# HG changeset patch # User Scott Morrison # Date 1275200017 25200 # Node ID 36eaa70caf05e805220a5620310c483cf553d0a6 # Parent f956f235213afa7eac54551326c8086d68659794# Parent 2252c53bd4490fc36e96f0fe3c857b15d390220c Automated merge with https://tqft.net/hg/blob/ diff -r f956f235213a -r 36eaa70caf05 text/a_inf_blob.tex --- a/text/a_inf_blob.tex Sat May 29 23:08:36 2010 -0700 +++ b/text/a_inf_blob.tex Sat May 29 23:13:37 2010 -0700 @@ -279,8 +279,14 @@ To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. \begin{thm} \label{thm:map-recon} -$\cB^\cT(M) \simeq C_*(\Maps(M\to T))$. +The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ is quasi-isomorphic to singular chains on maps from $M$ to $T$. +$$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ \end{thm} +\begin{rem} +\nn{This just isn't true, Lurie doesn't do this! I just heard this from Ricardo...} +Lurie has shown in \cite{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in \nn{a certain $E_n$ algebra constructed from $T$} recovers the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. This extra hypothesis is not surprising, in view of the idea that an $E_n$ algebra is roughly equivalent data as an $A_\infty$ $n$-category which is trivial at all but the topmost level. +\end{rem} + \begin{proof} We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. We then use \ref{extension_lemma_b} to show that $g$ induces isomorphisms on homology. diff -r f956f235213a -r 36eaa70caf05 text/evmap.tex --- a/text/evmap.tex Sat May 29 23:08:36 2010 -0700 +++ b/text/evmap.tex Sat May 29 23:13:37 2010 -0700 @@ -41,7 +41,8 @@ I lean toward the latter.} \medskip -The proof will occupy the the next several pages. +Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, and then give an outline of the method of proof. + Without loss of generality, we will assume $X = Y$. \medskip @@ -108,7 +109,7 @@ where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in this case a 0-blob diagram). Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ -(by \ref{disjunion} and \ref{bcontract}). +(by Properties \ref{property:disjoint-union} and \ref{property:contractibility}). Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$, there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$ such that @@ -153,8 +154,7 @@ \medskip -Now for the details. - +\begin{proof}[Proof of Proposition \ref{CHprop}.] Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$. Choose a metric on $X$. @@ -313,7 +313,7 @@ $G_*^{i,m}$. Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. -Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by families of homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}. +Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by families of homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{extension_lemma}. Recall that $h_j$ and also the homotopy connecting it to the identity do not increase supports. Define @@ -610,26 +610,10 @@ \end{itemize} -\nn{to be continued....} - -\noop{ - -\begin{lemma} - -\end{lemma} - -\begin{proof} - \end{proof} -} +\nn{to be continued....