author | Kevin Walker <kevin@canyon23.net> |
Mon, 23 Aug 2010 21:19:55 -0700 | |
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%!TEX root = ../blob1.tex |
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\section{Action of \texorpdfstring{$\CH{X}$}{C_*(Homeo(M))}} |
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\label{sec:evaluation} |
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\nn{new plan: use the sort-of-simplicial space version of |
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the blob complex. |
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first define it, then show it's hty equivalent to the other def, then observe that |
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$CH*$ acts. |
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maybe salvage some of the original version of this section as a subsection outlining |
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how one might proceed directly.} |
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In this section we extend the action of homeomorphisms on $\bc_*(X)$ |
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to an action of {\it families} of homeomorphisms. |
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That is, for each pair of homeomorphic manifolds $X$ and $Y$ |
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we define a chain map |
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\[ |
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e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
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\] |
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where $CH_*(X, Y) = C_*(\Homeo(X, Y))$, the singular chains on the space |
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of homeomorphisms from $X$ to $Y$. |
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(If $X$ and $Y$ have non-empty boundary, these families of homeomorphisms |
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are required to be fixed on the boundaries.) |
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See \S \ref{ss:emap-def} for a more precise statement. |
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The most convenient way to prove that maps $e_{XY}$ with the desired properties exist is to |
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introduce a homotopy equivalent alternate version of the blob complex $\btc_*(X)$ |
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which is more amenable to this sort of action. |
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Recall from Remark \ref{blobsset-remark} that blob diagrams |
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have the structure of a sort-of-simplicial set. |
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Blob diagrams can also be equipped with a natural topology, which converts this |
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sort-of-simplicial set into a sort-of-simplicial space. |
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Taking singular chains of this space we get $\btc_*(X)$. |
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The details are in \S \ref{ss:alt-def}. |
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For technical reasons we also show that requiring the blobs to be |
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embedded yields a homotopy equivalent complex. |
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Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct |
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the $CH_*$ actions directly in terms of $\bc_*(X)$. |
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This was our original approach, but working out the details created a nearly unreadable mess. |
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We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}. |
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\nn{should revisit above intro after this section is done} |
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\subsection{Alternative definitions of the blob complex} |
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\label{ss:alt-def} |
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\newcommand\sbc{\bc^{\cU}} |
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In this subsection we define a subcomplex (small blobs) and supercomplex (families of blobs) |
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of the blob complex, and show that they are both homotopy equivalent to $\bc_*(X)$. |
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\medskip |
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If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted |
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$\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
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For a general $k-chain$ $a\in \bc_k(X)$, define the support of $a$ to be the union |
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of the supports of the blob diagrams which appear in it. |
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If $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is |
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{\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$. |
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We will sometimes abuse language and talk about ``the" support of $f$, |
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again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that |
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$f$ is supported on $Y$. |
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If $f: M \cup (Y\times I) \to M$ is a collaring homeomorphism |
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(cf. end of \S \ref{ss:syst-o-fields}), |
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we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$. |
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Fix $\cU$, an open cover of $X$. |
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Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
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of all blob diagrams in which every blob is contained in some open set of $\cU$, |
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and moreover each field labeling a region cut out by the blobs is splittable |
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into fields on smaller regions, each of which is contained in some open set of $\cU$. |
