| author | Scott Morrison <scott@tqft.net> |
| Mon, 27 Sep 2010 20:57:02 -0700 | |
| changeset 561 | 77a80f91e214 |
| parent 560 | b138ee4a5938 |
| child 562 | 7725c999c3e7 |
| build.xml | file | annotate | diff | comparison | revisions | |
| code/signs.nb | file | annotate | diff | comparison | revisions | |
| diagrams/tempkw/zo1.pdf | file | annotate | diff | comparison | revisions | |
| diagrams/tempkw/zo3.pdf | file | annotate | diff | comparison | revisions | |
| diagrams/tempkw/zo4.pdf | file | annotate | diff | comparison | revisions | |
| diagrams/tempkw/zo5.pdf | file | annotate | diff | comparison | revisions | |
| gadgets-external.pdf | file | annotate | diff | comparison | revisions | |
| text/appendixes/smallblobs.tex | file | annotate | diff | comparison | revisions | |
| text/article_preamble.tex | file | annotate | diff | comparison | revisions | |
| text/comm_alg.tex | file | annotate | diff | comparison | revisions | |
| text/intro.tex | file | annotate | diff | comparison | revisions | |
| text/obsolete/comm_alg.tex | file | annotate | diff | comparison | revisions | |
| text/obsolete/smallblobs.tex | file | annotate | diff | comparison | revisions |
--- a/build.xml Fri Sep 24 15:32:55 2010 -0700 +++ b/build.xml Mon Sep 27 20:57:02 2010 -0700 @@ -41,7 +41,7 @@ <delete file="${arxivTarFile}"/> <delete file="${arxivTarFile}.gz"/> <tar destfile="${arxivTarFile}" basedir="." includes="**" - excludes="synctex.*,*.dvi,*.ps,*.pdf,*.png,${arxivTarFile},${arxivTarFile}.gz,sandbox.*,bibliography/**,papers/**,talks/**,diagrams/obsolete/**,.hg/**" + excludes="*.synctex*,*.dvi,*.ps,blob1.pdf,*.png,${arxivTarFile},${arxivTarFile}.gz,sandbox.*,bibliography/**,papers/**,talks/**,diagrams/obsolete/**,diagrams/latex2pdf/**,text/obsolete/**,.hg/**" /> <gzip src="${arxivTarFile}" destfile="${arxivTarFile}.gz"/> <delete file="${arxivTarFile}"/>
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--- a/text/appendixes/smallblobs.tex Fri Sep 24 15:32:55 2010 -0700 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,185 +0,0 @@ -%!TEX root = ../../blob1.tex -\section{The small blob complex} -\label{appendix:small-blobs} - -Before proving Theorem \ref{thm:small-blobs}, we need a lemma allowing us to choose a convenient action of families of diffeomorphisms. -Say that an open cover $\cV$ is strictly subordinate to $\cU$ if the closure of every open set of $\cV$ is contained in some open set of $\cU$. - -\begin{lem} -\label{lem:CH-small-blobs} -Fix an open cover $\cU$, and a sequence $\cV_k$ of open covers which are each strictly subordinate to $\cU$. For a given $k$, consider $\cG_k$ the subspace of $C_k(\Homeo(M)) \tensor \bc_*(M)$ spanned by $f \tensor b$, where $f:P^k \times M \to M$ is a $k$-parameter family of homeomorphisms such that for each $p \in P$, $f(p, -)$ makes $b$ small with respect to $\cV_k$. We can choose an up-to-homotopy representative $\ev$ of the chain map of Theorem \ref{thm:evaluation} which gives the action of families of homeomorphisms, which restricts to give a map -$$\ev : \cG_k \subset C_k(\Homeo(M)) \tensor \bc_*(M) \to \bc^{\cU}_*(M)$$ -for each $k$. -\end{lem} -\begin{rem} -We can't quite do the same with all $\cV_k$ just equal to $\cU$, but we can get by if we give ourselves arbitrarily little room to maneuver, by making the blobs we act on slightly smaller. -\end{rem} -\begin{proof} -This follows from Remark \ref{rem:for-small-blobs} following the proof of -Theorem \ref{thm:CH}. -\end{proof} - -\begin{proof}[Proof of Theorem \ref{thm:small-blobs}] -We begin by describing the homotopy inverse in small degrees, to illustrate the general technique. -We will construct a chain map $s: \bc_*(M) \to \bc^{\cU}_*(M)$ and a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ so that $\bdy h+h \bdy=i\circ s - \id$. The composition $s \circ i$ will just be the identity. - -On $0$-blobs, $s$ is just the identity; a blob diagram without any blobs is compatible with any open cover. -%\nn{KW: For some systems of fields this is not true. -%For example, consider a planar algebra with boxes of size greater than zero. -%So I think we should do the homotopy even in degree zero. -%But as noted above, maybe it's best to ignore this.} -Nevertheless, we'll begin introducing nomenclature at this point: for configuration $\beta$ of disjoint embedded balls in $M$ we'll associate a one parameter family of homeomorphisms $\phi_\beta : \Delta^1 \to \Homeo(M)$ (here $\Delta^m$ is the standard simplex $\setc{\mathbf{x} \in \Real^{m+1}}{\sum_{i=0}^m x_i = 1}$). For $0$-blobs, where $\beta = \eset$, all these homeomorphisms are just the identity. - -When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ ``makes $\beta$ small" if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$. Further, we'll say a homeomorphism ``makes $\beta$ $\epsilon$-small" if the image of each ball is contained in some open ball of radius $\epsilon$. - -On a $1$-blob $b$, with ball $\beta$, $s$ is defined as the sum of two terms. Essentially, the first term ``makes $\beta$ small", while the other term ``gets the boundary right". First, pick a one-parameter family $\phi_\beta : \Delta^1 \to \Homeo(M)$ of homeomorphisms, so $\phi_\beta(1,0)$ is the identity and $\phi_\beta(0,1)$ makes the ball $\beta$ small --- in fact, not just small with respect to $\cU$, but $\epsilon/2$-small, where $\epsilon > 0$ is such that every $\epsilon$-ball is contained in some open set of $\cU$. Next, pick a two-parameter family $\phi_{\eset \prec \beta} : \Delta^2 \to \Homeo(M)$ so that $\phi_{\eset \prec \beta}(0,x_1,x_2)$ makes the ball $\beta$ $\frac{3\epsilon}{4}$-small for all $x_1+x_2=1$, while $\phi_{\eset \prec \beta}(x_0,0,x_2) = \phi_\beta(x_0,x_2)$ and $\phi_{\eset \prec \beta}(x_0,x_1,0) = \phi_\eset(x_0,x_1)$. (It's perhaps not obvious that this is even possible --- see Lemma \ref{lem:extend-small-homeomorphisms} below.) We now define $s$ by -$$s(b) = \restrict{\phi_\beta}{x_0=0}(b) - \restrict{\phi_{\eset \prec \beta}}{x_0=0}(\bdy b).$$ -Here, $\restrict{\phi_\beta}{x_0=0} = \phi_\beta(0,1)$ is just a homeomorphism, which we apply to $b$, while $\restrict{\phi_{\eset \prec \beta}}{x_0=0}$ is a one parameter family of homeomorphisms which acts on the $0$-blob $\bdy b$ to give a $1$-blob. To be precise, this action is via the chain map identified in Lemma \ref{lem:CH-small-blobs} with $\cV_0$ the open cover by $\epsilon/2$-balls and $\cV_1$ the open cover by $\frac{3\epsilon}{4}$-balls. From this, it is immediate that $s(b) \in \bc^{\cU}_1(M)$, as desired. - -We now check that $s$, as defined so far, is a chain map, calculating -\begin{align*} -\bdy (s(b)) & = \restrict{\phi_\beta}{x_0=0}(\bdy b) - (\bdy \restrict{\phi_{\eset \prec \beta}}{x_0=0})(\bdy b) \\ - & = \restrict{\phi_\beta}{x_0=0}(\bdy b) - \restrict{\phi_\beta}{x_0=0}(\bdy b) + \restrict{\phi_\eset}{x_0=0}(\bdy b) \\ - & = \restrict{\phi_\eset}{x_0=0}(\bdy b) \\ - & = s(\bdy b) -\end{align*} -Next, we compute the compositions $s \circ i$ and $i \circ s$. If we start with a small $1$-blob diagram $b$, first include it up to the full blob complex then apply $s$, we get exactly back to $b$, at least assuming we adopt the convention that for any ball $\beta$ which is already small, we choose the families of homeomorphisms $\phi_\beta$ and $\phi_{\eset \prec \beta}$ to always be the identity. In the other direction, $i \circ s$, we will need to construct a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ for $*=0$ or $1$. - -The homotopy $h$ is defined by $$h(b) = \phi_\eset(b)$$ when $b$ is a $0$-blob (here $\phi_\eset$ is a one parameter family of homeomorphisms, so this is a $1$-blob), and $$h(b) = \phi_\beta(b) - \phi_{\eset \prec \beta}(\bdy b)$$ when $b$ is a $1$-blob (here $\beta$ is the ball in $b$, and the first term is the action of a one parameter family of homeomorphisms on a $1$-blob, and the second term is the action of a two parameter family of homeomorphisms on a $0$-blob, so both are $2$-blobs). We then calculate -\begin{align*} -(\bdy h+h \bdy)(b) & = \bdy (\phi_{\beta}(b) - \phi_{\eset \prec \beta}(\bdy b)) + \phi_\eset(\bdy b) \\ - & = \restrict{\phi_\beta}{x_0=0}(b) - \restrict{\phi_\beta}{x_1=0}(b) - \phi_\beta(\bdy b) - (\bdy \phi_{\eset \prec \beta})(\bdy b) + \phi_\eset(\bdy b) \\ - & = \restrict{\phi_\beta}{x_0=0}(b) - b - \phi_\beta(\bdy b) - \restrict{\phi_{\eset \prec \beta}}{x_0=0}(\bdy b) + \phi_\beta(\bdy b) - \phi_\eset(\bdy b) + \phi_\eset(\bdy b) \\ - & = \restrict{\phi_\beta}{x_0=0}(b) - b - \restrict{\phi_{\eset \prec \beta}}{x_0=0}(\bdy b) \\ - & = (i \circ s - \id)(b). -\end{align*} - - -In order to define $s$ on arbitrary blob diagrams, we first fix a sequence of strictly subordinate covers for $\cU$. First choose an $\epsilon > 0$ so every $\epsilon$ ball is contained in some open set of $\cU$. For $k \geq 1$, let $\cV_{k}$ be the open cover of $M$ by $\epsilon (1-2^{-k})$ balls, and $\cV_0 = \cU$. Certainly $\cV_k$ is strictly subordinate to $\cU$. We now chose the chain map $\ev$ provided by Lemma \ref{lem:CH-small-blobs} for the open covers $\cV_k$ strictly subordinate to $\cU$. Note that $\cV_1$ and $\cV_2$ have already implicitly appeared in the description above. - -Next, we choose a ``shrinking system" for $\left(\cU,\{\cV_k\}_{k \geq 1}\right)$, namely for each increasing sequence of blob configurations -$\beta_1 \prec \cdots \prec \beta_n$, an $n$ parameter family of diffeomorphisms -$\phi_{\beta_1 \prec \cdots \prec \beta_n} : \Delta^{n+1} \to \Diff{M}$, such that -\begin{itemize} -\item for any $x$ with $x_0 = 0$, $\phi_{\beta_1 \prec \cdots \prec \beta_n}(x)(\beta_1)$ is subordinate to $\cV_{n+1}$, and -\item for each $i = 1, \ldots, n$, -\begin{align*} -\phi_{\beta_1 \prec \cdots \prec \beta_n}&(x_0, \ldots, x_{i-1},0,x_{i+1},\ldots,x_n) = \\ &\phi_{\beta_1 \prec \cdots \prec \beta_{i-1} \prec \beta_{i+1} \prec \cdots \prec \beta_n}(x_0,\ldots, x_{i-1},x_{i+1},\ldots,x_n). -\end{align*} -\end{itemize} -Again, we've already made the choices for $\phi_{\beta}$ and for $\phi_{\eset \prec \beta}$, where $\beta$ is a single ball. It's not immediately obvious that it's possible to make such choices, but it follows readily from the following. - -\begin{lem} -\label{lem:extend-small-homeomorphisms} -Fix a collection of disjoint embedded balls $\beta$ in $M$ and some open cover $\cV$. Suppose we have a map $f : X \to \Homeo(M)$ on some compact $X$ such that for each $x \in \bdy X$, $f(x)$ makes $\beta$ $\cV$-small. Then we can extend $f$ to a map $\tilde{f} : X \times [0,1] \to \Homeo(M)$ so that $\tilde{f}(x,0) = f(x)$ and for every $x \in \bdy X \times [0,1] \cup X \times \{1\}$, $\tilde{f}(x)$ makes $\beta$ $\cV$-small. -\end{lem} -\begin{proof} -Fix a metric on $M$, and pick $\epsilon > 0$ so every $\epsilon$ ball in $M$ is contained in some open set of $\cV$. First construct a family of homeomorphisms $g_s : M \to M$, $s \in [1,\infty)$ so $g_1$ is the identity, and $g_s(\beta_i) \subset \beta_i$ and $\rad g_s(\beta_i) \leq \frac{1}{s} \rad \beta_i$ for each ball $\beta_i$. -There is some $K$ which uniformly bounds the expansion factors of all the homeomorphisms $f(x)$, that is $d(f(x)(a), f(x)(b)) < K d(a,b)$ for all $x \in X, a,b \in M$. Write $S=\epsilon^{-1} K \max_i \{\rad \beta_i\}$ (note that is $S<1$, we can just take $S=1$, as already $f(x)$ makes $\beta$ small for all $x$). Now define $\tilde{f}(t, x) = f(x) \compose g_{(S-1)t+1}$. - -If $x \in \bdy X$, then $g_{(S-1)t+1}(\beta_i) \subset \beta_i$, and by hypothesis $f(x)$ makes $\beta_i$ small, so $\tilde{f}(t, x)$ makes $\beta$ $\cV$-small for all $t \in [0,1]$. Alternatively, $\rad g_S(\beta_i) \leq \frac{1}{S} \rad \beta_i \leq \frac{\epsilon}{K}$, so $\rad \tilde{f}(1,x)(\beta_i) \leq \epsilon$, and so $\tilde{f}(1,x)$ makes $\beta$ $\cV$-small for all $x \in X$. -\end{proof} - -In fact, the application of this Lemma would allow us to choose the families of diffeomorphisms $\phi_{\beta_1 \prec \cdots \prec \beta_n}$ so that for any $x$ with $x_0 = 0$, $\phi_{\beta_1 \prec \cdots \prec \beta_n}(x)(\beta_1)$ is subordinate to any fixed open cover, for example $\cV_1$ (that is, the covering by $\epsilon/2$ balls), not just $\cV_{n+1}$, which is a weaker condition. Regardless, because of the way we have chosen the $\ev$ map, we only ensure that $\ev(\restrict{\phi_{\beta_1 \prec \cdots \prec \beta_n}}{x_0 = 0} \tensor \beta_1) \in \bc_{\deg \beta_1 + n}^{\cU}(M)$, so the distinction is not important. - -We now describe the general case. For a $k$-blob diagram $b \in \bc_k(M)$, denote by $b_\cS$ for $\cS \subset \{1, \ldots, k\}$ the blob diagram obtained by erasing the corresponding blobs. In particular, $b_\eset = b$, $b_{\{1,\ldots,k\}} \in \bc_0(M)$, and $\bdy b_\cS = \sum_{i \notin \cS} (-1)^{i+1+\card{\setc{j \in \cS}{j < i}}} b_{\cS \cup \{i\}}$. -Similarly, for a disjoint embedding of $k$ balls $\beta$ (that is, a blob diagram but without the labels on regions), $\beta_\cS$ denotes the result of erasing a subset of blobs. - -\newcommand{\length}[1]{\operatorname{length}(#1)} - -We've finally reached the point where we can define a map $s: \bc_*(M) \to \bc^{\cU}_*(M)$, and then a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ so that $dh+hd=i\circ s$. We have -$$s(b) = \sum_{m=0}^{k} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} (-1)^{\sigma(i)} \ev(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor b_i),$$ -where the sum is over sequences without repeats $i=(i_1,\ldots,i_m)$ in $\{1,\ldots,k\}$, with $0\leq m \leq k$ (we're using $\Delta$ here to indicate the generalized diagonal, where any two entries coincide), $\sigma(i)$ is defined blow, $i(b)$ denotes the increasing sequence of blob configurations -$$\beta_{(i_1,\ldots,i_m)} \prec \beta_{(i_2,\ldots,i_m)} \prec \cdots \prec \beta_{()},$$ -and, as usual, $b_i$ denotes $b$ with blobs $i_1, \ldots, i_m$ erased. -The homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ is similarly given by -$$h(b) = \sum_{m=0}^{k} \sum_{i} (-1)^{\sigma(i)} \ev(\phi_{i(b)}, b_i).$$ - - The signs $\sigma: \{1, \ldots, k\}^{m} \setminus \Delta \to \Integer/2\Integer$ are defined by -$$\sigma(i)= \lceil\frac{m-1}{2}\rceil +o(i) + \sum i$$ -where $o(i)$ is the number of transpositions required to bring $i$ into increasing order. Given a sequence $a \in \{1, \ldots, k\}^{m-1} \setminus \Delta$ and $1 \leq b \leq k$ with $b \not \in a$, denote by $a \!\downarrow_b\in \{1, \ldots, k-1\}^{m-1} \setminus \Delta$ the sequence obtained by reducing by 1 each entry of $a$ which is greater than $b$. We'll need the identities -\begin{align} -\sigma(ab) & = \sigma(a\!\downarrow_b) +m + b+1 \label{eq:sigma(ab)}\\ -% & = \sigma(a) + m+b + 1 + \card{\setc{x \in a}{x > b}} \\ -\intertext{and} -\sigma(ba) & = \sigma(a\!