author | Kevin Walker <kevin@canyon23.net> |
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\section{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} |
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\label{sec:evaluation} |
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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 restrict to a fixed homeomorphism on the boundaries.) |
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These actions (for various $X$ and $Y$) are compatible with gluing. |
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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 cone-product set. |
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Blob diagrams can also be equipped with a natural topology, which converts this |
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cone-product set into a cone-product 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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We also prove a useful result (Lemma \ref{small-blobs-b}) which says that we can assume that |
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blobs are small with respect to any fixed open cover. |
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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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% |
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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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More generally, we say that a chain $a\in \bc_k(X)$ is supported on $S$ if |
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$a = a'\bullet r$, where $a'\in \bc_k(S)$ and $r\in \bc_0(X\setmin S)$. |
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Similarly, 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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%Equivalently, $f = f'\bullet r$, where $f'\in CH_k(Y)$ and $r\in CH_0(X\setmin Y)$. |
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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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\medskip |
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Fix $\cU$, an open cover of $X$. |
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Define the ``small blob complex" $\bc^{\cU}_*(X)$ 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{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} |
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The inclusion $i: \bc^{\cU}_*(X) \into \bc_*(X)$ is a homotopy equivalence. |
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\end{lemma} |
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\begin{proof} |
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Since both complexes are free, it suffices to show that the inclusion induces |
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an isomorphism of homotopy groups. |
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To show that it suffices to show that for any finitely generated |
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pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that |
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\[ |
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(C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) |
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\] |
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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 |
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\[ |
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h\bd(x) + \bd h(x) + x \in \sbc_*(X) |
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\] |
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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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$\sbc_0(X) = \bc_0(X)$. |
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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 $s(b)\in \sbc_1(X)$ 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 stage of a decomposition |
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of $X$ where $B$ is embedded. |
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See Definition \ref{defn:configuration} and preceding discussion.) |
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It then follows from Corollary \ref{disj-union-contract} that we can choose |
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$h_1(b) \in \bc_2(X)$ 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 ball which is small with respect to $\cU$. |
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Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
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fine enough that a condition stated later in the proof is satisfied. |
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Let $b = (B, u, r)$, with $u = \sum a_i$ the label of $B$, and $a_i\in \bc_0(B)$. |
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Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions |
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specified at the end of this paragraph. |
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Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to |
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a slightly smaller submanifold of $B$. |
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Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. |
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Let $g$ be the last of the $g_j$'s. |
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Choose the sequence $\bar{f}_j$ so that |
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$g(B)$ is contained is an open set of $\cV_1$ and |
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$g_{j-1}(|f_j|)$ is also contained is an open set of $\cV_1$. |
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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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(more specifically, $|c_{ij}| = g_{j-1}(B)$) |
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and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. |
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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 $h_1(b) \in \bc_2(X)$ such that |
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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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Let $b\in C_2$ be a 2-blob diagram. |
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Let $B = |b|$, either a ball or a union of two balls. |
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By possibly working in a decomposition of $X$, we may assume that the ball(s) |
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of $B$ are disjointly embedded. |
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We will construct a 2-chain $s(b)\in \sbc_2(X)$ such that |
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\[ |
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\bd(s(b)) = \bd(h_1(\bd b) + b) = s(\bd b) |
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\] |
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and the support of $s(b)$ is contained in $B$. |
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It then follows from Corollary \ref{disj-union-contract} that we can choose |
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$h_2(b) \in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$. |
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Similarly to the construction of $h_1$ above, |
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$s(b)$ consists of a series of 2-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 |
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disjoint union of balls. |
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Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
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fine enough that a condition stated later in the proof is satisfied. |
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As before, choose a sequence of collar maps $f_j$ |
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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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Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
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Fix $j$. |
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We will construct a 2-chain $d_j$ such that $\bd d_j = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$. |
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Let $s(\bd b) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams |
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appearing in the boundaries of the $e_k$. |
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As in the construction of $h_1$, we can choose 1-blob diagrams $q_m$ such that |
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$\bd q_m = g_{j-1}(p_m) - g_j(p_m)$ and $|q_m|$ is contained in an open set of $\cV_1$. |
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If $x$ is a sum of $p_m$'s, we denote the corresponding sum of $q_m$'s by $q(x)$. |
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Now consider, for each $k$, $g_{j-1}(e_k) - q(\bd e_k)$. |
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This is a 1-chain whose boundary is $g_j(\bd e_k)$. |
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The support of $e_k$ is $g_{j-1}(V)$ for some $V\in \cV_1$, and |
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the support of $q(\bd e_k)$ is contained in a union $V'$ of finitely many open sets |
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of $\cV_1$, all of which contain the support of $f_j$. |
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We now reveal the mysterious condition (mentioned above) which $\cV_1$ satisfies: |
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the union of $g_{j-1}(V)$ and $V'$, for all of the finitely many instances |
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arising in the construction of $h_2$, lies inside a disjoint union of balls $U$ |
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such that each individual ball lies in an open set of $\cV_2$. |
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(In this case there are either one or two balls in the disjoint union.) |
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For any fixed open cover $\cV_2$ this condition can be satisfied by choosing $\cV_1$ |
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to be a sufficiently fine cover. |
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It follows from Corollary \ref{disj-union-contract} that we can choose |
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$x_k \in \bc_2(X)$ with $\bd x_k = g_{j-1}(e_k) - g_j(e_k) - q(\bd e_k)$ |
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and with $\supp(x_k) = U$. |
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We can now take $d_j \deq \sum x_k$. |
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It is clear that $\bd d_j = \sum (g_{j-1}(e_k) - g_j(e_k)) = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$, as desired. |
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We now define |
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\[ |
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s(b) = \sum d_j + g(b), |
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\] |
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where $g$ is the composition of all the $f_j$'s. |
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It is easy to verify that $s(b) \in \sbc_2$, $\supp(s(b)) = \supp(b)$, and |
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$\bd(s(b)) = s(\bd b)$. |
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If follows that we can choose $h_2(b)\in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$. |
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This completes the definition of $h_2$. |
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The general case $h_l$ is similar. |
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When constructing the analogue of $x_k$ above, we will need to find a disjoint union of balls $U$ |
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which contains finitely many open sets from $\cV_{l-1}$ |
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such that each ball is contained in some open set of $\cV_l$. |
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For sufficiently fine $\cV_{l-1}$ this will be possible. |
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Since $C_*$ is finite, the process terminates after finitely many, say $r$, steps. |
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We take $\cV_r = \cU$. |
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\end{proof} |
517 | 213 |
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