author | Scott Morrison <scott@tqft.net> |
Mon, 15 Nov 2010 09:49:04 -0800 | |
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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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\[ |
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\bd(h_1(b)) = s(b) - b . |
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\] |
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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 $j$. |
526 | 170 |
We will construct a 2-chain $d_j$ such that $\bd d_j = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$. |
171 |
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 $e_k$. |
173 |
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 |
181 |
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 $\cV_1$ |
188 |
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. |
211 |
We take $\cV_r = \cU$. |
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\end{proof} |
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\medskip |
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Next we define the cone-product space version of the blob complex, $\btc_*(X)$. |
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First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
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We give $\BD_k$ the finest topology such that |
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\begin{itemize} |
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\item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
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\item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
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\item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
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where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
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$\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
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\end{itemize} |
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We can summarize the above by saying that in the typical continuous family |
539 | 229 |
$P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map |
230 |
$P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. |
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536 | 231 |
We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, |
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if we did allow this it would not affect the truth of the claims we make below. |
539 | 233 |
In particular, such a definition of $\btc_*(X)$ would result in a homotopy equivalent complex. |
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Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |
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whose $(i,j)$ entry is $C_j(\BD_i)$, the singular $j$-chains on the space of $i$-blob diagrams. |
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The vertical boundary of the double complex, |
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denoted $\bd_t$, is the singular boundary, and the horizontal boundary, denoted $\bd_b$, is |
539 | 239 |
the blob boundary. Following the usual sign convention, we have $\bd = \bd_b + (-1)^i \bd_t$. |
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We will regard $\bc_*(X)$ as the subcomplex $\btc_{*0}(X) \sub \btc_{**}(X)$. |
520 | 242 |
The main result of this subsection is |
243 |
||
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\begin{lemma} \label{lem:bc-btc} |
520 | 245 |
The inclusion $\bc_*(X) \sub \btc_*(X)$ is a homotopy equivalence |
246 |
\end{lemma} |
|
247 |
||
248 |
Before giving the proof we need a few preliminary results. |
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\begin{lemma} \label{bt-contract} |
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$\btc_*(B^n)$ is contractible (acyclic in positive degrees). |
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\end{lemma} |
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\begin{proof} |
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We will construct a contracting homotopy $h: \btc_*(B^n)\to \btc_{*+1}(B^n)$. |
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255 |
|
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We will assume a splitting $s:H_0(\btc_*(B^n))\to \btc_0(B^n)$ |
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of the quotient map $q:\btc_0(B^n)\to H_0(\btc_*(B^n))$. |
549 | 258 |
Let $\rho = s\circ q$. |
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For $x\in \btc_{ij}$ with $i\ge 1$ define |
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\[ |
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h(x) = e(x) , |
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\] |
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where |
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\[ |
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e: \btc_{ij}\to\btc_{i+1,j} |
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\] |
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268 |
adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams. |
539 | 269 |
Note that for fixed $i$, $e$ is a chain map, i.e. $\bd_t e = e \bd_t$. |
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A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron. |
549 | 272 |
We define $r(y)\in \btc_{0j}$ to be the constant function $\rho\circ y : P\to \BD_0$. |