\downarrow_b) + b \notag \\ - & = \sigma(a) + m+b + 1+\card{\setc{x \in a}{x < b}}. \label{eq:sigma(ba)} -\end{align} - -Before completing the proof, we unpack this definition for $b \in \bc_2(M)$, a $2$-blob. We'll write $\beta$ for the underlying balls (either nested or disjoint). -Now $s$ is the sum of $5$ terms, split into three groups depending on with the length of the sequence $i$ is $0, 1$ or $2$. Thus -\begin{align*} -s(b) & = (-1)^{\sigma()} \restrict{\phi_{\beta}}{x_0 = 0}(b) + \\ - & \quad + (-1)^{\sigma(1)} \restrict{\phi_{\beta_1 \prec \beta}}{x_0 = 0}(b_1) + (-1)^{\sigma(2)} \restrict{\phi_{\beta_2 \prec \beta}}{x_0 = 0}(b_2) + \\ - & \quad + (-1)^{\sigma(12)} \restrict{\phi_{\eset \prec \beta_2 \prec \beta}}{x_0 = 0}(b_{12}) + (-1)^{\sigma(21)} \restrict{\phi_{\eset \prec \beta_1 \prec \beta}}{x_0 = 0}(b_{12}). -\end{align*} - -As in the $k=1$ case, the first term, corresponding to $i(b) = \eset$, makes the all balls in $\beta$ $\cV_1$-small. However, if this were the only term $s$ would not be a chain map, because we have no control over $\restrict{\phi_{\beta}}{x_0 = 0}(\bdy b)$. This necessitates the other terms, which fix the boundary at successively higher codimensions. - -It may be useful to look at Figure \ref{fig:erectly-a-tent-badly} to help understand the arrangement. The red, blue and orange $2$-cells there correspond to the $m=0$, $m=1$ and $m=2$ terms respectively, while the $3$-cells (only one of each type is shown) correspond to the terms in the homotopy $h$. -\begin{figure}[!ht] -$$\mathfig{0.5}{smallblobs/tent}$$ -\caption{``Erecting a tent badly.'' We know where we want to send a simplex, and each of the iterated boundary components. However, these do not agree, and we need to stitch the pieces together. Note that these diagrams don't exactly match the situation in the text: a $k$-simplex has $k+1$ boundary components, while a $k$-blob has $k$ boundary terms.} -\label{fig:erectly-a-tent-badly} -\end{figure} - -Now -\begin{align*} -\bdy s(b) & = (-1)^{\sigma()} \restrict{\phi_{\beta}}{x_0 = 0}(\bdy b) + \\ - & \quad + (-1)^{\sigma(1)} \left( \restrict{\phi_{\beta}}{x_0 = 0}(b_1) - \restrict{\phi_{\beta_1}}{x_0 = 0}(b_1) - \restrict{\phi_{\beta_1 \prec \beta}}{x_0 = 0}(b_{12}) \right) + \\ - & \quad + (-1)^{\sigma(2)} \left( \restrict{\phi_{\beta}}{x_0 = 0}(b_2) - \restrict{\phi_{\beta_2}}{x_0 = 0}(b_2) - \restrict{\phi_{\beta_2 \prec \beta}}{x_0 = 0}(b_{12}) \right) + \\ - & \quad + (-1)^{\sigma(12)} \left( \restrict{\phi_{\beta_2 \prec \beta}}{x_0 = 0}(b_{12}) - \restrict{\phi_{\eset \prec \beta}}{x_0 = 0}(b_{12}) + \restrict{\phi_{\eset \prec \beta_2}}{x_0 = 0}(b_{12}) \right) + \\ - & \quad + (-1)^{\sigma(21)} \left( \restrict{\phi_{\beta_1 \prec \beta}}{x_0 = 0}(b_{12}) - \restrict{\phi_{\eset \prec \beta}}{x_0 = 0}(b_{12}) + \restrict{\phi_{\eset \prec \beta_1}}{x_0 = 0}(b_{12}) \right), \\ -\intertext{while} -s(\bdy(b)) & = s(b_1) - s(b_2) \\ - & = \restrict{\phi_{\beta_1}}{x_0=0}(b_1) - \restrict{\phi_{\eset \prec \beta_1}}{x_0=0}(b_{12}) - \restrict{\phi_{\beta_2}}{x_0=0}(b_2) + \restrict{\phi_{\eset \prec \beta_2}}{x_0=0}(b_{12}) . -\end{align*} -This gives what we want, since $\sigma() = 0,\sigma(1)=1, \sigma(2)=0, \sigma(21)=1$ and $\sigma(12)=0$. - -We now return to the general case. Certainly, the image of $s$ in fact lies in $\bc^{\cU}_*(M)$: each of the families of diffeomorphisms $\phi_{i(b)}$ has been chosen so with $x_0=0$ they pointwise make $b_i$ sufficiently small that the $\ev$ map we've chosen has image in $\bc^{\cU}_*(M)$. - -We need to check that $s$ is a chain map. -\begin{align*} -\bdy(s(b)) & = \sum_{m=0}^{k} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} (-1)^{\sigma(i)} \ev\left(\bdy(\restrict{\phi_{i(b)}}{x_0 = 0})\tensor b_i\right) + (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor \bdy b_i\right) \\ -\intertext{and begin by expanding out $\bdy(\restrict{\phi_{i(b)}}{x_0 = 0})$,} - & = \sum_{m=0}^{k} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} \Bigg(\sum_{p=1}^{m+1} (-1)^{\sigma(i)+p+1} \ev\left(\restrict{\phi_{i(b)}}{x_0 = x_p = 0}\tensor b_i\right) \Bigg) + \\ - & \qquad + (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor \bdy b_i\right) -\end{align*} -Now, write $s_{p-1,p}(i)$ to indicate the sequence obtained from $i$ by transposing its $p-1$-th and $p$-th entries and note that for $2 \leq p \leq m$, -\begin{align*} -\restrict{\phi_{i(b)}}{x_0=x_p=0} & = \restrict{\phi_{\beta_{i_1\cdots i_m} \prec \beta_{i_2 \cdots i_m} \prec \cdots \prec \beta}}{x_0=x_p=0} \\ - & = \restrict{\phi_{\beta_{i_1\cdots i_m} \prec \beta_{i_2 \cdots i_m} \prec \cdots \prec \beta_{i_{p-1} i_p \cdots i_m} \prec \beta_{i_{p+1} \cdots i_m} \prec \cdots \prec \beta}}{x_0=0} \\ - & = \restrict{\phi_{s_{p-1,p}(i)(b)}}{x_0=x_p=0}. -\end{align*} -Since $\sigma(i) = - \sigma(s_{p-1,p}(i))$, we can cancel out in pairs all the terms above except those with $p=1$ or $p=m+1$. Thus -\begin{align*} -\bdy(s(b)) & = \sum_{m=0}^{k} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} \Bigg((-1)^{\sigma(i)} \ev\left(\restrict{\phi_{\rest(i)(b)}}{x_0 = 0}\tensor b_i\right) + (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{\most(i)\!\downarrow_{i_m}(b_{i_m})}}{x_0 = 0}\tensor b_i\right)\Bigg) + \\ - & \qquad + (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor \bdy b_i\right) -\end{align*} -where we use the notations $\rest(i_1 i_2 \cdots i_m) = (i_2 \cdots i_m)$ and $\most(i_1 \cdots i_{m-1} i_m) = (i_1 \cdots i_{m-1})$. Next, we note that $b_i = (b_{i_1})_{\rest(i)} = (b_{i_m})_{\most(i)}$, and then rewrite the sum of $i$ as a double sum over $i_1$ and $\rest(i)$, with $i = i_1\rest(i)$, for the first term, and as a double sum over $\most(i)$ and $i_m$, with $i = \most(i)i_m$, for the second term. -\begin{align*} -\bdy(s(b)) & = \sum_{m=0}^{k} \Bigg( \sum_{\rest(i) \in \{1, \ldots, k\}^{m-1} \setminus \Delta} \sum_{\substack{i_1 = 1 \\ i_1 \not\in \rest(i)}}^{k} (-1)^{\sigma(i_1\rest(i))} \ev\left(\restrict{\phi_{\rest(i)(b)}}{x_0 = 0}\tensor b_{i_1\rest(i)}\right) \Bigg)+ \\ - & \qquad \Bigg( \sum_{\most(i) \in \{1, \ldots, k\}^{m-1} \setminus \Delta} \sum_{\substack{i_m = 1 \\ i_1 \not\in \most(i)}}^{k} (-1)^{\sigma(\most(i) i_m) + m} \ev\left(\restrict{\phi_{\most(i)\!\downarrow_{i_m}(b_{i_m})}}{x_0 = 0}\tensor b_{\most(i) i_m}\right)\Bigg) + \\ - & \qquad \Bigg( \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor \bdy b_i\right)\Bigg) -\end{align*} -We will show that the first and third rows cancel, and that the second row gives with $s(\bdy b)$. -First, -\begin{align*} -\sum_{\substack{i_1 = 1 \\ i_1 \not\in \rest(i)}}^{k} (-1)^{\sigma(i_1\rest(i))} b_{i_1\rest(i)} & = - \sum_{\substack{i_1 = 1 \\ i_1 \not\in \rest(i)}}^{k} (-1)^{\sigma(\rest(i))+m+1+i_1 + \card{\setc{x \in \rest(i)}{x<i_1}}} (b_{\rest(i)})_{i_1} \\ -\intertext{(using Equation \eqref{eq:sigma(ba)})} -& = (-1)^{\sigma(\rest(i))+m} \bdy (b_{\rest(i)}), -\end{align*} -and this cancels exactly with the term indexed by $\rest(i)$ (with a value of $m$ off by one) in the third row. -The second row gives -\begin{align*} -& \sum_{m=0}^k \sum_{\most(i) \in \{1, \ldots, k\}^{m-1} \setminus \Delta} \sum_{\substack{i_m = 1 \\ i_1 \not\in \most(i)}}^{k} (-1)^{\sigma(\most(i) i_m) + m} \ev\left(\restrict{\phi_{\most(i)\!\downarrow_{i_m}(b_{i_m})}}{x_0 = 0}\tensor b_{\most(i) i_m}\right) \\ -& \quad = \sum_{m=0}^{k-1} \sum_{q=1}^k \sum_{i \in \{1, \ldots, k-1\}^m\setminus \Delta} (-1)^{\sigma(i)+q+1} \ev\left(\restrict{\phi_{i(b_{q})}}{x_0 = 0}\tensor (b_q)_i\right) \\ -\intertext{(here we've used Equation \eqref{eq:sigma(ab)} and renamed $i_m$ to $q$ and $\most(i)$ to $i$, as well as shifted $m$ by one), which is just} -& \quad = \sum_{q=1}^k (-1)^{q+1} s(b_q) \\ -& \quad = s(\bdy b). -\end{align*} - -Finally, the calculation that $\bdy h+h \bdy=i\circ s - \id$ is very similar, and we omit it. -\end{proof} - \ No newline at end of file