} -%\nn{say something about associativity here} - - - - - diff -r f956f235213a -r 36eaa70caf05 text/ncat.tex --- a/text/ncat.tex Sat May 29 23:08:36 2010 -0700 +++ b/text/ncat.tex Sat May 29 23:13:37 2010 -0700 @@ -86,6 +86,7 @@ Morphisms are modeled on balls, so their boundaries are modeled on spheres: \begin{axiom}[Boundaries (spheres)] +\label{axiom:spheres} For each $0 \le k \le n-1$, we have a functor $\cC_k$ from the category of $k$-spheres and homeomorphisms to the category of sets and bijections. @@ -735,7 +736,7 @@ (actions of homeomorphisms); define $k$-cat $\cC(\cdot\times W)$} -Recall that Axiom \ref{} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction. +Recall that Axiom \ref{axiom:spheres} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction. \begin{lem} For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$ diff -r f956f235213a -r 36eaa70caf05 text/smallblobs.tex --- a/text/smallblobs.tex Sat May 29 23:08:36 2010 -0700 +++ b/text/smallblobs.tex Sat May 29 23:13:37 2010 -0700 @@ -88,14 +88,12 @@ \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 -$$s(b) = \sum_{i} (-1)^{\sigma(i)} \ev(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor b_i)$$ -where the sum is over sequences $i=(i_1,\ldots,i_m)$ in $\{1,\ldots,k\}$, with $0\leq m \leq k$, $\sigma(i)$ is something to do with $i$, $i(b)$ denotes the increasing sequence of blob configurations +$$s(b) = \sum_{m=0}^{k} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} (-1)^{\sigma(i)} \ev(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor b_i),$$ +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 something to do with $i$, $i(b)$ 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. We'll also write -$$s(b) = \sum_{m=0}^{k} \sum_{\length{i}=m} (-1)^{\sigma(i)} \ev(\restrict{\phi_{i(b)}}{x_0 = 0} \tensor b_i),$$ -where we arrange the sum according to the length of $i$. +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 -$$h(b) = \sum_{i} (-1)^{\sigma(i)} \ev(\phi_{i(b)}, b_i).$$ +$$h(b) = \sum_{m=0}^{k} \sum_{i} (-1)^{\sigma(i)} \ev(\phi_{i(b)}, b_i).$$ 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 @@ -125,17 +123,43 @@ 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*} -\nn{that does indeed work, modulo signs} +\nn{that does indeed work, modulo signs, with $\sigma() = 1,\sigma(1)=-1, \sigma(2)=1, \sigma(21)=-1, \sigma(12)=1$} -We need to check that $s$ is a chain map, and that \todo{} the image of $s$ in fact lies in $\bc^{\cU}_*(M)$. Calculate +We need to check that $s$ is a chain map, and that \todo{} the image of $s$ in fact lies in $\bc^{\cU}_*(M)$. +We first do some preliminary calculations, and introduce yet more notation. For $i \in \{1, \ldots, k\}^{m} \setminus \Delta$ and $1 \leq p \leq m$, we'll denote by $i \setminus i_p$ the sequence in $\{1, \ldots, k-1\}^{m-1} \setminus \Delta$ obtained by deleting the $p$-th entry of $i$, and reducing all entries which are greater than $i_p$ by one. Conversely, for $i \in \{1, \ldots, k-1\}^{m-1} \setminus \Delta$, $1 \leq p \leq m$ and $1 \leq q \leq k$, we'll denote by $i \ll_p q$ the sequence in $\{1, \ldots, k\}^{m} \setminus \Delta$ obtained by increasing any entries of $i$ which are at least $q$ by one, and inserting $q$ as the $p$-th entry, shifting later entries to the right. Note the natural bijection between the sets +\begin{align} +\setc{(i,p)}{i \in \{1, \ldots, k\}^{m} \setminus \Delta, 1 \leq p \leq m} & \iso \setc{(i,p,q)}{i \in \{1, \ldots, k-1\}^{m-1} \setminus \Delta, 1 \leq p \leq m, 1 \leq q \leq k} \notag \\ +\intertext{given by} +(i, p) & \mapsto (i \setminus i_p, p, i_p) \label{eq:reindexing-bijection} \\ +(i \ll_p q, p) & \mapsfrom (i,p,q) \notag +\end{align} +which we will