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Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking |
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the same value (namely $r(y(p))$, for any $p\in P$). |
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Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
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whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
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Now define, for $y\in \btc_{0j}$, |
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\[ |
560 | 279 |
h(y) = e(y - r(y)) - c(r(y)) . |
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\] |
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|
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We must now verify that $h$ does the job it was intended to do. |
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For $x\in \btc_{ij}$ with $i\ge 2$ we have |
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\begin{align*} |
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\bd h(x) + h(\bd x) &= \bd(e(x)) + e(\bd x) && \\ |
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&= \bd_b(e(x)) + (-1)^{i+1} \bd_t(e(x)) + e(\bd_b x) + (-1)^i e(\bd_t x) && \\ |
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&= \bd_b(e(x)) + e(\bd_b x) && \text{(since $\bd_t(e(x)) = e(\bd_t x)$)} \\ |
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&= x . && |
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\end{align*} |
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For $x\in \btc_{1j}$ we have |
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\begin{align*} |
560 | 292 |
\bd h(x) + h(\bd x) &= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x - r(\bd_b x)) - c(r(\bd_b x)) - e(\bd_t x) && \\ |
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&= \bd_b(e(x)) + e(\bd_b x) && \text{(since $r(\bd_b x) = 0$)} \\ |
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&= x . && |
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\end{align*} |
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For $x\in \btc_{0j}$ with $j\ge 1$ we have |
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\begin{align*} |
560 | 298 |
\bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) - \bd_t(e(x - r(x))) + \bd_t(c(r(x))) + |
299 |
e(\bd_t x - r(\bd_t x)) - c(r(\bd_t x)) \\ |
|
300 |
&= x - r(x) + \bd_t(c(r(x))) - c(r(\bd_t x)) \\ |
|
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&= x - r(x) + r(x) \\ |
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&= x. |
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\end{align*} |
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Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram, |
560 | 305 |
as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$ |
306 |
and $\bd_t(c(r(x))) - c(r(\bd_t x)) = r(x)$. |
|
539 | 307 |
|
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For $x\in \btc_{00}$ we have |
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\begin{align*} |
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\bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(c(r(x))) \\ |
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&= x - r(x) + r(x) - r(x)\\ |
539 | 312 |
&= x - r(x). \qedhere |
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\end{align*} |
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\end{proof} |
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315 |
|
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\begin{lemma} \label{btc-prod} |
523 | 317 |
For manifolds $X$ and $Y$, we have $\btc_*(X\du Y) \simeq \btc_*(X)\otimes\btc_*(Y)$. |
318 |
\end{lemma} |
|
319 |
\begin{proof} |
|
539 | 320 |
This follows from the Eilenberg-Zilber theorem and the fact that |
321 |
\begin{align*} |
|
322 |
\BD_k(X\du Y) & \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . \qedhere |
|
323 |
\end{align*} |
|
523 | 324 |
\end{proof} |
325 |
||
326 |
For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} |
|
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if there exists $a'\in \btc_k(S)$ |
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and $r\in \btc_0(X\setmin S)$ such that $a = a'\bullet r$. |
523 | 329 |
|
330 |
\newcommand\sbtc{\btc^{\cU}} |
|
331 |
Let $\cU$ be an open cover of $X$. |
|
332 |
Let $\sbtc_*(X)\sub\btc_*(X)$ be the subcomplex generated by |
|
333 |
$a\in \btc_*(X)$ such that there is a decomposition $X = \cup_i D_i$ |
|
334 |
such that each $D_i$ is a ball contained in some open set of $\cU$ and |
|
335 |
$a$ is splittable along this decomposition. |
|
336 |
In other words, $a$ can be obtained by gluing together pieces, each of which |
|
337 |
is small with respect to $\cU$. |
|
338 |
||
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\begin{lemma} \label{small-top-blobs} |
|
340 |
For any open cover $\cU$ of $X$, the inclusion $\sbtc_*(X)\sub\btc_*(X)$ |
|
341 |
is a homotopy equivalence. |
|
342 |
\end{lemma} |
|
343 |
\begin{proof} |
|
344 |
This follows from a combination of Lemma \ref{extension_lemma_c} and the techniques of |
|
345 |
the proof of Lemma \ref{small-blobs-b}. |
|
346 |
||
347 |
It suffices to show that we can deform a finite subcomplex $C_*$ of $\btc_*(X)$ into $\sbtc_*(X)$ |
|
348 |
(relative to any designated subcomplex of $C_*$ already in $\sbtc_*(X)$). |
|
349 |
The first step is to replace families of general blob diagrams with families that are |
|
350 |