--- a/text/article_preamble.tex Fri Sep 24 15:32:55 2010 -0700 +++ b/text/article_preamble.tex Mon Sep 27 20:57:02 2010 -0700 @@ -20,6 +20,8 @@ \usetikzlibrary{decorations,decorations.pathreplacing} \usetikzlibrary{fit,calc,through} +\pgfrealjobname{blob1} + \newtheorem{example}[prop]{Example} \usepackage{color}
--- a/text/comm_alg.tex Fri Sep 24 15:32:55 2010 -0700 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,193 +0,0 @@ -%!TEX root = ../blob1.tex - -\section{Commutative algebras as \texorpdfstring{$n$}{n}-categories} -\label{sec:comm_alg} - -If $C$ is a commutative algebra it -can also be thought of as an $n$-category whose $j$-morphisms are trivial for -$j<n$ and whose $n$-morphisms are $C$. -The goal of this appendix is to compute -$\bc_*(M^n, C)$ for various commutative algebras $C$. - -Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative -algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with -coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). -This possibility was suggested to us by Thomas Tradler. - - -\medskip - -Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. - -Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ -unlabeled points in $M$. -Note that $\Sigma^0(M)$ is a point. -Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. - -Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$. - -\begin{prop} \label{sympowerprop} -$\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. -\end{prop} - -\begin{proof} -We will use acyclic models (\S \ref{sec:moam}). -Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ -satisfying the conditions of Theorem \ref{moam-thm}. -If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a -finite unordered collection of points of $M$ with multiplicities, which is -a point in $\Sigma^\infty(M)$. -Define $R(b)_*$ to be the singular chain complex of this point. -If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). -The path components of $\Sigma^\infty(D)$ are contractible, and these components are indexed -by the numbers of points in each component of $D$. -We may assume that the blob labels $u$ have homogeneous $t$ degree in $k[t]$, and so -$u$ picks out a component $X \sub \Sigma^\infty(D)$. -The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, -and using this point we can embed $X$ in $\Sigma^\infty(M)$. -Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a -subspace of $\Sigma^\infty(M)$. -It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from -Theorem \ref{moam-thm}. -Thus we have defined (up to homotopy) a map from -$\bc_*(M, k[t])$ to $C_*(\Sigma^\infty(M))$. - -Next we define a map going the other direction. -First we replace $C_*(\Sigma^\infty(M))$ with a homotopy equivalent -subcomplex $S_*$ of small simplices. -Roughly, we define $c\in C_*(\Sigma^\infty(M))$ to be small if the -corresponding track of points in $M$ -is contained in a disjoint union of balls. -Because there could be different, inequivalent choices of such balls, we must a bit more careful. -\nn{this runs into the same issues as in defining evmap. -either refer there for details, or use the simp-space-ish version of the blob complex, -which makes things easier here.} - -\nn{...} - - -We will define, for each simplex $c$ of $S_*$, a contractible subspace -$R(c)_* \sub \bc_*(M, k[t])$. -If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and -$\Sigma^\infty(M)$ described above. -Now let $c$ be an $i$-simplex of $S_*$. -Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. -We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ -is homotopy equivalent to the subcomplex of small simplices. -How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. -Let $T\sub M$ be the ``track" of $c$ in $M$. -\nn{do we need to define this precisely?} -Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. -\nn{need to say more precisely how small} -Define $R(c)_*$ to be $\bc_*(D; k[t]) \sub \bc_*(M; k[t])$. -This is contractible by Proposition \ref{bcontract}. -We can arrange that the boundary/inclusion condition is satisfied if we start with -low-dimensional simplices and work our way up. -\nn{need to be more precise} - -\nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity -(for this, might need a lemma that says we can assume that blob diameters are small)} -\end{proof} - - -\begin{prop} \label{ktchprop} -The above maps are compatible with the evaluation map actions of $C_*(\Homeo(M))$. -\end{prop} - -\begin{proof} -The actions agree in degree 0, and both are compatible with gluing. -(cf. uniqueness statement in Theorem \ref{thm:CH}.) -\nn{if Theorem \ref{thm:CH} is rewritten/rearranged, make sure uniqueness discussion is properly referenced from here} -\end{proof} - -\medskip - -In view of Theorem \ref{thm:hochschild}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, -and that the cyclic homology of $k[t]$ is related to the action of rotations -on $C_*(\Sigma^\infty(S^1), k)$. -\nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} -Let us check this directly. - -The algebra $k[t]$ has a resolution -$k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, -which has coinvariants $k[t] \xrightarrow{0} k[t]$. -So we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. -(See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: -$HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one. - -We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. -The fixed points of this flow are the equally spaced configurations. -This defines a deformation retraction from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). -The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, -and the holonomy of the $\Delta^{j-1}$ bundle -over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices. - -In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and -of course $\Sigma^0(S^1)$ is a point. -Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ -and is zero for $i\ge 2$. -Note that the $j$-grading here matches with the $t$-grading on the algebraic side. - -By Proposition \ref{ktchprop}, -the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. -Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. -If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree -0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. -The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even -degrees and 0 in odd degrees. -This agrees with the calculation in \cite[\S 3.1.7]{MR1600246}. - -\medskip - -Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. -Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations -of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable. -The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers -corresponding to the number of points of each color of a configuration. -A proof similar to that of \ref{sympowerprop} shows that - -\begin{prop} -$\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. -\end{prop} - -According to \cite[3.2.2]{MR1600246}, -\[ - HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . -\] -Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. -We will content ourselves with the case $k = \z$. -One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the -same color repel each other and points of different colors do not interact. -This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent -to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple -corresponding to $X$. -The homology calculation we desire follows easily from this. - -%\nn{say something about cyclic homology in this case? probably not necessary.} - -\medskip - -Next we consider the case $C$ is the truncated polynomial -algebra $k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. -Define $\Delta_l \sub \Sigma^\infty(M)$ to be configurations of points in $M$ with $l$ or -more of the points coinciding. - -\begin{prop} -$\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ -(relative singular chains with coefficients in $k$). -\end{prop} - -\begin{proof} -\nn{...} -\end{proof} - -\medskip -\hrule -\medskip - -Still to do: -\begin{itemize} -\item compare the topological computation for truncated polynomial algebra with \cite{MR1600246} -\item multivariable truncated polynomial algebras (at least mention them) -\end{itemize} -
