diff -r 352389c6ddcf -r edf8798ef477 text/evmap.tex --- a/text/evmap.tex Fri Aug 27 10:58:21 2010 -0700 +++ b/text/evmap.tex Fri Aug 27 15:36:21 2010 -0700 @@ -3,14 +3,6 @@ \section{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} \label{sec:evaluation} - -\nn{new plan: use the sort-of-simplicial space version of -the blob complex. -first define it, then show it's hty equivalent to the other def, then observe that -$CH*$ acts. -maybe salvage some of the original version of this section as a subsection outlining -how one might proceed directly.} - In this section we extend the action of homeomorphisms on $\bc_*(X)$ to an action of {\it families} of homeomorphisms. That is, for each pair of homeomorphic manifolds $X$ and $Y$ @@ -36,12 +28,12 @@ For technical reasons we also show that requiring the blobs to be embedded yields a homotopy equivalent complex. -Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct -the $CH_*$ actions directly in terms of $\bc_*(X)$. -This was our original approach, but working out the details created a nearly unreadable mess. -We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}. - -\nn{should revisit above intro after this section is done} +%Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct +%the $CH_*$ actions directly in terms of $\bc_*(X)$. +%This was our original approach, but working out the details created a nearly unreadable mess. +%We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}. +% +%\nn{should revisit above intro after this section is done} \subsection{Alternative definitions of the blob complex} @@ -75,15 +67,21 @@ 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 of $\cU$. -\begin{lemma}[Small blobs] \label{small-blobs-b} +\begin{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. \end{lemma} \begin{proof} It suffices to show that for any finitely generated pair of subcomplexes -$(C_*, D_*) \sub (\bc_*(X), \sbc_*(X))$ +\[ + (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) +\] we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ -and $x + h\bd(x) + \bd h(X) \in \sbc_*(X)$ for all $x\in C_*$. +and +\[ + x + h\bd(x) + \bd h(X) \in \sbc_*(X) +\] +for all $x\in C_*$. For simplicity we will assume that all fields are splittable into small pieces, so that $\sbc_0(X) = \bc_0$. @@ -225,7 +223,7 @@ We will regard $\bc_*(X)$ as the subcomplex $\btc_{*0}(X) \sub \btc_{**}(X)$. The main result of this subsection is -\begin{lemma} \label{lem:bt-btc} +\begin{lemma} \label{lem:bc-btc} The inclusion $\bc_*(X) \sub \btc_*(X)$ is a homotopy equivalence \end{lemma} @@ -297,7 +295,7 @@ \end{align*} \end{proof} -\begin{lemma} +\begin{lemma} \label{btc-prod} For manifolds $X$ and $Y$, we have $\btc_*(X\du Y) \simeq \btc_*(X)\otimes\btc_*(Y)$. \end{lemma} \begin{proof} @@ -342,7 +340,7 @@ \end{proof} -\begin{proof}[Proof of \ref{lem:bt-btc}] +\begin{proof}[Proof of \ref{lem:bc-btc}] Armed with the above lemmas, we can now proceed similarly to the proof of \ref{small-blobs-b}. It suffices to show that for any finitely generated pair of subcomplexes @@ -357,10 +355,102 @@ Since $\bc_0(X) = \btc_0(X)$, we can take $h_0 = 0$. +Let $b \in C_1$ be a generator. +Since $b$ is supported in a disjoint union of balls, +we can find $s(b)\in \bc_1$ with $\bd (s(b)) = \bd b$ +(by \ref{disj-union-contract}), and also $h_1(b) \in \btc_2$ +such that $\bd (h_1(b)) = s(b) - b$ +(by \ref{bt-contract} and \ref{btc-prod}). + +Now let $b$ be a generator of $C_2$. +If $\cU$ is fine enough, there is a disjoint union of balls $V$ +on which $b + h_1(\bd b)$ is supported. +Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2$, we can find +$s(b)\in \bc_2$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by \ref{disj-union-contract}). +By \ref{bt-contract} and \ref{btc-prod}, we can now find +$h_2(b) \in \btc_3$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ + +The general case, $h_k$, is similar. +\end{proof} + +The proof of \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion +$\bc_*(X)\sub \btc_*(X)$. +One might ask for more: a contractible set of possible homotopy inverses, or at least an +$m$-connected set for arbitrarily large $m$. +The latter can be achieved with finer control over the various +choices of disjoint unions of balls in the above proofs, but we will not pursue this here. + -\nn{...} +\subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} +\label{ss:emap-def} + +Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of +the space of homeomorphisms +between the $n$-manifolds $X$ and $Y$ +(any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). +We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. +(For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general +than simplices --- they can be based on any linear polyhedron. +\nn{be more restrictive here? does more need to be said?}) + +\begin{thm} \label{thm:CH} +For $n$-manifolds $X$ and $Y$ there is a chain map +\eq{ + e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , +} +well-defined up to homotopy, +such that +\begin{enumerate} +\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of +$\Homeo(X, Y)$ on $\bc_*(X)$ described in Property (\ref{property:functoriality}), and +\item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, +the following diagram commutes up to homotopy +\begin{equation*} +\xymatrix@C+2cm{ + CH_*(X, Y) \otimes \bc_*(X) + \ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & + \bc_*(Y)\ar[d]^{\gl} \\ + CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) +} +\end{equation*} +\end{enumerate} +\end{thm} + +\begin{proof} +In light of Lemma \ref{lem:bc-btc}, it suffices to prove the theorem with +$\bc_*$ replaced by $\btc_*$. +And in fact for $\btc_*$ we get a sharper result: we can omit +the ``up to homotopy" qualifiers. + +Let $f\in CH_k(X, Y)$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, +$a:Q^j \to \BD_i(X)$. +Define $e_{XY}(f\ot a)\in \btc_{i,j+k}(Y)$ by +\begin{align*} + e_{XY}(f\ot a) : P\times Q &\to \BD_i(Y) \\ + (p,q) &\mapsto f(p)(a(q)) . +\end{align*} +It is clear that this agrees with the previously defined $CH_0(X, Y)$ action on $\btc_*$, +and it is also easy to see that the diagram in item 2 of the statement of the theorem +commutes on the nose. +\end{proof} + + +\begin{thm} +\label{thm:CH-associativity} +The $CH_*(X, Y)$ actions defined above are associative. +That is, the following diagram commutes up to homotopy: +\[ \xymatrix{ +& CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ +CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ +& CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & +} \] +Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition +of homeomorphisms. +\end{thm} +\begin{proof} +The corresponding diagram for $\btc_*$ commutes on the nose. \end{proof} @@ -369,9 +459,7 @@ -\subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} -\label{ss:emap-def} - +\noop{ \subsection{[older version still hanging around]} @@ -1042,3 +1130,5 @@ We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers, to construct a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. \end{proof} + +} % end \noop