small with respect to $\cU$. |
|
351 |
This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families. |
|
352 |
Each such family is homotopic to a sum families which can be a ``lifted" to $\Homeo(X)$. |
|
353 |
That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$. |
|
354 |
(We are ignoring a complication related to twig blob labels, which might vary |
|
355 |
independently of $g$, but this complication does not affect the conclusion we draw here.) |
|
356 |
We now apply Lemma \ref{extension_lemma_c} to get families which are supported |
|
357 |
on balls $D_i$ contained in open sets of $\cU$. |
|
358 |
\end{proof} |
|
359 |
||
360 |
||
539 | 361 |
\begin{proof}[Proof of Lemma \ref{lem:bc-btc}] |
362 |
Armed with the above lemmas, we can now proceed similarly to the proof of Lemma \ref{small-blobs-b}. |
|
523 | 363 |
|
364 |
It suffices to show that for any finitely generated pair of subcomplexes |
|
365 |
$(C_*, D_*) \sub (\btc_*(X), \bc_*(X))$ |
|
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366 |
we can find a homotopy $h:C_*\to \btc_{*+1}(X)$ such that $h(D_*) \sub \bc_{*+1}(X)$ |
539 | 367 |
and $x + h\bd(x) + \bd h(x) \in \bc_*(X)$ for all $x\in C_*$. |
523 | 368 |
|
369 |
By Lemma \ref{small-top-blobs}, we may assume that $C_* \sub \btc_*^\cU(X)$ for some |
|
370 |
cover $\cU$ of our choosing. |
|
371 |
We choose $\cU$ fine enough so that each generator of $C_*$ is supported on a disjoint union of balls. |
|
372 |
(This is possible since the original $C_*$ was finite and therefore had bounded dimension.) |
|
373 |
||
374 |
Since $\bc_0(X) = \btc_0(X)$, we can take $h_0 = 0$. |
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375 |
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Let $b \in C_1$ be a generator. |
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377 |
Since $b$ is supported in a disjoint union of balls, |
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we can find $s(b)\in \bc_1$ with $\bd (s(b)) = \bd b$ |
539 | 379 |
(by Corollary \ref{disj-union-contract}), and also $h_1(b) \in \btc_2(X)$ |
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such that $\bd (h_1(b)) = s(b) - b$ |
539 | 381 |
(by Lemmas \ref{bt-contract} and \ref{btc-prod}). |
524
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382 |
|
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Now let $b$ be a generator of $C_2$. |
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If $\cU$ is fine enough, there is a disjoint union of balls $V$ |
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385 |
on which $b + h_1(\bd b)$ is supported. |
539 | 386 |
Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2(X)$, we can find |
387 |
$s(b)\in \bc_2(X)$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by Corollary \ref{disj-union-contract}). |
|
388 |
By Lemmas \ref{bt-contract} and \ref{btc-prod}, we can now find |
|
389 |
$h_2(b) \in \btc_3(X)$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ |
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390 |
|
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The general case, $h_k$, is similar. |
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392 |
\end{proof} |
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393 |
|
539 | 394 |
The proof of Lemma \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion |
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$\bc_*(X)\sub \btc_*(X)$. |
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396 |
One might ask for more: a contractible set of possible homotopy inverses, or at least an |
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397 |
$m$-connected set for arbitrarily large $m$. |
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398 |
The latter can be achieved with finer control over the various |
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399 |
choices of disjoint unions of balls in the above proofs, but we will not pursue this here. |
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400 |
|
523 | 401 |
|
402 |
||
403 |
||
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404 |
\subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} |
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405 |
\label{ss:emap-def} |
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406 |
|
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407 |
Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
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408 |
the space of homeomorphisms |
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409 |
between the $n$-manifolds $X$ and $Y$ |
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410 |
(any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
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411 |
We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
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412 |
(For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
549 | 413 |
than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).) |
524
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414 |
|
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415 |
\begin{thm} \label{thm:CH} \label{thm:evaluation}% |
524
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416 |
For $n$-manifolds $X$ and $Y$ there is a chain map |
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417 |
\eq{ |
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418 |
e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) , |