--- a/text/intro.tex Fri Sep 24 15:32:55 2010 -0700 +++ b/text/intro.tex Mon Sep 27 20:57:02 2010 -0700 @@ -93,6 +93,7 @@ \begin{figure}[!ht] {\center +\beginpgfgraphicnamed{gadgets-external}% \begin{tikzpicture}[align=center,line width = 1.5pt] \newcommand{\xa}{2} \newcommand{\xb}{8} @@ -129,6 +130,7 @@ \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); \draw[<->] (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); \end{tikzpicture} +\endpgfgraphicnamed{gadgets-external}% } \caption{The main gadgets and constructions of the paper.}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/text/obsolete/comm_alg.tex Mon Sep 27 20:57:02 2010 -0700 @@ -0,0 +1,193 @@ +%!TEX root = ../blob1.tex + +\section{Commutative algebras as \texorpdfstring{$n$}{n}-categories} +\label{sec:comm_alg} + +If $C$ is a commutative algebra it +can also be thought of as an $n$-category whose $j$-morphisms are trivial for +$j<n$ and whose $n$-morphisms are $C$. +The goal of this appendix is to compute +$\bc_*(M^n, C)$ for various commutative algebras $C$. + +Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative +algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with +coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). +This possibility was suggested to us by Thomas Tradler. + + +\medskip + +Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. + +Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ +unlabeled points in $M$. +Note that $\Sigma^0(M)$ is a point. +Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. + +Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$. + +\begin{prop} \label{sympowerprop} +$\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. +\end{prop} + +\begin{proof} +We will use acyclic models (\S \ref{sec:moam}). +Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ +satisfying the conditions of Theorem \ref{moam-thm}. +If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a +finite unordered collection of points of $M$ with multiplicities, which is +a point in $\Sigma^\infty(M)$. +Define $R(b)_*$ to be the singular chain complex of this point. +If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). +The path components of $\Sigma^\infty(D)$ are contractible, and these components are indexed +by the numbers of points in each component of $D$. +We may assume that the blob labels $u$ have homogeneous $t$ degree in $k[t]$, and so +$u$ picks out a component $X \sub \Sigma^\infty(D)$. +The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, +and using this point we can embed $X$ in $\Sigma^\infty(M)$. +Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a +subspace of $\Sigma^\infty(M)$. +It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from +Theorem \ref{moam-thm}. +Thus we have defined (up to homotopy) a map from +$\bc_*(M, k[t])$ to $C_*(\Sigma^\infty(M))$. + +Next we define a map going the other direction. +First we replace $C_*(\Sigma^\infty(M))$ with a homotopy equivalent +subcomplex $S_*$ of small simplices. +Roughly, we define $c\in C_*(\Sigma^\infty(M))$ to be small if the +corresponding track of points in $M$ +is contained in a disjoint union of balls. +Because there could be different, inequivalent choices of such balls, we must a bit more careful. +\nn{this runs into the same issues as in defining evmap. +either refer there for details, or use the simp-space-ish version of the blob complex, +which makes things easier here.} + +\nn{...} + + +We will define, for each simplex $c$ of $S_*$, a contractible subspace +$R(c)_* \sub \bc_*(M, k[t])$. +If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and +$\Sigma^\infty(M)$ described above. +Now let $c$ be an $i$-simplex of $S_*$. +Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. +We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ +is homotopy equivalent to the subcomplex of small simplices. +How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. +Let $T\sub M$ be the ``track" of $c$ in $M$. +\nn{do we need to define this precisely?} +Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. +\nn{need to say more precisely how small} +Define $R(c)_*$ to be $\bc_*(D; k[t]) \sub \bc_*(M; k[t])$. +This is contractible by Proposition \ref{bcontract}. +We can arrange that the boundary/inclusion condition is satisfied if we start with +low-dimensional simplices and work our way up. +\nn{need to be more precise} + +\nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity +(for this, might need a lemma that says we can assume that blob diameters are small)} +\end{proof} + + +\begin{prop} \label{ktchprop} +The above maps are compatible with the evaluation map actions of $C_*(\Homeo(M))$. +\end{prop} + +\begin{proof} +The actions agree in degree 0, and both are compatible with gluing. +(cf. uniqueness statement in Theorem \ref{thm:CH}.) +\nn{if Theorem \ref{thm:CH} is rewritten/rearranged, make sure uniqueness discussion is properly referenced from here} +\end{proof} + +\medskip + +In view of Theorem \ref{thm:hochschild}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, +and that the cyclic homology of $k[t]$ is related to the action of rotations +on $C_*(\Sigma^\infty(S^1), k)$. +\nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} +Let us check this directly. + +The algebra $k[t]$ has a resolution +$k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, +which has coinvariants $k[t] \xrightarrow{0} k[t]$. +So we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. +(See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: +$HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one. + +We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. +The fixed points of this flow are the equally spaced configurations. +This defines a deformation retraction from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). +The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, +and the holonomy of the $\Delta^{j-1}$ bundle +over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices. + +In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and +of course $\Sigma^0(S^1)$ is a point. +Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ +and is zero for $i\ge 2$. +Note that the $j$-grading here matches with the $t$-grading on the algebraic side. + +By Proposition \ref{ktchprop}, +the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. +Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. +If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree +0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. +The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even +degrees and 0 in odd degrees. +This agrees with the calculation in \cite[\S 3.1.7]{MR1600246}. + +\medskip + +Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. +Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations +of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable. +The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers +corresponding to the number of points of each color of a configuration. +A proof similar to that of \ref{sympowerprop} shows that + +\begin{prop} +$\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. +\end{prop} + +According to \cite[3.2.2]{MR1600246}, +\[ + HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . +\] +Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. +We will content ourselves with the case $k = \z$. +One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the +same color repel each other and points of different colors do not interact. +This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent +to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple +corresponding to $X$. +The homology calculation we desire follows easily from this. + +%\nn{say something about cyclic homology in this case? probably not necessary.