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\begin{thm}[Small blobs] \label{thm:small-blobs-xx} |
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The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
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\end{thm} |
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\begin{proof} |
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It suffices to show that for any finitely generated pair of subcomplexes |
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$(C_*, D_*) \sub (\bc_*(X), \sbc_*(X))$ |
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we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
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and $x + h\bd(x) + \bd h(X) \in \sbc_*(X)_*(X)$ for all $x\in C_*$. |
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For simplicity we will assume that all fields are splittable into small pieces, so that |
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$\sbc_0(X) = \bc_0$. |
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(This is true for all of the examples presented in this paper.) |
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Accordingly, we define $h_0 = 0$. |
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Next we define $h_1$. |
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Let $b\in C_1$ be a 1-blob diagram. |
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Let $B$ be the blob of $b$. |
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We will construct a 1-chain $s(b)\in \sbc_1$ such that $\bd(s(b)) = \bd b$ |
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and the support of $s(b)$ is contained in $B$. |
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(If $B$ is not embedded in $X$, then we implicitly work in some term of a decomposition |
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of $X$ where $B$ is embedded. |
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See \ref{defn:configuration} and preceding discussion.) |
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It then follows from \ref{disj-union-contract} that we can choose |
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$h_1(b) \in \bc_1(X)$ such that $\bd(h_1(b)) = s(b) - b$. |
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Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
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of small collar maps, plus a shrunken version of $b$. |
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The composition of all the collar maps shrinks $B$ to a sufficiently small ball. |
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Let $\cV_1$ be an auxiliary open cover of $X$, satisfying conditions specified below. |
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Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. |
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Choose a series of collar maps $f_j:\bc_0(B)\to\bc_0(B)$ such that each has support |
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contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
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yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
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\nn{need to say this better; maybe give fig} |
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Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
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There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
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and $\bd c_{ij} = g_j(a_i) = g_{j-1}(a_i)$. |
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Define |
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\[ |
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s(b) = \sum_{i,j} c_{ij} + g(b) |
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\] |
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and choose $h_1(b) \in \bc_1(X)$ such that |
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\[ |
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\bd(h_1(b)) = s(b) - b . |
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\] |
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Next we define $h_2$. |
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\nn{...} |
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\end{proof} |
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\subsection{Action of \texorpdfstring{$\CH{X}$}{C_*(Homeo(M))}} |
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\label{ss:emap-def} |
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\subsection{[older version still hanging around]} |
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\label{ss:old-evmap-remnants} |
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\nn{should comment at the start about any assumptions about smooth, PL etc.} |
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\nn{should maybe mention alternate def of blob complex (sort-of-simplicial space instead of |
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sort-of-simplicial set) where this action would be easy} |
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Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
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the space of homeomorphisms |
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between the $n$-manifolds $X$ and $Y$ (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
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We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
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(For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
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than simplices --- they can be based on any linear polyhedron. |
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\nn{be more restrictive here? does more need to be said?}) |
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\begin{thm} \label{thm:CH} |
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For $n$-manifolds $X$ and $Y$ there is a chain map |
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\eq{ |
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e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) |
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} |
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such that |
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\begin{enumerate} |