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419 |
} |
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420 |
well-defined up to homotopy, |
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421 |
such that |
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422 |
\begin{enumerate} |
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423 |
\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
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424 |
$\Homeo(X, Y)$ on $\bc_*(X)$ described in Property \ref{property:functoriality}, and |
524
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425 |
\item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
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426 |
the following diagram commutes up to homotopy |
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427 |
\begin{equation*} |
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428 |
\xymatrix@C+2cm{ |
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429 |
CH_*(X, Y) \otimes \bc_*(X) |
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430 |
\ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & |
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431 |
\bc_*(Y)\ar[d]^{\gl} \\ |
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432 |
CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
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433 |
} |
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434 |
\end{equation*} |
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435 |
\end{enumerate} |
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436 |
\end{thm} |
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437 |
|
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438 |
\begin{proof} |
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439 |
In light of Lemma \ref{lem:bc-btc}, it suffices to prove the theorem with |
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440 |
$\bc_*$ replaced by $\btc_*$. |
539 | 441 |
In fact, for $\btc_*$ we get a sharper result: we can omit |
524
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442 |
the ``up to homotopy" qualifiers. |
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443 |
|
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444 |
Let $f\in CH_k(X, Y)$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, |
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445 |
$a:Q^j \to \BD_i(X)$. |
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446 |
Define $e_{XY}(f\ot a)\in \btc_{i,j+k}(Y)$ by |
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447 |
\begin{align*} |
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448 |
e_{XY}(f\ot a) : P\times Q &\to \BD_i(Y) \\ |
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449 |
(p,q) &\mapsto f(p)(a(q)) . |
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450 |
\end{align*} |
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451 |
It is clear that this agrees with the previously defined $CH_0(X, Y)$ action on $\btc_*$, |
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452 |
and it is also easy to see that the diagram in item 2 of the statement of the theorem |
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453 |
commutes on the nose. |
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454 |
\end{proof} |
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455 |
|
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456 |
|
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457 |
\begin{thm} |
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458 |
\label{thm:CH-associativity} |
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459 |
The $CH_*(X, Y)$ actions defined above are associative. |
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460 |
That is, the following diagram commutes up to homotopy: |
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461 |
\[ \xymatrix{ |
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462 |
& CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
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463 |
CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
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464 |
& CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
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465 |
} \] |
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466 |
Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
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467 |
of homeomorphisms. |
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468 |
\end{thm} |
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469 |
\begin{proof} |
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470 |
The corresponding diagram for $\btc_*$ commutes on the nose. |
523 | 471 |
\end{proof} |
472 |
||
473 |
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520 | 477 |
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478 |
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479 |
\noop{ |
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480 |
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481 |
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482 |
\subsection{[older version still hanging around]} |
513 | 483 |
\label{ss:old-evmap-remnants} |
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484 |
|
246 | 485 |
\nn{should comment at the start about any assumptions about smooth, PL etc.} |
486 |
||
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487 |