} + +\medskip + +Next we consider the case $C$ is the truncated polynomial +algebra $k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. +Define $\Delta_l \sub \Sigma^\infty(M)$ to be configurations of points in $M$ with $l$ or +more of the points coinciding. + +\begin{prop} +$\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ +(relative singular chains with coefficients in $k$). +\end{prop} + +\begin{proof} +\nn{...} +\end{proof} + +\medskip +\hrule +\medskip + +Still to do: +\begin{itemize} +\item compare the topological computation for truncated polynomial algebra with \cite{MR1600246} +\item multivariable truncated polynomial algebras (at least mention them) +\end{itemize} +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/text/obsolete/smallblobs.tex Mon Sep 27 20:57:02 2010 -0700 @@ -0,0 +1,185 @@ +%!TEX root = ../../blob1.tex +\section{The small blob complex} +\label{appendix:small-blobs} + +Before proving Theorem \ref{thm:small-blobs}, we need a lemma allowing us to choose a convenient action of families of diffeomorphisms. +Say that an open cover $\cV$ is strictly subordinate to $\cU$ if the closure of every open set of $\cV$ is contained in some open set of $\cU$. + +\begin{lem} +\label{lem:CH-small-blobs} +Fix an open cover $\cU$, and a sequence $\cV_k$ of open covers which are each strictly subordinate to $\cU$. For a given $k$, consider $\cG_k$ the subspace of $C_k(\Homeo(M)) \tensor \bc_*(M)$ spanned by $f \tensor b$, where $f:P^k \times M \to M$ is a $k$-parameter family of homeomorphisms such that for each $p \in P$, $f(p, -)$ makes $b$ small with respect to $\cV_k$. We can choose an up-to-homotopy representative $\ev$ of the chain map of Theorem \ref{thm:evaluation} which gives the action of families of homeomorphisms, which restricts to give a map +$$\ev : \cG_k \subset C_k(\Homeo(M)) \tensor \bc_*(M) \to \bc^{\cU}_*(M)$$ +for each $k$. +\end{lem} +\begin{rem} +We can't quite do the same with all $\cV_k$ just equal to $\cU$, but we can get by if we give ourselves arbitrarily little room to maneuver, by making the blobs we act on slightly smaller. +\end{rem} +\begin{proof} +This follows from Remark \ref{rem:for-small-blobs} following the proof of +Theorem \ref{thm:CH}. +\end{proof} + +\begin{proof}[Proof of Theorem \ref{thm:small-blobs}] +We begin by describing the homotopy inverse in small degrees, to illustrate the general technique. +We will construct a chain map $s: \bc_*(M) \to \bc^{\cU}_*(M)$ and a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ so that $\bdy h+h \bdy=i\circ s - \id$. The composition $s \circ i$ will just be the identity. + +On $0$-blobs, $s$ is just the identity; a blob diagram without any blobs is compatible with any open cover. +%\nn{KW: For some systems of fields this is not true. +%For example, consider a planar algebra with boxes of size greater than zero. +%So I think we should do the homotopy even in degree zero. +%But as noted above, maybe it's best to ignore this.} +Nevertheless, we'll begin introducing nomenclature at this point: for configuration $\beta$ of disjoint embedded balls in $M$ we'll associate a one parameter family of homeomorphisms $\phi_\beta : \Delta^1 \to \Homeo(M)$ (here $\Delta^m$ is the standard simplex $\setc{\mathbf{x} \in \Real^{m+1}}{\sum_{i=0}^m x_i = 1}$). For $0$-blobs, where $\beta = \eset$, all these homeomorphisms are just the identity. + +When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ ``makes $\beta$ small" if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$. Further, we'll say a homeomorphism ``makes $\beta$ $\epsilon$-small" if the image of each ball is contained in some open ball of radius $\epsilon$. + +On a $1$-blob $b$, with ball $\beta$, $s$ is defined as the sum of two terms. Essentially, the first term ``makes $\beta$ small", while the other term ``gets the boundary right". First, pick a one-parameter family $\phi_\beta : \Delta^1 \to \Homeo(M)$ of homeomorphisms, so $\phi_\beta(1,0)$ is the identity and $\phi_\beta(0,1)$ makes the ball $\beta$ small --- in fact, not just small with respect to $\cU$, but $\epsilon/2$-small, where $\epsilon > 0$ is such that every $\epsilon$-ball is contained in some open set of $\cU$. Next, pick a two-parameter family $\phi_{\eset \prec \beta} : \Delta^2 \to \Homeo(M)$ so that $\phi_{\eset \prec \beta}(0,x_1,x_2)$ makes the ball $\beta$ $\frac{3\epsilon}{4}$-small for all $x_1+x_2=1$, while $\phi_{\eset \prec \beta}(x_0,0,x_2) = \phi_\beta(x_0,x_2)$ and $\phi_{\eset \prec \beta}(x_0,x_1,0) = \phi_\eset(x_0,x_1)$. (It's perhaps not obvious that this is even possible --- see Lemma \ref{lem:extend-small-homeomorphisms} below.) We now define $s$ by +$$s(b) = \restrict{\phi_\beta}{x_0=0}(b) - \restrict{\phi_{\eset \prec \beta}}{x_0=0}(\bdy b).$$ +Here, $\restrict{\phi_\beta}{x_0=0} = \phi_\beta(0,1)$ is just a homeomorphism, which we apply to $b$, while $\restrict{\phi_{\eset \prec \beta}}{x_0=0}$ is a one parameter family of homeomorphisms which acts on the $0$-blob $\bdy b$ to give a $1$-blob. To be precise, this action is via the chain map identified in Lemma \ref{lem:CH-small-blobs} with $\cV_0$ the open cover by $\epsilon/2$-balls and $\cV_1$ the open cover by $\frac{3\epsilon}{4}$-balls. From this, it is immediate that $s(b) \in \bc^{\cU}_1(M)$, as desired. + +We now check that $s$, as defined so far, is a chain map, calculating +\begin{align*} +\bdy (s(b)) & = \restrict{\phi_\beta}{x_0=0}(\bdy b) - (\bdy \restrict{\phi_{\eset \prec \beta}}{x_0=0})(\bdy b) \\ + & = \restrict{\phi_\beta}{x_0=0}(\bdy b) - \restrict{\phi_\beta}{x_0=0}(\bdy b) + \restrict{\phi_\eset}{x_0=0}(\bdy b) \\ + & = \restrict{\phi_\eset}{x_0=0}(\bdy b) \\ + & = s(\bdy b) +\end{align*} +Next, we compute the compositions $s \circ i$ and $i \circ s$. If we start with a small $1$-blob diagram $b$, first include it up to the full blob complex then apply $s$, we get exactly back to $b$, at least assuming we adopt the convention that for any ball $\beta$ which is already small, we choose the families of homeomorphisms $\phi_\beta$ and $\phi_{\eset \prec \beta}$ to always be the identity. In the other direction, $i \circ s$, we will need to construct a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ for $*=0$ or $1$. + +The homotopy $h$ is defined by $$h(b) = \phi_\eset(b)$$ when $b$ is a $0$-blob (here $\phi_\eset$ is a one parameter family of homeomorphisms, so