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\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
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$\Homeo(X, Y)$ on $\bc_*(X)$ described in Property (\ref{property:functoriality}), and |
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\item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
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the following diagram commutes up to homotopy |
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\begin{equation*} |
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\xymatrix@C+2cm{ |
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CH_*(X, Y) \otimes \bc_*(X) |
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\ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & |
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\bc_*(Y)\ar[d]^{\gl} \\ |
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CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
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} |
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\end{equation*} |
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\end{enumerate} |
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Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$ |
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satisfying the above two conditions which is $m$-connected. In particular, this means that the choice of chain map above is unique up to homotopy. |
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\end{thm} |
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\begin{rem} |
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Note that the statement doesn't quite give uniqueness up to iterated homotopy. We fully expect that this should actually be the case, but haven't been able to prove this. |
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\end{rem} |
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|
70 | 191 |
|
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Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
193 |
and then give an outline of the method of proof. |
|
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Without loss of generality, we will assume $X = Y$. |
196 |
||
197 |
\medskip |
|
198 |
||
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Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) |
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and let $S \sub X$. |
70 | 201 |
We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
345 | 202 |
$x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if |
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there is a family of homeomorphisms $f' : P \times S \to S$ and a ``background" |
236 | 204 |
homeomorphism $f_0 : X \to X$ so that |
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\begin{align*} |
70 | 206 |
f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
207 |
\intertext{and} |
|
208 |
f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
|
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\end{align*} |
70 | 210 |
Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
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(So when we talk about ``the" support of a family, there is some ambiguity, |
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but this ambiguity will not matter to us.) |
70 | 213 |
|
214 |
Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
|
236 | 215 |
A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
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{\it adapted to $\cU$} |
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if the support of $f$ is contained in the union |
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of at most $k$ of the $U_\alpha$'s. |
70 | 219 |
|
220 |
\begin{lemma} \label{extension_lemma} |
|
236 | 221 |
Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
222 |
Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$. |
|
70 | 223 |
Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
224 |
\end{lemma} |
|
225 |
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The proof will be given in \S\ref{sec:localising}. |
70 | 227 |
|
228 |
\medskip |
|
229 |
||
437 | 230 |
Before diving into the details, we outline our strategy for the proof of Theorem \ref{thm:CH}. |
236 | 231 |
Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
246 | 232 |
We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that |
233 |
\begin{itemize} |
|
70 | 234 |
\item $V$ is homeomorphic to a disjoint union of balls, and |
235 |
\item $\supp(p) \cup \supp(b) \sub V$. |
|
246 | 236 |
\end{itemize} |
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(Recall that $\supp(b)$ is defined to be the union of the blobs of the diagram $b$.) |
246 | 238 |
|
239 |
Assuming that $p\ot b$ is localizable as above, |
|
240 |
let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$. |
|
70 | 241 |
We then have a factorization |
242 |
\[ |
|
243 |
p = \gl(q, r), |
|
244 |
\] |
|
236 | 245 |
where $q \in CH_k(V, V')$ and $r \in CH_0(W, W')$. |
73 | 246 |
We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$. |
70 | 247 |
According to the commutative diagram of the proposition, we must have |
248 |
\[ |
|
73 | 249 |
e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = |
250 |
gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) . |
|
251 |
\] |
|
430 | 252 |
Since $r$ is a 0-parameter family of homeomorphisms, we must have |
73 | 253 |
\[ |
254 |
e_{WW'}(r\otimes b_W) = r(b_W), |
|
70 | 255 |
\] |
236 | 256 |
where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in |
73 | 257 |
this case a 0-blob diagram). |
258 |
Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ |
|
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(by Properties \ref{property:disjoint-union} and \ref{property:contractibility}). |
73 | 260 |
Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$, |
83 | 261 |
there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$ |
73 | 262 |
such that |
263 |
\[ |
|
264 |
\bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
|
265 |
\] |
|
266 |
||
267 |
Thus the conditions of the proposition determine (up to homotopy) the evaluation |
|
246 | 268 |
map for localizable generators $p\otimes b$. |
73 | 269 |
On the other hand, Lemma \ref{extension_lemma} allows us to homotope |
246 | 270 |
arbitrary generators to sums of localizable generators. |
73 | 271 |
This (roughly) establishes the uniqueness part of the proposition. |
272 |
To show existence, we must show that the various choices involved in constructing |
|
273 |