\nn{should maybe mention alternate def of blob complex (cone-product space instead of |
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488 |
cone-product set) where this action would be easy} |
447 | 489 |
|
236 | 490 |
Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of |
491 |
the space of homeomorphisms |
|
430 | 492 |
between the $n$-manifolds $X$ and $Y$ (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$). |
249 | 493 |
We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$. |
494 |
(For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general |
|
70 | 495 |
than simplices --- they can be based on any linear polyhedron. |
249 | 496 |
\nn{be more restrictive here? does more need to be said?}) |
70 | 497 |
|
437 | 498 |
\begin{thm} \label{thm:CH} |
70 | 499 |
For $n$-manifolds $X$ and $Y$ there is a chain map |
500 |
\eq{ |
|
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501 |
e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) |
70 | 502 |
} |
244
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503 |
such that |
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504 |
\begin{enumerate} |
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505 |
\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of |
437 | 506 |
$\Homeo(X, Y)$ on $\bc_*(X)$ described in Property (\ref{property:functoriality}), and |
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507 |
\item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, |
70 | 508 |
the following diagram commutes up to homotopy |
430 | 509 |
\begin{equation*} |
510 |
\xymatrix@C+2cm{ |
|
236 | 511 |
CH_*(X, Y) \otimes \bc_*(X) |
430 | 512 |
\ar[r]_(.6){e_{XY}} \ar[d]^{\gl \otimes \gl} & |
513 |
\bc_*(Y)\ar[d]^{\gl} \\ |
|
514 |
CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}} & \bc_*(Y\sgl) |
|
515 |
} |
|
516 |
\end{equation*} |
|
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517 |
\end{enumerate} |
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518 |
Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$ |
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519 |
satisfying the above two conditions which is $m$-connected. In particular, |
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520 |
this means that the choice of chain map above is unique up to homotopy. |
437 | 521 |
\end{thm} |
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522 |
\begin{rem} |
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523 |
Note that the statement doesn't quite give uniqueness up to iterated homotopy. |
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524 |
We fully expect that this should actually be the case, but haven't been able to prove this. |
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525 |
\end{rem} |
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526 |
|
70 | 527 |
|
345 | 528 |
Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, |
529 |
and then give an outline of the method of proof. |
|
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530 |
|
70 | 531 |
Without loss of generality, we will assume $X = Y$. |
532 |
||
533 |
\medskip |
|
534 |
||
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535 |
Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$) |
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536 |
and let $S \sub X$. |
70 | 537 |
We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
345 | 538 |
$x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if |
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539 |
there is a family of homeomorphisms $f' : P \times S \to S$ and a ``background" |
236 | 540 |
homeomorphism $f_0 : X \to X$ so that |
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541 |
\begin{align*} |
70 | 542 |
f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
543 |
\intertext{and} |
|
544 |
f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
|
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545 |
\end{align*} |
70 | 546 |
Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
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547 |
(So when we talk about ``the" support of a family, there is some ambiguity, |
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548 |
but this ambiguity will not matter to us.) |
70 | 549 |
|
550 |
Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
|
236 | 551 |
A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
245
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552 |
{\it adapted to $\cU$} |
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553 |
if the support of $f$ is contained in the union |
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554 |
of at most $k$ of the $U_\alpha$'s. |
70 | 555 |
|
556 |
\begin{lemma} \label{extension_lemma} |
|
236 | 557 |
Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
558 |
Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$. |
|
70 | 559 |
Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
560 |
\end{lemma} |
|
561 |
||
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562 |
The proof will be given in \S\ref{sec:localising}. |
70 | 563 |
|
564 |
\medskip |
|
565 |
||
437 | 566 |
Before diving into the details, we outline our strategy for the proof of Theorem \ref{thm:CH}. |
236 | 567 |
Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
246 | 568 |
We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that |
569 |
\begin{itemize} |
|
70 | 570 |
\item $V$ is homeomorphic to a disjoint union of balls, and |
571 |