this is a $1$-blob), and $$h(b) = \phi_\beta(b) - \phi_{\eset \prec \beta}(\bdy b)$$ when $b$ is a $1$-blob (here $\beta$ is the ball in $b$, and the first term is the action of a one parameter family of homeomorphisms on a $1$-blob, and the second term is the action of a two parameter family of homeomorphisms on a $0$-blob, so both are $2$-blobs). We then calculate +\begin{align*} +(\bdy h+h \bdy)(b) & = \bdy (\phi_{\beta}(b) - \phi_{\eset \prec \beta}(\bdy b)) + \phi_\eset(\bdy b) \\ + & = \restrict{\phi_\beta}{x_0=0}(b) - \restrict{\phi_\beta}{x_1=0}(b) - \phi_\beta(\bdy b) - (\bdy \phi_{\eset \prec \beta})(\bdy b) + \phi_\eset(\bdy b) \\ + & = \restrict{\phi_\beta}{x_0=0}(b) - b - \phi_\beta(\bdy b) - \restrict{\phi_{\eset \prec \beta}}{x_0=0}(\bdy b) + \phi_\beta(\bdy b) - \phi_\eset(\bdy b) + \phi_\eset(\bdy b) \\ + & = \restrict{\phi_\beta}{x_0=0}(b) - b - \restrict{\phi_{\eset \prec \beta}}{x_0=0}(\bdy b) \\ + & = (i \circ s - \id)(b). +\end{align*} + + +In order to define $s$ on arbitrary blob diagrams, we first fix a sequence of strictly subordinate covers for $\cU$. First choose an $\epsilon > 0$ so every $\epsilon$ ball is contained in some open set of $\cU$. For $k \geq 1$, let $\cV_{k}$ be the open cover of $M$ by $\epsilon (1-2^{-k})$ balls, and $\cV_0 = \cU$. Certainly $\cV_k$ is strictly subordinate to $\cU$. We now chose the chain map $\ev$ provided by Lemma \ref{lem:CH-small-blobs} for the open covers $\cV_k$ strictly subordinate to $\cU$. Note that $\cV_1$ and $\cV_2$ have already implicitly appeared in the description above. + +Next, we choose a ``shrinking system" for $\left(\cU,\{\cV_k\}_{k \geq 1}\right)$, namely for each increasing sequence of blob configurations +$\beta_1 \prec \cdots \prec \beta_n$, an $n$ parameter family of diffeomorphisms +$\phi_{\beta_1 \prec \cdots \prec \beta_n} : \Delta^{n+1} \to \Diff{M}$, such that +\begin{itemize} +\item for any $x$ with $x_0 = 0$, $\phi_{\beta_1 \prec \cdots \prec \beta_n}(x)(\beta_1)$ is subordinate to $\cV_{n+1}$, and +\item for each $i = 1, \ldots, n$, +\begin{align*} +\phi_{\beta_1 \prec \cdots \prec \beta_n}&(x_0, \ldots, x_{i-1},0,x_{i+1},\ldots,x_n) = \\ &\phi_{\beta_1 \prec \cdots \prec \beta_{i-1} \prec \beta_{i+1} \prec \cdots \prec \beta_n}(x_0,\ldots, x_{i-1},x_{i+1},\ldots,x_n). +\end{align*} +\end{itemize} +Again, we've already made the choices for $\phi_{\beta}$ and for $\phi_{\eset \prec \beta}$, where $\beta$ is a single ball. It's not immediately obvious that it's possible to make such choices, but it follows readily from the following. + +\begin{lem} +\label{lem:extend-small-homeomorphisms} +Fix a collection of disjoint embedded balls $\beta$ in $M$ and some open cover $\cV$. Suppose we have a map $f : X \to \Homeo(M)$ on some compact $X$ such that for each $x \in \bdy X$, $f(x)$ makes $\beta$ $\cV$-small. Then we can extend $f$ to a map $\tilde{f} : X \times [0,1] \to \Homeo(M)$ so that $\tilde{f}(x,0) = f(x)$ and for every $x \in \bdy X \times [0,1] \cup X \times \{1\}$, $\tilde{f}(x)$ makes $\beta$ $\cV$-small. +\end{lem} +\begin{proof} +Fix a metric on $M$, and pick $\epsilon > 0$ so every $\epsilon$ ball in $M$ is contained in some open set of $\cV$. First construct a family of homeomorphisms $g_s : M \to M$, $s \in [1,\infty)$ so $g_1$ is the identity, and $g_s(\beta_i) \subset \beta_i$ and $\rad g_s(\beta_i) \leq \frac{1}{s} \rad \beta_i$ for each ball $\beta_i$. +There is some $K$ which uniformly bounds the expansion factors of all the homeomorphisms $f(x)$, that is $d(f(x)(a), f(x)(b)) < K d(a,b)$ for all $x \in X, a,b \in M$. Write $S=\epsilon^{-1} K \max_i \{\rad \beta_i\}$ (note that is $S<1$, we can just take $S=1$, as already $f(x)$ makes $\beta$ small for all $x$). Now define $\tilde{f}(t, x) = f(x) \compose g_{(S-1)t+1}$. + +If $x \in \bdy X$, then $g_{(S-1)t+1}(\beta_i) \subset \beta_i$, and by hypothesis $f(x)$ makes $\beta_i$ small, so $\tilde{f}(t, x)$ makes $\beta$ $\cV$-small for all $t \in [0,1]$. Alternatively, $\rad g_S(\beta_i) \leq \frac{1}{S} \rad \beta_i \leq \frac{\epsilon}{K}$, so $\rad \tilde{f}(1,x)(\beta_i) \leq \epsilon$, and so $\tilde{f}(1,x)$ makes $\beta$ $\cV$-small for all $x \in X$. +\end{proof} + +In fact, the application of this Lemma would allow us to choose the families of diffeomorphisms $\phi_{\beta_1 \prec \cdots \prec \beta_n}$ so that for any $x$ with $x_0 = 0$, $\phi_{\beta_1 \prec \cdots \prec \beta_n}(x)(\beta_1)$ is subordinate to any fixed open cover, for example $\cV_1$ (that is, the covering by $\epsilon/2$ balls), not just $\cV_{n+1}$, which is a weaker condition. Regardless, because of the way we have chosen the $\ev$ map, we only ensure that $\ev(\restrict{\phi_{\beta_1 \prec \cdots \prec \beta_n}}{x_0 = 0} \tensor \beta_1) \in \bc_{\deg \beta_1 + n}^{\cU}(M)$, so the distinction is not important. + +We now describe the general case. For a $k$-blob diagram $b \in \bc_k(M)$, denote by $b_\cS$ for $\cS \subset \{1, \ldots, k\}$ the blob diagram obtained by erasing the corresponding blobs. In particular, $b_\eset = b$, $b_{\{1,\ldots,k\}} \in \bc_0(M)$, and $\bdy b_\cS = \sum_{i \notin \cS} (-1)^{i+1+\card{\setc{j \in \cS}{j < i}}} b_{\cS \cup \{i\}}$. +Similarly, for a disjoint embedding of $k$ balls $\beta$ (that is, a blob diagram but without the labels on regions), $\beta_\cS$ denotes the result of erasing a subset of blobs. + +\newcommand{\length}[1]{\operatorname{length}(#1)} + +We've finally reached the point where we can define a map $s: \bc_*(M) \to \bc^{\cU}_*(M)$, and then a homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ so that $dh+hd=i\circ s$. We have +$$s(b) = \sum_{m=0}^{k} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} (-1)^{\sigma(i)} \ev(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor b_i),$$ +where the sum is over sequences without repeats $i=(i_1,\ldots,i_m)$ in $\{1,\ldots,k\}$, with $0\leq m \leq k$ (we're using $\Delta$ here to indicate the generalized diagonal, where any two entries coincide), $\sigma(i)$ is defined blow, $i(b)$ denotes the increasing sequence of blob configurations +$$\beta_{(i_1,\ldots,i_m)} \prec \beta_{(i_2,\ldots,i_m)} \prec \cdots \prec \beta_{()},$$ +and, as usual, $b_i$ denotes $b$ with blobs $i_1, \ldots, i_m$ erased. +The homotopy $h:\bc_*(M) \to \bc_{*+1}(M)$ is similarly given by +$$h(b) = \sum_{m=0}^{k} \sum_{i} (-1)^{\sigma(i)} \ev(\phi_{i(b)}, b_i).$$ + + The signs $\sigma: \{1, \ldots, k\}^{m} \setminus \Delta \to \Integer/2\Integer$ are defined by +$$\sigma(i)= \lceil\frac{m-1}{2}\rceil +o(i) + \sum i$$ +where $o(i)$ is the number of transpositions required to bring $i$ into increasing order. Given a sequence $a \in \{1, \ldots, k\}^{m-1} \setminus \Delta$ and $1 \leq b \leq k$ with $b \not \in a$, denote by $a \!\downarrow_b\in \{1, \ldots, k-1\}^{m-1} \setminus \Delta$ the sequence obtained by reducing by 1 each entry of $a$ which is greater than $b$. We'll need the identities +\begin{align} +\sigma(ab) & = \sigma(a\!\downarrow_b) +m + b+1 \label{eq:sigma(ab)}\\ +% & = \sigma(a) + m+b + 1 + \card{\setc{x \in a}{x > b}} \\ +\intertext{and} +\sigma(ba) & = \sigma(a\!