evaluation maps in this way affect the final answer only by a homotopy. |
|
274 |
||
246 | 275 |
Now for a little more detail. |
276 |
(But we're still just motivating the full, gory details, which will follow.) |
|
434 | 277 |
Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of $X$ by balls of radius $\gamma$. |
246 | 278 |
By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families |
279 |
$p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls. |
|
280 |
For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough |
|
281 |
$p\ot b$ must be localizable. |
|
282 |
On the other hand, for fixed $k$ and $\gamma$ there exist $p$ and $b$ such that $p\ot b$ is not localizable, |
|
283 |
and for fixed $\gamma$ and $b$ there exist non-localizable $p\ot b$ for sufficiently large $k$. |
|
284 |
Thus we will need to take an appropriate limit as $\gamma$ approaches zero. |
|
83 | 285 |
|
246 | 286 |
The construction of $e_X$, as outlined above, depends on various choices, one of which |
287 |
is the choice, for each localizable generator $p\ot b$, |
|
288 |
of disjoint balls $V$ containing $\supp(p)\cup\supp(b)$. |
|
289 |
Let $V'$ be another disjoint union of balls containing $\supp(p)\cup\supp(b)$, |
|
430 | 290 |
and assume that there exists yet another disjoint union of balls $W$ containing |
246 | 291 |
$V\cup V'$. |
292 |
Then we can use $W$ to construct a homotopy between the two versions of $e_X$ |
|
293 |
associated to $V$ and $V'$. |
|
294 |
If we impose no constraints on $V$ and $V'$ then such a $W$ need not exist. |
|
295 |
Thus we will insist below that $V$ (and $V'$) be contained in small metric neighborhoods |
|
296 |
of $\supp(p)\cup\supp(b)$. |
|
297 |
Because we want not mere homotopy uniqueness but iterated homotopy uniqueness, |
|
298 |
we will similarly require that $W$ be contained in a slightly larger metric neighborhood of |
|
299 |
$\supp(p)\cup\supp(b)$, and so on. |
|
300 |
||
83 | 301 |
|
437 | 302 |
\begin{proof}[Proof of Theorem \ref{thm:CH}.] |
430 | 303 |
We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$. |
83 | 304 |
|
305 |
Choose a metric on $X$. |
|
434 | 306 |
Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
83 | 307 |
(e.g.\ $\ep_i = 2^{-i}$). |
308 |
Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
|
309 |
converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
|
88 | 310 |
Let $\phi_l$ be an increasing sequence of positive numbers |
430 | 311 |
satisfying the inequalities of Lemma \ref{xx2phi} below. |
236 | 312 |
Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ |
83 | 313 |
define |
314 |
\[ |
|
88 | 315 |
N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|). |
83 | 316 |
\] |
247 | 317 |
In other words, for each $i$ |
318 |
we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
|
88 | 319 |
by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling |
320 |
the size of the buffers around $|p|$. |
|
83 | 321 |
|
236 | 322 |
Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$. |
323 |
Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
|
83 | 324 |
= \deg(p) + \deg(b)$. |
430 | 325 |
We say $p\ot b$ is in $G_*^{i,m}$ exactly when either (a) $\deg(p) = 0$ or (b) |
84 | 326 |
there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ |
83 | 327 |
is homeomorphic to a disjoint union of balls and |
328 |
\[ |
|
84 | 329 |
N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) |
434 | 330 |
\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) , |
83 | 331 |
\] |
430 | 332 |
and further $\bd(p\ot b) \in G_*^{i,m}$. |
83 | 333 |
We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. |
334 |
||
335 |
Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. |
|
73 | 336 |
|
83 | 337 |
As sketched above and explained in detail below, |
338 |
$G_*^{i,m}$ is a subcomplex where it is easy to define |
|
339 |
the evaluation map. |
|
84 | 340 |
The parameter $m$ controls the number of iterated homotopies we are able to construct |
87 | 341 |
(see Lemma \ref{m_order_hty}). |
83 | 342 |
The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
236 | 343 |
$CH_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}). |
83 | 344 |
|
249 | 345 |
Next we define a chain map (dependent on some choices) $e_{i,m}: G_*^{i,m} \to \bc_*(X)$. |
346 |
(When the domain is clear from context we will drop the subscripts and write |
|
347 |
simply $e: G_*^{i,m} \to \bc_*(X)$). |
|
83 | 348 |
Let $p\ot b \in G_*^{i,m}$. |
349 |
If $\deg(p) = 0$, define |
|
350 |
\[ |
|
351 |
e(p\ot b) = p(b) , |
|
352 |
\] |
|
236 | 353 |
where $p(b)$ denotes the obvious action of the homeomorphism(s) $p$ on the blob diagram $b$. |
83 | 354 |
For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined |
355 |
$e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$. |
|
84 | 356 |
Choose $V = V_0$ as above so that |
83 | 357 |
\[ |
84 | 358 |
N_{i,k}(p\ot b) \subeq V \subeq N_{i,k+1}(p\ot b) . |
83 | 359 |
\] |
84 | 360 |
Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V^j$ be the choice of neighborhood |
83 | 361 |
of $|p_j|\cup |b_j|$ made at the preceding stage of the induction. |
362 |
For all $j$, |
|
363 |
\[ |
|
88 | 364 |
V^j \subeq N_{i,k}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . |
83 | 365 |
\] |
366 |
(The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
|
367 |
We therefore have splittings |
|
368 |
\[ |
|
247 | 369 |
p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' , |
83 | 370 |
\] |
236 | 371 |
where $p' \in CH_*(V)$, $p'' \in CH_*(X\setmin V)$, |
84 | 372 |
$b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, |
86 | 373 |
$f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$. |
236 | 374 |
(Note that since the family of homeomorphisms $p$ is constant (independent of parameters) |
86 | 375 |
near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
83 | 376 |
unambiguous.) |
86 | 377 |
We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
378 |
%We also have that $\deg(b'') = 0 = \deg(p'')$. |
|
84 | 379 |
Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
402 | 380 |
This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility} and the fact that isotopic fields |
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381 |
differ by a local relation. |
83 | 382 |
Finally, define |
383 |
\[ |
|
384 |
e(p\ot b) \deq x' \bullet p''(b'') . |
|
385 |
\] |
|
73 | 386 |
|
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Note that above we are essentially using the method of acyclic models \nn{\S \ref{sec:moam}}. |
84 | 388 |
For each generator $p\ot b$ we specify the acyclic (in positive degrees) |
389 |
