# HG changeset patch # User Scott Morrison # Date 1280524763 25200 # Node ID 6712876d73e074ae90e34fda8c661186f7db194b # Parent cc44e5ed2db10422995f0f4b7854b2f96d597e5d# Parent c221d8331f30f7d72a61ca77ff3f36d0bec7b49e Automated merge with https://tqft.net/hg/blob/ diff -r c221d8331f30 -r 6712876d73e0 text/appendixes/moam.tex --- a/text/appendixes/moam.tex Fri Jul 30 14:19:11 2010 -0700 +++ b/text/appendixes/moam.tex Fri Jul 30 14:19:23 2010 -0700 @@ -1,4 +1,54 @@ %!TEX root = ../../blob1.tex \section{The method of acyclic models} \label{sec:moam} -\todo{...} \ No newline at end of file + +Let $F_*$ and $G_*$ be chain complexes. +Assume $F_k$ has a basis $\{x_{kj}\}$ +(that is, $F_*$ is free and we have specified a basis). +(In our applications, $\{x_{kj}\}$ will typically be singular $k$-simplices or +$k$-blob diagrams.) +For each basis element $x_{kj}$ assume we have specified a ``target" $D^{kj}_*\sub G_*$. + +We say that a chain map $f:F_*\to G_*$ is {\it compatible} with the above data (basis and targets) +if $f(x_{kj})\in D^{kj}_*$ for all $k$ and $j$. +Let $\Compat(D^\bullet_*)$ denote the subcomplex of maps from $F_*$ to $G_*$ +such that the image of each higher homotopy applied to $x_{kj}$ lies in $D^{kj}_*$. + +\begin{thm}[Acyclic models] \label{moam-thm} +Suppose +\begin{itemize} +\item $D^{k-1,l}_* \sub D^{kj}_*$ whenever $x_{k-1,l}$ occurs in $\bd x_{kj}$ +with non-zero coefficient; +\item $D^{0j}_0$ is non-empty for all $j$; and +\item $D^{kj}_*$ is $(k{-}1)$-acyclic (i.e.\ $H_{k-1}(D^{kj}_*) = 0$) for all $k,j$ . +\end{itemize} +Then $\Compat(D^\bullet_*)$ is non-empty. +If, in addition, +\begin{itemize} +\item $D^{kj}_*$ is $m$-acyclic for $k\le m \le k+i$ and for all $k,j$, +\end{itemize} +then $\Compat(D^\bullet_*)$ is $i$-connected. +\end{thm} + +\begin{proof} +(Sketch) +This is a standard result; see, for example, \nn{need citations}. + +We will build a chain map $f\in \Compat(D^\bullet_*)$ inductively. +Choose $f(x_{0j})\in D^{0j}_0$ for all $j$ +(possible since $D^{0j}_0$ is non-empty). +Choose $f(x_{1j})\in D^{1j}_1$ such that $\bd f(x_{1j}) = f(\bd x_{1j})$ +(possible since $D^{0l}_* \sub D^{1j}_*$ for each $x_{0l}$ in $\bd x_{1j}$ +and $D^{1j}_*$ is 0-acyclic). +Continue in this way, choosing $f(x_{kj})\in D^{kj}_k$ such that $\bd f(x_{kj}) = f(\bd x_{kj})$ +We have now constructed $f\in \Compat(D^\bullet_*)$, proving the first claim of the theorem. + +Now suppose that $D^{kj}_*$ is $k$-acyclic for all $k$ and $j$. +Let $f$ and $f'$ be two chain maps (0-chains) in $\Compat(D^\bullet_*)$. +Using a technique similar to above we can construct a homotopy (1-chain) in $\Compat(D^\bullet_*)$ +between $f$ and $f'$. +Thus $\Compat(D^\bullet_*)$ is 0-connected. +Similarly, if $D^{kj}_*$ is $(k{+}i)$-acyclic then we can show that $\Compat(D^\bullet_*)$ is $i$-connected. +\end{proof} + +\nn{do we also need some version of ``backwards" acyclic models? probably} diff -r c221d8331f30 -r 6712876d73e0 text/comm_alg.tex --- a/text/comm_alg.tex Fri Jul 30 14:19:11 2010 -0700 +++ b/text/comm_alg.tex Fri Jul 30 14:19:23 2010 -0700 @@ -31,24 +31,9 @@ \end{prop} \begin{proof} -%To define the chain maps between the two complexes we will use the following lemma: -% -%\begin{lemma} -%Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with -%a basis (e.g.\ blob diagrams or singular simplices). -%For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ -%such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$. -%Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that -%$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). -%\end{lemma} -% -%\begin{proof} -%\nn{easy, but should probably write the details eventually} -%\nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that} -%\end{proof} -We will use acyclic models \nn{need ref}. +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 \nn{need ref}. +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 $\Sigma^\infty(M)$. @@ -63,15 +48,29 @@ 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 -\nn{need ref, or state condition}. +Theorem \ref{moam-thm}. Thus we have defined (up to homotopy) a map from -$\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$. +$\bc_*(M, k[t])$ to $C_*(\Sigma^\infty(M))$. -Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace -$R(c)_* \sub \bc_*(M^n, k[t])$. +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 $\Sigma^j(M)$. +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. @@ -80,7 +79,7 @@ \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 $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$. +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. @@ -92,12 +91,13 @@ \begin{prop} \label{ktchprop} -The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. +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 @@ -108,16 +108,16 @@ \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 Koszul resolution +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 $k[t] \xrightarrow{0} k[t]$. -This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. +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 map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). +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. @@ -128,7 +128,7 @@ and is zero for $i\ge 2$. Note that the $j$-grading here matches with the $t$-grading on the algebraic side. -By xxxx and Proposition \ref{ktchprop}, +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 @@ -163,7 +163,7 @@ corresponding to $X$. The homology calculation we desire follows easily from this. -\nn{say something about cyclic homology in this case? probably not necessary.} +%\nn{say something about cyclic homology in this case? probably not necessary.} \medskip diff -r c221d8331f30 -r 6712876d73e0 text/kw_macros.tex --- a/text/kw_macros.tex Fri Jul 30 14:19:11 2010 -0700 +++ b/text/kw_macros.tex Fri Jul 30 14:19:23 2010 -0700 @@ -60,7 +60,7 @@ % \DeclareMathOperator{\pr}{pr} etc. \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} -\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}; +\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}; \DeclareMathOperator*{\colim}{colim} \DeclareMathOperator*{\hocolim}{hocolim}