use in a moment to re-index a summation. + +We then calculate \begin{align*} -\bdy(s(b)) & = \sum_{m=0}^{k} \sum_{\length{i}=m} (-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) \\ - & = \sum_{m=0}^{k} \sum_{\length{i}=m}(-1)^{\sigma(i)} \ev\left(\sum_{p=1}^m \pm \restrict{\phi_{(i\setminus i_p)(b)}}{x_0 = 0})\tensor b_i\right) + (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0}\tensor \sum_{\substack{q=1 \\ q \not\in i}}^k \pm b_{i \cup \{q\}}\right) \\ -\intertext{which we telescope as} - & = \ev \left( \restrict{\phi_\beta}{x_0=0} \tensor \sum_{q=1}^k \pm b_{\{q\}}\right) + \\ - & \qquad + \sum_{m=1}^{k-1} \sum_{\length{i}=m} \Bigg( \sum_{q=1}^{m+1} \sum_{\substack{i^+ \\ i = i^+ \setminus i^+_q}} (-1)^{\sigma(i^+)} \ev\left(\sum_{p=1}^m \pm \restrict{\phi_{i(b)}}{x_0 = 0})\tensor b_{i^+}\right) + \\ - & \qquad \qquad + (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0}\tensor \sum_{\substack{q=1 \\ q \not\in i}}^k \pm b_{i \cup \{q\}}\right)\Bigg) \\ - & \qquad + (-1)^k \sum_{\length{i}=k}(-1)^{\sigma(i)} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0}\tensor \sum_{\substack{q=1 \\ q \not\in i}}^k \pm b_{i \cup \{q\}}\right)\\ +\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) \\ + & = \sum_{m=0}^{k} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} \Bigg(\sum_{p=1}^m (-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*} + +\nn{Crap follows:} +\begin{align*} + & = \sum_{m=0}^{k} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta}(-1)^{\sigma(i)} \ev\left(\sum_{p=1}^m (-1)^{p+1} \restrict{\phi_{(i\setminus i_p)(b_{i_p})}}{x_0 = 0})\tensor (b_{i_p})_{(i \setminus i_p)}\right) + \\ + & \qquad \qquad \qquad + (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0}\tensor \sum_{\substack{q=1 \\ q \not\in i}}^k (-1)^{q+1+\card{\setc{r}{i_r < q}}} b_{i \cup \{q\}}\right). +\end{align*} +Notice the first term vanishes when $m=0$, and the second term vanishes when $m=k$, so it is convenient to rearrange the terms according to the degree of the family of diffeomorphisms. We obtain +\begin{align*} +\bdy(s(b)) & = \sum_{m=0}^{k-1} \sum_{i \in \{1, \ldots, k\}^{m+1} \setminus \Delta}(-1)^{\sigma(i) + p + 1} \ev\left(\sum_{p=1}^{m+1} \restrict{\phi_{(i\setminus i_p)(b_{i_p})}}{x_0 = 0})\tensor (b_{i_p})_{(i \setminus i_p)}\right) + \\ + & \qquad \sum_{m=0}^{k-1} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0}\tensor \sum_{\substack{q=1 \\ q \not\in i}}^k (-1)^{q+1+\card{\setc{r}{i_r < q}}} b_{i \cup \{q\}}\right) \\ +\intertext{then reindex the first sum using the bijection from Equation \eqref{eq:reindexing-bijection}, giving} + & = \sum_{m=0}^{k-1} \sum_{i \in \{1, \ldots, k-1\}^{m} \setminus \Delta} \sum_{p=1}^{m+1} \sum_{q=1}^k (-1)^{\sigma(i \ll_p q) + p + 1} \ev\left( \restrict{\phi_{i(b_{q})}}{x_0 = 0})\tensor (b_{q})_{i}\right) + \\ + & \qquad \sum_{m=0}^{k-1} \sum_{i \in \{1, \ldots, k\}^{m} \setminus \Delta} (-1)^{\sigma(i) + m} \ev\left(\restrict{\phi_{i(b)}}{x_0 = 0}\tensor \sum_{\substack{q=1 \\ q \not\in i}}^k (-1)^{q+1+\card{\setc{r}{i_r < q}}} b_{i \cup \{q\}}\right) \\ +\end{align*} + +On the other hand, we have +\begin{align*} +s(\bdy b) & = \sum_{q=1}^k (-1)^{q+1} s(b_q) \\ + & = \sum_{q=1}^k (-1)^{q+1} \sum_{m=0}^{k-1} \sum_{i \in \{1, \ldots, k-1\}^{m} \setminus \Delta} (-1)^{\sigma(i)} \ev(\restrict{\phi_{i(b_q)}}{x_0 = 0} \tensor (b_q)_i). \end{align*} \todo{to be continued...}