cT(M)$ generated by all |
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407 |
pairs $(b, \ol{K})$, where $b$ is a generator appearing in an iterated boundary of $a$ |
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and $\ol{K}$ is an index of the homotopy colimit $\cB^\cT(M)$. |
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Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
Scott Morrison <scott@tqft.net>
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409 |
(See the proof of Theorem \ref{thm:product} for more details.) |
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410 |
The same proof as of Lemma \ref{lem:d-a-acyclic} shows that $D(a)$ is acyclic. |
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411 |
By the usual acyclic models nonsense, there is a (unique up to homotopy) |
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412 |
map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$. |
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413 |
Furthermore, we may choose $\phi$ such that for all $a$ |
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414 |
\[ |
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415 |
\phi(a) = (a, K) + r |
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416 |
\] |
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417 |
where $(a, K) \in C^0$ and $r\in \bigoplus_{j\ge 1} C^j$. |
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418 |
|
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419 |
It is now easy to see that $\psi\circ\phi$ is the identity on the nose. |
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420 |
Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity. |
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a02a6158f3bd
Breaking up 'properties' in the intro into smaller subsections, converting many properties back to theorems, and numbering according to where they occur in the text. Not completely done, e.g. the action map which needs statements made consistent.
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parents:
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diff
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|
421 |
(See the proof of Theorem \ref{thm:product} for more details.) |
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422 |
\end{proof} |
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423 |
|
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424 |
\noop{ |
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425 |
% old proof (just start): |
212 | 426 |
We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
325 | 427 |
We then use Lemma \ref{extension_lemma_c} to show that $g$ induces isomorphisms on homology. |
212 | 428 |
|
429 |
Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
|
430 |
$j$-fold mapping cylinders, $j \ge 0$. |
|
431 |
So, as an abelian group (but not as a chain complex), |
|
432 |
\[ |
|
433 |
\cB^\cT(M) = \bigoplus_{j\ge 0} C^j, |
|
434 |
\] |
|
435 |
where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders. |
|
436 |
||
437 |
Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by |
|
438 |
decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms |
|
439 |
of $\cT$. |
|
440 |
Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs |
|
441 |
$(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous |
|
442 |
maps from the $n{-}1$-skeleton of $K$ to $T$. |
|
443 |
The summand indexed by $(K, \vphi)$ is |
|
444 |
\[ |
|
445 |
\bigotimes_b D_*(b, \vphi), |
|
446 |
\] |
|
447 |
where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes |
|
448 |
chains of maps from $b$ to $T$ compatible with $\vphi$. |
|
449 |
We can take the product of these chains of maps to get a chains of maps from |
|
450 |
all of $M$ to $K$. |
|
451 |
This defines $g$ on $C^0$. |
|
452 |
||
453 |
We define $g(C^j) = 0$ for $j > 0$. |
|
454 |
It is not hard to see that this defines a chain map from |
|
455 |
$\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
|
456 |
||
457 |
\nn{...} |
|
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458 |
} |
211 | 459 |
|
212 | 460 |
\nn{maybe should also mention version where we enrich over |
325 | 461 |
spaces rather than chain complexes;} |
211 | 462 |
|
463 |
\medskip |
|
113 | 464 |
\hrule |
465 |
\medskip |
|
466 |
||
467 |
\nn{to be continued...} |
|
468 |
\medskip |
|
325 | 469 |
\nn{still to do: general maps} |
113 | 470 |
|
134 | 471 |
\todo{} |
472 |
Various citations we might want to make: |
|
473 |
\begin{itemize} |
|
474 |
\item \cite{MR2061854} McClure and Smith's review article |
|
475 |
\item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) |
|
476 |
\item \cite{MR0236922,MR0420609} Boardman and Vogt |
|
477 |
\item \cite{MR1256989} definition of framed little-discs operad |
|
478 |
\end{itemize} |
|
479 |
||
480 |