target complex $\bc_*(p(V)) \bullet p''(b'')$. |
|
390 |
||
391 |
The definition of $e: G_*^{i,m} \to \bc_*(X)$ depends on two sets of choices: |
|
392 |
The choice of neighborhoods $V$ and the choice of inverse boundaries $x'$. |
|
88 | 393 |
The next lemma shows that up to (iterated) homotopy $e$ is independent |
84 | 394 |
of these choices. |
88 | 395 |
(Note that independence of choices of $x'$ (for fixed choices of $V$) |
396 |
is a standard result in the method of acyclic models.) |
|
84 | 397 |
|
88 | 398 |
%\begin{lemma} |
399 |
%Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
|
400 |
%different choices of $x'$ at each step. |
|
401 |
%(Same choice of $V$ at each step.) |
|
402 |
%Then $e$ and $\tilde{e}$ are homotopic via a homotopy in $\bc_*(p(V)) \bullet p''(b'')$. |
|
403 |
%Any two choices of such a first-order homotopy are second-order homotopic, and so on, |
|
404 |
%to arbitrary order. |
|
405 |
%\end{lemma} |
|
84 | 406 |
|
88 | 407 |
%\begin{proof} |
408 |
%This is a standard result in the method of acyclic models. |
|
409 |
%\nn{should we say more here?} |
|
410 |
%\nn{maybe this lemma should be subsumed into the next lemma. probably it should.} |
|
411 |
%\end{proof} |
|
84 | 412 |
|
87 | 413 |
\begin{lemma} \label{m_order_hty} |
84 | 414 |
Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
415 |
different choices of $V$ (and hence also different choices of $x'$) at each step. |
|
416 |
If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
|
417 |
If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. |
|
430 | 418 |
Continuing, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy. |
84 | 419 |
\end{lemma} |
420 |
||
421 |
\begin{proof} |
|
422 |
We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$. |
|
430 | 423 |
The chain maps $e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$ |
84 | 424 |
to be zero there. |
425 |
Assume inductively that $h$ has been defined for degrees less than $k$. |
|
426 |
Let $p\ot b$ be a generator of degree $k$. |
|
427 |
Choose $V_1$ as in the definition of $G_*^{i,m}$ so that |
|
428 |
\[ |
|
429 |
N_{i,k+1}(p\ot b) \subeq V_1 \subeq N_{i,k+2}(p\ot b) . |
|
430 |
\] |
|
431 |
There are splittings |
|
432 |
\[ |
|
433 |
p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 , |
|
434 |
\;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 , |
|
435 |
\] |
|
236 | 436 |
where $p'_1 \in CH_*(V_1)$, $p''_1 \in CH_*(X\setmin V_1)$, |
84 | 437 |
$b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$, |
438 |
$f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$. |
|
88 | 439 |
Inductively, $\bd f'_1 = 0$ and $f_1'' = p_1''(b_1'')$. |
84 | 440 |
Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$. |
441 |
Define |
|
442 |
\[ |
|
443 |
h(p\ot b) \deq x'_1 \bullet p''_1(b''_1) . |
|
444 |
\] |
|
445 |
This completes the construction of the first-order homotopy when $m \ge 1$. |
|
446 |
||
447 |
The $j$-th order homotopy is constructed similarly, with $V_j$ replacing $V_1$ above. |
|
448 |
\end{proof} |
|
449 |
||
450 |
Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
|
249 | 451 |
$e_{i,m}$ and $e_{i,m+1}$. |
452 |
An easy variation on the above lemma shows that |
|
453 |
the restrictions of $e_{i,m}$ and $e_{i,m+1}$ to $G_*^{i,m+1}$ are $m$-th |
|
84 | 454 |
order homotopic. |
455 |
||
236 | 456 |
Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
85 | 457 |
$G_*^{i,m}$. |
458 |
Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
|
459 |
Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
|
345 | 460 |
Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is |
461 |
spanned by families of homeomorphisms with support compatible with $\cU_j$, |
|
462 |
as described in Lemma \ref{extension_lemma}. |
|
86 | 463 |
Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
85 | 464 |
supports. |
465 |
Define |
|
466 |
\[ |
|
467 |
g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
|
468 |
\] |
|
469 |
The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
|
470 |
$g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
|
247 | 471 |
(depending on $b$, $\deg(p)$ and $m$). |
472 |
%(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
|
85 | 473 |
|
87 | 474 |
\begin{lemma} \label{Gim_approx} |
236 | 475 |
Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$. |
85 | 476 |
Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
255 | 477 |
there exists another constant $j_{ibmn}$ such that for all $j \ge j_{ibmn}$ and all $p\in CH_n(X)$ |
85 | 478 |
we have $g_j(p)\ot b \in G_*^{i,m}$. |
479 |
\end{lemma} |
|
480 |
||
255 | 481 |
For convenience we also define $k_{bmp} = k_{bmn}$ |
482 |
and $j_{ibmp} = j_{ibmn}$ where $n=\deg(p)$. |
|
254 | 483 |
Note that we may assume that |
484 |
\[ |
|
485 |
k_{bmp} \ge k_{alq} |
|
486 |
\] |
|
487 |
for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
|
255 | 488 |
Additionally, we may assume that |
489 |
\[ |
|
490 |
j_{ibmp} \ge j_{ialq} |
|
491 |
\] |
|
492 |
for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
|
493 |
||
254 | 494 |
|
85 | 495 |
\begin{proof} |
430 | 496 |
|
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|
497 |
There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ the set $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . |
434 | 498 |
(Here we are using the fact that the blobs are |
499 |
piecewise smooth or piecewise-linear and that $\bd c$ is collared.) |
|
90 | 500 |
We need to consider all such $c$ because all generators appearing in |
247 | 501 |
iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
85 | 502 |
|
86 | 503 |
Let $r = \deg(b)$ and |
504 |
\[ |
|
90 | 505 |
t = r+n+m+1 = \deg(p\ot b) + m + 1. |
86 | 506 |
\] |
85 | 507 |
|
508 |
Choose $k = k_{bmn}$ such that |
|
509 |
\[ |
|
248 | 510 |
t\ep_k < \lambda |
85 | 511 |
\] |
512 |
and |
|
513 |
\[ |
|
90 | 514 |
n\cdot (2 (\phi_t + 1) \delta_k) < \ep_k . |
85 | 515 |
\] |
516 |
Let $i \ge k_{bmn}$. |
|
517 |
Choose $j = j_i$ so that |
|
518 |
\[ |
|
90 | 519 |
\gamma_j < \delta_i |
520 |
\] |
|
521 |
and also so that $\phi_t \gamma_j$ is less than the constant $\rho(M)$ of Lemma \ref{xxzz11}. |
|
522 |
||
236 | 523 |
Let $j \ge j_i$ and $p\in CH_n(X)$. |
90 | 524 |
Let $q$ be a generator appearing in $g_j(p)$. |
525 |
Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$, |
|
526 |
which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$. |
|
527 |
We must show that $q\ot b \in G_*^{i,m}$, which means finding neighborhoods |
|
528 |
$V_0,\ldots,V_m \sub X$ of $|q|\cup |b|$ such that each $V_j$ |
|
529 |
is homeomorphic to a disjoint union of balls and |
|