\item $\supp(p) \cup \supp(b) \sub V$. |
|
246 | 572 |
\end{itemize} |
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573 |
(Recall that $\supp(b)$ is defined to be the union of the blobs of the diagram $b$.) |
246 | 574 |
|
575 |
Assuming that $p\ot b$ is localizable as above, |
|
576 |
let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$. |
|
70 | 577 |
We then have a factorization |
578 |
\[ |
|
579 |
p = \gl(q, r), |
|
580 |
\] |
|
236 | 581 |
where $q \in CH_k(V, V')$ and $r \in CH_0(W, W')$. |
73 | 582 |
We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$. |
70 | 583 |
According to the commutative diagram of the proposition, we must have |
584 |
\[ |
|
73 | 585 |
e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = |
586 |
gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) . |
|
587 |
\] |
|
430 | 588 |
Since $r$ is a 0-parameter family of homeomorphisms, we must have |
73 | 589 |
\[ |
590 |
e_{WW'}(r\otimes b_W) = r(b_W), |
|
70 | 591 |
\] |
236 | 592 |
where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in |
73 | 593 |
this case a 0-blob diagram). |
594 |
Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ |
|
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595 |
(by Properties \ref{property:disjoint-union} and \ref{property:contractibility}). |
73 | 596 |
Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$, |
83 | 597 |
there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$ |
73 | 598 |
such that |
599 |
\[ |
|
600 |
\bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
|
601 |
\] |
|
602 |
||
603 |
Thus the conditions of the proposition determine (up to homotopy) the evaluation |
|
246 | 604 |
map for localizable generators $p\otimes b$. |
73 | 605 |
On the other hand, Lemma \ref{extension_lemma} allows us to homotope |
246 | 606 |
arbitrary generators to sums of localizable generators. |
73 | 607 |
This (roughly) establishes the uniqueness part of the proposition. |
608 |
To show existence, we must show that the various choices involved in constructing |
|
609 |
evaluation maps in this way affect the final answer only by a homotopy. |
|
610 |
||
246 | 611 |
Now for a little more detail. |
612 |
(But we're still just motivating the full, gory details, which will follow.) |
|
434 | 613 |
Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of $X$ by balls of radius $\gamma$. |
246 | 614 |
By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families |
615 |
$p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls. |
|
616 |
For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough |
|
617 |
$p\ot b$ must be localizable. |
|
618 |
On the other hand, for fixed $k$ and $\gamma$ there exist $p$ and $b$ such that $p\ot b$ is not localizable, |
|
619 |
and for fixed $\gamma$ and $b$ there exist non-localizable $p\ot b$ for sufficiently large $k$. |
|
620 |
Thus we will need to take an appropriate limit as $\gamma$ approaches zero. |
|
83 | 621 |
|
246 | 622 |
The construction of $e_X$, as outlined above, depends on various choices, one of which |
623 |
is the choice, for each localizable generator $p\ot b$, |
|
624 |
of disjoint balls $V$ containing $\supp(p)\cup\supp(b)$. |
|
625 |
Let $V'$ be another disjoint union of balls containing $\supp(p)\cup\supp(b)$, |
|
430 | 626 |
and assume that there exists yet another disjoint union of balls $W$ containing |
246 | 627 |
$V\cup V'$. |
628 |
Then we can use $W$ to construct a homotopy between the two versions of $e_X$ |
|
629 |
associated to $V$ and $V'$. |
|
630 |
If we impose no constraints on $V$ and $V'$ then such a $W$ need not exist. |
|
631 |
Thus we will insist below that $V$ (and $V'$) be contained in small metric neighborhoods |
|
632 |
of $\supp(p)\cup\supp(b)$. |
|
633 |
Because we want not mere homotopy uniqueness but iterated homotopy uniqueness, |
|
634 |
we will similarly require that $W$ be contained in a slightly larger metric neighborhood of |
|
635 |
$\supp(p)\cup\supp(b)$, and so on. |
|
636 |
||
83 | 637 |
|
437 | 638 |
\begin{proof}[Proof of Theorem \ref{thm:CH}.] |
430 | 639 |
We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$. |
83 | 640 |
|
641 |
Choose a metric on $X$. |
|
434 | 642 |
Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
83 | 643 |
(e.g.\ $\ep_i = 2^{-i}$). |
644 |
Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
|
645 |
converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
|
88 | 646 |
Let $\phi_l$ be an increasing sequence of positive numbers |
430 | 647 |
satisfying the inequalities of Lemma \ref{xx2phi} below. |
236 | 648 |
Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ |
83 | 649 |
define |
650 |
\[ |
|
88 | 651 |
N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|). |
83 | 652 |
\] |
247 | 653 |
In other words, for each $i$ |
654 |
we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
|
88 | 655 |
by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling |
656 |
the size of the buffers around $|p|$. |
|
83 | 657 |
|
236 | 658 |
Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$. |
659 |
Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
|
83 | 660 |
= \deg(p) + \deg(b)$. |
430 | 661 |
We say $p\ot b$ is in $G_*^{i,m}$ exactly when either (a) $\deg(p) = 0$ or (b) |
84 | 662 |
there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ |
83 | 663 |
is homeomorphic to a disjoint union of balls and |
664 |
\[ |
|
84 | 665 |
N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) |
434 | 666 |
\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) , |