\downarrow_b) + b \notag \\ + & = \sigma(a) + m+b + 1+\card{\setc{x \in a}{x < b}}. \label{eq:sigma(ba)} +\end{align} + +Before completing the proof, we unpack this definition for $b \in \bc_2(M)$, a $2$-blob. We'll write $\beta$ for the underlying balls (either nested or disjoint). +Now $s$ is the sum of $5$ terms, split into three groups depending on with the length of the sequence $i$ is $0, 1$ or $2$. Thus +\begin{align*} +s(b) & = (-1)^{\sigma()} \restrict{\phi_{\beta}}{x_0 = 0}(b) + \\ + & \quad + (-1)^{\sigma(1)} \restrict{\phi_{\beta_1 \prec \beta}}{x_0 = 0}(b_1) + (-1)^{\sigma(2)} \restrict{\phi_{\beta_2 \prec \beta}}{x_0 = 0}(b_2) + \\ + & \quad + (-1)^{\sigma(12)} \restrict{\phi_{\eset \prec \beta_2 \prec \beta}}{x_0 = 0}(b_{12}) + (-1)^{\sigma(21)} \restrict{\phi_{\eset \prec \beta_1 \prec \beta}}{x_0 = 0}(b_{12}). +\end{align*} + +As in the $k=1$ case, the first term, corresponding to $i(b) = \eset$, makes the all balls in $\beta$ $\cV_1$-small. However, if this were the only term $s$ would not be a chain map, because we have no control over $\restrict{\phi_{\beta}}{x_0 = 0}(\bdy b)$. This necessitates the other terms, which fix the boundary at successively higher codimensions. + +It may be useful to look at Figure \ref{fig:erectly-a-tent-badly} to help understand the arrangement. The red, blue and orange $2$-cells there correspond to the $m=0$, $m=1$ and $m=2$ terms respectively, while the $3$-cells (only one of each type is shown) correspond to the terms in the homotopy $h$. +\begin{figure}[!ht] +$$\mathfig{0.5}{smallblobs/tent}$$ +\caption{``Erecting a tent badly.'' We know where we want to send a simplex, and each of the iterated boundary components. However, these do not agree, and we need to stitch the pieces together. Note that these diagrams don't exactly match the situation in the text: a $k$-simplex has $k+1$ boundary components, while a $k$-blob has $k$ boundary terms.} +\label{fig:erectly-a-tent-badly} +\end{figure} + +Now +\begin{align*} +\bdy s(b) & = (-1)^{\sigma()} \restrict{\phi_{\beta}}{x_0 = 0}(\bdy b) + \\ + & \quad + (-1)^{\sigma(1)} \left( \restrict{\phi_{\beta}}{x_0 = 0}(b_1) - \restrict{\phi_{\beta_1}}{x_0 = 0}(b_1) - \restrict{\phi_{\beta_1 \prec \beta}}{x_0 = 0}(b_{12}) \right) + \\ + & \quad + (-1)^{\sigma(2)} \left( \restrict{\phi_{\beta}}{x_0 = 0}(b_2) - \restrict{\phi_{\beta_2}}{x_0 = 0}(b_2) - \restrict{\phi_{\beta_2 \prec \beta}}{x_0 = 0}(b_{12}) \right) + \\ + & \quad + (-1)^{\sigma(12)} \left( \restrict{\phi_{\beta_2 \prec \beta}}{x_0 = 0}(b_{12}) - \restrict{\phi_{\eset \prec \beta}}{x_0 = 0}(b_{12}) + \restrict{\phi_{\eset \prec \beta_2}}{x_0 = 0}(b_{12}) \right) + \\ + & \quad + (-1)^{\sigma(21)} \left( \restrict{\phi_{\beta_1 \prec \beta}}{x_0 = 0}(b_{12}) - \restrict{\phi_{\eset \prec \beta}}{x_0 = 0}(b_{12}) + \restrict{\phi_{\eset \prec \beta_1}}{x_0 = 0}(b_{12}) \right), \\ +\intertext{while} +s(\bdy(b)) & = s(b_1) - s(b_2) \\ + & = \restrict{\phi_{\beta_1}}{x_0=0}(b_1) - \restrict{\phi_{\eset \prec \beta_1}}{x_0=0}(b_{12}) - \restrict{\phi_{\beta_2}}{x_0=0}(b_2) + \restrict{\phi_{\eset \prec \beta_2}}{x_0=0}(b_{12}) . +\end{align*} +This gives what we want, since $\sigma() = 0,\sigma(1)=1, \sigma(2)=0, \sigma(21)=1$ and $\sigma(12)=0$. + +We now return to the general case. Certainly, the image of $s$ in fact lies in $\bc^{\cU}_*(M)$: each of the families of diffeomorphisms $\phi_{i(b)}$ has been chosen so with $x_0=0$ they pointwise make $b_i$ sufficiently small that the $\ev$ map we've chosen has image in $\bc^{\cU}_*(M)$. + +We need to check that $s$ is a chain map. +\begin{align*} +\bdy(s(b)) & = \sum_{m=0}^{k} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} (-1)^{\sigma(i)} \ev\left(\bdy(\restrict{\phi_{i(b)}}{x_0 = 0})\tensor b_i\right) + (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor \bdy b_i\right) \\ +\intertext{and begin by expanding out $\bdy(\restrict{\phi_{i(b)}}{x_0 = 0})$,} + & = \sum_{m=0}^{k} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} \Bigg(\sum_{p=1}^{m+1} (-1)^{\sigma(i)+p+1} \ev\left(\restrict{\phi_{i(b)}}{x_0 = x_p = 0}\tensor b_i\right) \Bigg) + \\ + & \qquad + (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor \bdy b_i\right) +\end{align*} +Now, write $s_{p-1,p}(i)$ to indicate the sequence obtained from $i$ by transposing its $p-1$-th and $p$-th entries and note that for $2 \leq p \leq m$, +\begin{align*} +\restrict{\phi_{i(b)}}{x_0=x_p=0} & = \restrict{\phi_{\beta_{i_1\cdots i_m} \prec \beta_{i_2 \cdots i_m} \prec \cdots \prec \beta}}{x_0=x_p=0} \\ + & = \restrict{\phi_{\beta_{i_1\cdots i_m} \prec \beta_{i_2 \cdots i_m} \prec \cdots \prec \beta_{i_{p-1} i_p \cdots i_m} \prec \beta_{i_{p+1} \cdots i_m} \prec \cdots \prec \beta}}{x_0=0} \\ + & = \restrict{\phi_{s_{p-1,p}(i)(b)}}{x_0=x_p=0}. +\end{align*} +Since $\sigma(i) = - \sigma(s_{p-1,p}(i))$, we can cancel out in pairs all the terms above except those with $p=1$ or $p=m+1$. Thus +\begin{align*} +\bdy(s(b)) & = \sum_{m=0}^{k} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} \Bigg((-1)^{\sigma(i)} \ev\left(\restrict{\phi_{\rest(i)(b)}}{x_0 = 0}\tensor b_i\right) + (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{\most(i)\!\downarrow_{i_m}(b_{i_m})}}{x_0 = 0}\tensor b_i\right)\Bigg) + \\ + & \qquad + (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor \bdy b_i\right) +\end{align*} +where we use the notations $\rest(i_1 i_2 \cdots i_m) = (i_2 \cdots i_m)$ and $\most(i_1 \cdots i_{m-1} i_m) = (i_1 \cdots i_{m-1})$. Next, we note that $b_i = (b_{i_1})_{\rest(i)} = (b_{i_m})_{\most(i)}$, and then rewrite the sum of $i$ as a double sum over $i_1$ and $\rest(i)$, with $i = i_1\rest(i)$, for the first term, and as a double sum over $\most(i)$ and $i_m$, with $i = \most(i)i_m$, for the second term. +\begin{align*} +\bdy(s(b)) & = \sum_{m=0}^{k} \Bigg( \sum_{\rest(i) \in \{1, \ldots, k\}^{m-1} \setminus \Delta} \sum_{\substack{i_1 = 1 \\ i_1 \not\in \rest(i)}}^{k} (-1)^{\sigma(i_1\rest(i))} \ev\left(\restrict{\phi_{\rest(i)(b)}}{x_0 = 0}\tensor b_{i_1\rest(i)}\right) \Bigg)+ \\ + & \qquad \Bigg( \sum_{\most(i) \in \{1, \ldots, k\}^{m-1} \setminus \Delta} \sum_{\substack{i_m = 1 \\ i_1 \not\in \most(i)}}^{k} (-1)^{\sigma(\most(i) i_m) + m} \ev\left(\restrict{\phi_{\most(i)\!\downarrow_{i_m}(b_{i_m})}}{x_0 = 0}\tensor b_{\most(i) i_m}\right)\Bigg) + \\ + & \qquad \Bigg( \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor \bdy b_i\right)\Bigg) +\end{align*} +We will show that the first and third rows cancel, and that the second row gives with $s(\bdy b)$. +First, +\begin{align*} +\sum_{\substack{i_1 = 1 \\ i_1 \not\in \rest(i)}}^{k} (-1)^{\sigma(i_1\rest(i))} b_{i_1\rest(i)} & = + \sum_{\substack{i_1 = 1 \\ i_1 \not\in \rest(i)}}^{k} (-1)^{\sigma(\rest(i))+m+1+i_1 + \card{\setc{x \in \rest(i)}{x<i_1}}} (b_{\rest(i)})_{i_1} \\ +\intertext{(using Equation \eqref{eq:sigma(ba)})} +& = (-1)^{\sigma(\rest(i))+m} \bdy (b_{\rest(i)}), +\end{align*} +and this cancels exactly with the term indexed by $\rest(i)$ (with a value of $m$ off by one) in the third row. +The second row gives +\begin{align*} +& \sum_{m=0}^k \sum_{\most(i) \in \{1, \ldots, k\}^{m-1} \setminus \Delta} \sum_{\substack{i_m = 1 \\ i_1 \not\in \most(i)}}^{k} (-1)^{\sigma(\most(i) i_m) + m} \ev\left(\restrict{\phi_{\most(i)\!\downarrow_{i_m}(b_{i_m})}}{x_0 = 0}\tensor b_{\most(i) i_m}\right) \\ +& \quad = \sum_{m=0}^{k-1} \sum_{q=1}^k \sum_{i \in \{1, \ldots, k-1\}^m\setminus \Delta} (-1)^{\sigma(i)+q+1} \ev\left(\restrict{\phi_{i(b_{q})}}{x_0 = 0}\tensor (b_q)_i\right) \\ +\intertext{(here we've used Equation \eqref{eq:sigma(ab)} and renamed $i_m$ to $q$ and $\most(i)$ to $i$, as well as shifted $m$ by one), which is just} +& \quad = \sum_{q=1}^k (-1)^{q+1} s(b_q) \\ +& \quad = s(\bdy b). +\end{align*} + +Finally, the calculation that $\bdy h+h \bdy=i\circ s - \id$ is very similar, and we omit it. +\end{proof} + \ No newline at end of file