530 |
\[ |
|
531 |
N_{i,n}(q\ot b) \subeq V_0 \subeq N_{i,n+1}(q\ot b) |
|
532 |
\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,t}(q\ot b) . |
|
533 |
\] |
|
248 | 534 |
Recall that |
535 |
\[ |
|
536 |
N_{i,a}(q\ot b) \deq \Nbd_{a\ep_i}(|b|) \cup \Nbd_{\phi_a\delta_i}(|q|). |
|
537 |
\] |
|
90 | 538 |
By repeated applications of Lemma \ref{xx2phi} we can find neighborhoods $U_0,\ldots,U_m$ |
539 |
of $|q|$, each homeomorphic to a disjoint union of balls, with |
|
540 |
\[ |
|
541 |
\Nbd_{\phi_{n+l} \delta_i}(|q|) \subeq U_l \subeq \Nbd_{\phi_{n+l+1} \delta_i}(|q|) . |
|
85 | 542 |
\] |
248 | 543 |
The inequalities above guarantee that |
544 |
for each $0\le l\le m$ we can find $u_l$ with |
|
90 | 545 |
\[ |
546 |
(n+l)\ep_i \le u_l \le (n+l+1)\ep_i |
|
547 |
\] |
|
548 |
such that each component of $U_l$ is either disjoint from $\Nbd_{u_l}(|b|)$ or contained in |
|
549 |
$\Nbd_{u_l}(|b|)$. |
|
550 |
This is because there are at most $n$ components of $U_l$, and each component |
|
551 |
has radius $\le (\phi_t + 1) \delta_i$. |
|
552 |
It follows that |
|
553 |
\[ |
|
554 |
V_l \deq \Nbd_{u_l}(|b|) \cup U_l |
|
555 |
\] |
|
556 |
is homeomorphic to a disjoint union of balls and satisfies |
|
557 |
\[ |
|
558 |
N_{i,n+l}(q\ot b) \subeq V_l \subeq N_{i,n+l+1}(q\ot b) . |
|
559 |
\] |
|
86 | 560 |
|
90 | 561 |
The same argument shows that each generator involved in iterated boundaries of $q\ot b$ |
562 |
is in $G_*^{i,m}$. |
|
86 | 563 |
\end{proof} |
564 |
||
430 | 565 |
In the next three lemmas, which provide the estimates needed above, we have made no effort to optimize the various bounds. |
86 | 566 |
(The bounds are, however, optimal in the sense of minimizing the amount of work |
567 |
we do. Equivalently, they are the first bounds we thought of.) |
|
568 |
||
569 |
We say that a subset $S$ of a metric space has radius $\le r$ if $S$ is contained in |
|
570 |
some metric ball of radius $r$. |
|
571 |
||
572 |
\begin{lemma} |
|
573 |
Let $S \sub \ebb^n$ (Euclidean $n$-space) have radius $\le r$. |
|
574 |
Then $\Nbd_a(S)$ is homeomorphic to a ball for $a \ge 2r$. |
|
575 |
\end{lemma} |
|
576 |
||
577 |
\begin{proof} \label{xxyy2} |
|
578 |
Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
|
89 | 579 |
Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. |
580 |
Let $z\in \Nbd_a(S) \setmin B_r(y)$. |
|
581 |
Consider the triangle |
|
494
cb76847c439e
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Scott Morrison <scott@tqft.net>
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diff
changeset
|
582 |
with vertices $z$, $y$ and $s$ with $s\in S$ such that $z \in B_a(s)$. |
89 | 583 |
The length of the edge $yz$ is greater than $r$ which is greater |
584 |
than the length of the edge $ys$. |
|
585 |
It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), |
|
586 |
which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is, |
|
587 |
which implies that these points are also in $\Nbd_a(S)$. |
|
588 |
Hence $\Nbd_a(S)$ is star-shaped with respect to $y$. |
|
589 |
\end{proof} |
|
590 |
||
591 |
If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$, |
|
592 |
the same result holds, so long as $a$ is not too large: |
|
430 | 593 |
\nn{replace this with a PL version} |
89 | 594 |
|
595 |
\begin{lemma} \label{xxzz11} |
|
596 |
Let $M$ be a compact Riemannian manifold. |
|
597 |
Then there is a constant $\rho(M)$ such that for all |
|
598 |
subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$, |
|
599 |
$\Nbd_a(S)$ is homeomorphic to a ball. |
|
600 |
\end{lemma} |
|
601 |
||
602 |
\begin{proof} |
|
603 |
Choose $\rho = \rho(M)$ such that $3\rho/2$ is less than the radius of injectivity of $M$, |
|
604 |
and also so that for any point $y\in M$ the geodesic coordinates of radius $3\rho/2$ around |
|
605 |
$y$ distort angles by only a small amount. |
|
606 |
Now the argument of the previous lemma works. |
|
85 | 607 |
\end{proof} |
608 |
||
609 |
||
89 | 610 |
|
611 |
\begin{lemma} \label{xx2phi} |
|
612 |
Let $S \sub M$ be contained in a union (not necessarily disjoint) |
|
86 | 613 |
of $k$ metric balls of radius $r$. |
89 | 614 |
Let $\phi_1, \phi_2, \ldots$ be an increasing sequence of real numbers satisfying |
615 |
$\phi_1 \ge 2$ and $\phi_{i+1} \ge \phi_i(2\phi_i + 2) + \phi_i$. |
|
616 |
For convenience, let $\phi_0 = 0$. |
|
248 | 617 |
Assume also that $\phi_k r \le \rho(M)$, |
618 |
where $\rho(M)$ is as in Lemma \ref{xxzz11}. |
|
89 | 619 |
Then there exists a neighborhood $U$ of $S$, |
620 |
homeomorphic to a disjoint union of balls, such that |
|
86 | 621 |
\[ |
89 | 622 |
\Nbd_{\phi_{k-1} r}(S) \subeq U \subeq \Nbd_{\phi_k r}(S) . |
86 | 623 |
\] |
624 |
\end{lemma} |
|
625 |
||
626 |
\begin{proof} |
|
89 | 627 |
For $k=1$ this follows from Lemma \ref{xxzz11}. |
628 |
Assume inductively that it holds for $k-1$. |
|
86 | 629 |
Partition $S$ into $k$ disjoint subsets $S_1,\ldots,S_k$, each of radius $\le r$. |
89 | 630 |
By Lemma \ref{xxzz11}, each $\Nbd_{\phi_{k-1} r}(S_i)$ is homeomorphic to a ball. |
631 |
If these balls are disjoint, let $U$ be their union. |
|
632 |
Otherwise, assume WLOG that $S_{k-1}$ and $S_k$ are distance less than $2\phi_{k-1}r$ apart. |
|
633 |
Let $R_i = \Nbd_{\phi_{k-1} r}(S_i)$ for $i = 1,\ldots,k-2$ |
|
634 |
and $R_{k-1} = \Nbd_{\phi_{k-1} r}(S_{k-1})\cup \Nbd_{\phi_{k-1} r}(S_k)$. |
|
635 |
Each $R_i$ is contained in a metric ball of radius $r' \deq (2\phi_{k-1}+2)r$. |
|
91 | 636 |
Note that the defining inequality of the $\phi_i$ guarantees that |
637 |
\[ |
|
638 |
\phi_{k-1}r' = \phi_{k-1}(2\phi_{k-1}+2)r \le \phi_k r \le \rho(M) . |
|
639 |
\] |
|
89 | 640 |
By induction, there is a neighborhood $U$ of $R \deq \bigcup_i R_i$, |
641 |
homeomorphic to a disjoint union |
|
642 |
of balls, and such that |
|
86 | 643 |
\[ |
89 | 644 |
U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |
86 | 645 |
\] |
89 | 646 |
where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
86 | 647 |
\end{proof} |
648 |
||
430 | 649 |
|
650 |
We now return to defining the chain maps $e_X$. |
|
651 |
||
70 | 652 |
|
254 | 653 |
Let $R_*$ be the chain complex with a generating 0-chain for each non-negative |
654 |
integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$. |
|
358
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|
655 |
(So $R_*$ is a simplicial version of the non-negative reals.) |
254 | 656 |
Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$ |
657 |
by $\iota_j$. |
|
658 |
Define a map (homotopy equivalence) |
|
250 | 659 |
\[ |
254 | 660 |
\sigma: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to CH_*(X, X)\ot \bc_*(X) |
250 | 661 |
\] |
254 | 662 |
as follows. |
663 |
On $R_0\ot CH_*(X, X) \otimes \bc_*(X)$ we define |
|
664 |
\[ |
|
665 |
\sigma(j\ot p\ot b) = g_j(p)\ot b . |
|
666 |
\] |
|
255 | 667 |
On $R_1\ot CH_*(X, X) \otimes \bc_*(X)$ we define |
668 |
\[ |
|
669 |
\sigma(\iota_j\ot p\ot b) = f_j(p)\ot b , |
|
670 |
\] |
|
671 |
where $f_j$ is the homotopy from $g_j$ to $g_{j+1}$. |
|
86 | 672 |
|
254 | 673 |
Next we specify subcomplexes $G^m_* \sub R_*\ot CH_*(X, X) \otimes \bc_*(X)$ on which we will eventually |
674 |
define a version of the action map $e_X$. |
|
255 | 675 |
A generator $j\ot p\ot b$ is defined to be in $G^m_*$ if $j\ge j_{kbmp}$, where |
254 | 676 |
$k = k_{bmp}$ is the constant from Lemma \ref{Gim_approx}. |
255 | 677 |
Similarly $\iota_j\ot p\ot b$ is in $G^m_*$ if $j\ge j_{kbmp}$. |
254 | 678 |
The inequality following Lemma \ref{Gim_approx} guarantees that $G^m_*$ is indeed a subcomplex |
679 |
and that $G^m_* \sup G^{m+1}_*$. |
|
250 | 680 |
|
254 | 681 |
It is easy to see that each $G^m_*$ is homotopy equivalent (via the inclusion map) |
682 |
to $R_*\ot CH_*(X, X) \otimes \bc_*(X)$ |
|
683 |
and hence to $CH_*(X, X) \otimes \bc_*(X)$, and furthermore that the homotopies are well-defined |
|
684 |
up to a contractible set of choices. |
|
250 | 685 |
|
254 | 686 |
Next we define a map |
687 |
\[ |
|
688 |
e_m : G^m_* \to \bc_*(X) . |
|
689 |
\] |
|
255 | 690 |
Let $p\ot b$ be a generator of $G^m_*$. |
691 |
Each $g_j(p)\ot b$ or $f_j(p)\ot b$ is a linear combination of generators $q\ot c$, |
|
692 |
where $\supp(q)\cup\supp(c)$ is contained in a disjoint union of balls satisfying |
|
693 |
various conditions specified above. |
|
694 |
As in the construction of the maps $e_{i,m}$ above, |
|
695 |
it suffices to specify for each such $q\ot c$ a disjoint union of balls |
|
696 |
$V_{qc} \sup \supp(q)\cup\supp(c)$, such that $V_{qc} \sup V_{q'c'}$ |
|
697 |
whenever $q'\ot c'$ appears in the boundary of $q\ot c$. |
|
698 |
||
699 |
Let $q\ot c$ be a summand of $g_j(p)\ot b$, as above. |
|
700 |
Let $i$ be maximal such that $j\ge j_{ibmp}$ |
|
701 |
(notation as in Lemma \ref{Gim_approx}). |
|
702 |
Then $q\ot c \in G^{i,m}_*$ and we choose $V_{qc} \sup \supp(q)\cup\supp(c)$ |
|
703 |
such that |
|
704 |
\[ |
|
705 |
N_{i,d}(q\ot c) \subeq V_{qc} \subeq N_{i,d+1}(q\ot c) , |
|
706 |
\] |
|
707 |
where $d = \deg(q\ot c)$. |
|
708 |
Let $\tilde q = f_j(q)$. |
|
709 |
The summands of $f_j(p)\ot b$ have the form $\tilde q \ot c$, |
|
710 |
where $q\ot c$ is a summand of $g_j(p)\ot b$. |
|
711 |
Since the homotopy $f_j$ does not increase supports, we also have that |
|
712 |
\[ |
|
713 |
V_{qc} \sup \supp(\tilde q) \cup \supp(c) . |
|
714 |
\] |
|
715 |
So we define $V_{\tilde qc} = V_{qc}$. |
|
716 |
||
717 |
It is now easy to check that we have $V_{qc} \sup V_{q'c'}$ |
|
718 |
whenever $q'\ot c'$ appears in the boundary of $q\ot c$. |
|
719 |
As in the construction of the maps $e_{i,m}$ above, |
|
720 |
this allows us to construct a map |
|
721 |
\[ |
|
722 |
e_m : G^m_* \to \bc_*(X) |
|
723 |
\] |
|
724 |
which is well-defined up to homotopy. |
|
725 |
As in the proof of Lemma \ref{m_order_hty}, we can show that the map is well-defined up |
|
726 |
to $m$-th order homotopy. |
|
727 |
Put another way, we have specified an $m$-connected subcomplex of the complex of |
|
728 |
all maps $G^m_* \to \bc_*(X)$. |
|
729 |
On $G^{m+1}_* \sub G^m_*$ we have defined two maps, $e_m$ and $e_{m+1}$. |
|
730 |
One can similarly (to the proof of Lemma \ref{m_order_hty}) show that |
|
731 |
these two maps agree up to $m$-th order homotopy. |
|
732 |
More precisely, one can show that the subcomplex of maps containing the various |
|
733 |
$e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$. |
|
253
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evmap; about to delete a few paragraphs, but committing just so there's
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251
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|
734 |
|
358
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CH_* action -- gluing compatibility
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|
735 |
\medskip |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
736 |
|
8589275ac65b
CH_* action -- gluing compatibility
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parents:
357
diff
changeset
|
737 |
Next we show that the action maps are compatible with gluing. |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
738 |
Let $G^m_*$ and $\ol{G}^m_*$ be the complexes, as above, used for defining |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
739 |
the action maps $e_{X\sgl}$ and $e_X$. |
8589275ac65b
CH_* action -- gluing compatibility
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parents:
357
diff
changeset
|
740 |
The gluing map $X\sgl\to X$ induces a map |
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
741 |
\[ |
430 | 742 |
\gl: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) , |
358
8589275ac65b
CH_* action -- gluing compatibility
Kevin Walker <kevin@canyon23.net>
parents:
357
diff
changeset
|
743 |
\] |
8589275ac65b
CH_* action -- gluing compatibility
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parents:
357
diff
changeset
|
744 |
and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. |
437 | 745 |
From this it follows that the diagram in the statement of Theorem \ref{thm:CH} commutes. |
358
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diff
changeset
|
746 |
|
430 | 747 |
\todo{this paragraph isn't very convincing, or at least I don't see what's going on} |
358
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diff
changeset
|
748 |
Finally we show that the action maps defined above are independent of |
8589275ac65b
CH_* action -- gluing compatibility
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parents:
357
diff
changeset
|
749 |
the choice of metric (up to iterated homotopy). |
359
6224e50c9311
metric independence for homeo action (proof done now)
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parents:
358
diff
changeset
|
750 |
The arguments are very similar to ones given above, so we only sketch them. |
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metric independence for homeo action (proof done now)
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parents:
358
diff
changeset
|
751 |
Let $g$ and $g'$ be two metrics on $X$, and let $e$ and $e'$ be the corresponding |
6224e50c9311
metric independence for homeo action (proof done now)
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parents:
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diff
changeset
|
752 |
actions $CH_*(X, X) \ot \bc_*(X)\to\bc_*(X)$. |
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metric independence for homeo action (proof done now)
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diff
changeset
|
753 |
We must show that $e$ and $e'$ are homotopic. |
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metric independence for homeo action (proof done now)
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parents:
358
diff
changeset
|
754 |
As outlined in the discussion preceding this proof, |
6224e50c9311
metric independence for homeo action (proof done now)
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parents:
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diff
changeset
|
755 |
this follows from the facts that both $e$ and $e'$ are compatible |
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358
diff
changeset
|
756 |
with gluing and that $\bc_*(B^n)$ is contractible. |
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diff
changeset
|
757 |
As above, we define a subcomplex $F_*\sub CH_*(X, X) \ot \bc_*(X)$ generated |
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metric independence for homeo action (proof done now)
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diff
changeset
|
758 |
by $p\ot b$ such that $|p|\cup|b|$ is contained in a disjoint union of balls. |
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759 |
Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. |
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760 |
We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. |
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|
761 |
Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
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762 |
up to second order homotopy, and so on. |
430 | 763 |
|
437 | 764 |
This completes the proof of Theorem \ref{thm:CH}. |
84 | 765 |
\end{proof} |
766 |
||
767 |
||
396
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|
768 |
\begin{rem*} |
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|
769 |
\label{rem:for-small-blobs} |
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|
770 |
For the proof of Lemma \ref{lem:CH-small-blobs} below we will need the following observation on the action constructed above. |
368
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|
771 |
Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. |
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|
772 |
Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each |
385
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|
773 |
of which has support close to $p(t,|b|)$ for some $t\in P$. |
430 | 774 |
More precisely, the support of the generators is contained in the union of a small neighborhood |
775 |
of $p(t,|b|)$ with some small balls. |
|
385
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|
776 |
(Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
396
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|
777 |
\end{rem*} |
385
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|
778 |
|
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|
779 |
|
437 | 780 |
\begin{thm} |
781 |
\label{thm:CH-associativity} |
|
357
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782 |
The $CH_*(X, Y)$ actions defined above are associative. |
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|
783 |
That is, the following diagram commutes up to homotopy: |
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|
784 |
\[ \xymatrix{ |
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|
785 |
& CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
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|
786 |
CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
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|
787 |
& CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
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|
788 |
} \] |
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|
789 |
Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
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|
790 |
of homeomorphisms. |
437 | 791 |
\end{thm} |
70 | 792 |
|
357
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|
793 |
\begin{proof} |
437 | 794 |
The strategy of the proof is similar to that of Theorem \ref{thm:CH}. |
357
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|
795 |
We will identify a subcomplex |
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|
796 |
\[ |
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|
797 |
G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) |
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|
798 |
\] |
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|
799 |
where it is easy to see that the two sides of the diagram are homotopic, then |
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|
800 |
show that there is a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. |
70 | 801 |
|
357
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|
802 |
Let $p\ot q\ot b$ be a generator of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$. |
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|
803 |
By definition, $p\ot q\ot b\in G_*$ if there is a disjoint union of balls in $X$ which |
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|
804 |
contains $|p| \cup p\inv(|q|) \cup |b|$. |
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|
805 |
(If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of |
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|
806 |
$p(x, \cdot)\inv(|q|)$.) |
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|
807 |
|
437 | 808 |
As in the proof of Theorem \ref{thm:CH}, we can construct a homotopy |
357
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|
809 |
between the upper and lower maps restricted to $G_*$. |
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|
810 |
This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$, |
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|
811 |
that they are compatible with gluing, and the contractibility of $\bc_*(X)$. |
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|
812 |
|
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|
813 |
We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers, |
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|
814 |
to construct a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. |
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|
815 |
\end{proof} |