83 | 667 |
\] |
430 | 668 |
and further $\bd(p\ot b) \in G_*^{i,m}$. |
83 | 669 |
We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. |
670 |
||
671 |
Note that $G_*^{i,m+1} \subeq G_*^{i,m}$. |
|
73 | 672 |
|
83 | 673 |
As sketched above and explained in detail below, |
674 |
$G_*^{i,m}$ is a subcomplex where it is easy to define |
|
675 |
the evaluation map. |
|
84 | 676 |
The parameter $m$ controls the number of iterated homotopies we are able to construct |
87 | 677 |
(see Lemma \ref{m_order_hty}). |
83 | 678 |
The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
236 | 679 |
$CH_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}). |
83 | 680 |
|
249 | 681 |
Next we define a chain map (dependent on some choices) $e_{i,m}: G_*^{i,m} \to \bc_*(X)$. |
682 |
(When the domain is clear from context we will drop the subscripts and write |
|
683 |
simply $e: G_*^{i,m} \to \bc_*(X)$). |
|
83 | 684 |
Let $p\ot b \in G_*^{i,m}$. |
685 |
If $\deg(p) = 0$, define |
|
686 |
\[ |
|
687 |
e(p\ot b) = p(b) , |
|
688 |
\] |
|
236 | 689 |
where $p(b)$ denotes the obvious action of the homeomorphism(s) $p$ on the blob diagram $b$. |
83 | 690 |
For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined |
691 |
$e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$. |
|
84 | 692 |
Choose $V = V_0$ as above so that |
83 | 693 |
\[ |
84 | 694 |
N_{i,k}(p\ot b) \subeq V \subeq N_{i,k+1}(p\ot b) . |
83 | 695 |
\] |
84 | 696 |
Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V^j$ be the choice of neighborhood |
83 | 697 |
of $|p_j|\cup |b_j|$ made at the preceding stage of the induction. |
698 |
For all $j$, |
|
699 |
\[ |
|
88 | 700 |
V^j \subeq N_{i,k}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . |
83 | 701 |
\] |
702 |
(The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
|
703 |
We therefore have splittings |
|
704 |
\[ |
|
247 | 705 |
p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' , |
83 | 706 |
\] |
236 | 707 |
where $p' \in CH_*(V)$, $p'' \in CH_*(X\setmin V)$, |
84 | 708 |
$b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, |
86 | 709 |
$f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$. |
236 | 710 |
(Note that since the family of homeomorphisms $p$ is constant (independent of parameters) |
86 | 711 |
near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
83 | 712 |
unambiguous.) |
86 | 713 |
We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
714 |
%We also have that $\deg(b'') = 0 = \deg(p'')$. |
|
84 | 715 |
Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
544
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|
716 |
This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility} |
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|
717 |
and the fact that isotopic fields differ by a local relation. |
83 | 718 |
Finally, define |
719 |
\[ |
|
720 |
e(p\ot b) \deq x' \bullet p''(b'') . |
|
721 |
\] |
|
73 | 722 |
|
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|
723 |
Note that above we are essentially using the method of acyclic models \nn{\S \ref{sec:moam}}. |
84 | 724 |
For each generator $p\ot b$ we specify the acyclic (in positive degrees) |
725 |
target complex $\bc_*(p(V)) \bullet p''(b'')$. |
|
726 |
||
727 |
The definition of $e: G_*^{i,m} \to \bc_*(X)$ depends on two sets of choices: |
|
728 |
The choice of neighborhoods $V$ and the choice of inverse boundaries $x'$. |
|
88 | 729 |
The next lemma shows that up to (iterated) homotopy $e$ is independent |
84 | 730 |
of these choices. |
88 | 731 |
(Note that independence of choices of $x'$ (for fixed choices of $V$) |
732 |
is a standard result in the method of acyclic models.) |
|
84 | 733 |
|
88 | 734 |
%\begin{lemma} |
735 |
%Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
|
736 |
%different choices of $x'$ at each step. |
|
737 |
%(Same choice of $V$ at each step.) |
|
738 |
%Then $e$ and $\tilde{e}$ are homotopic via a homotopy in $\bc_*(p(V)) \bullet p''(b'')$. |
|
739 |
%Any two choices of such a first-order homotopy are second-order homotopic, and so on, |
|
740 |
%to arbitrary order. |
|
741 |
%\end{lemma} |
|
84 | 742 |
|
88 | 743 |
%\begin{proof} |
744 |
%This is a standard result in the method of acyclic models. |
|
745 |
%\nn{should we say more here?} |
|
746 |
%\nn{maybe this lemma should be subsumed into the next lemma. probably it should.} |
|
747 |
%\end{proof} |
|
84 | 748 |
|
87 | 749 |
\begin{lemma} \label{m_order_hty} |
84 | 750 |
Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
751 |
different choices of $V$ (and hence also different choices of $x'$) at each step. |
|
752 |
If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
|
753 |
If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. |
|
430 | 754 |
Continuing, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy. |
84 | 755 |
\end{lemma} |
756 |
||
757 |
\begin{proof} |
|
758 |
We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$. |
|
430 | 759 |
The chain maps $e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$ |
84 | 760 |
to be zero there. |
761 |
Assume inductively that $h$ has been defined for degrees less than $k$. |
|
762 |
Let $p\ot b$ be a generator of degree $k$. |
|
763 |
Choose $V_1$ as in the definition of $G_*^{i,m}$ so that |
|
764 |
\[ |
|
765 |
N_{i,k+1}(p\ot b) \subeq V_1 \subeq N_{i,k+2}(p\ot b) . |
|
766 |
\] |
|
767 |
There are splittings |
|
768 |
\[ |
|
769 |
p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 , |
|
770 |
\;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 , |
|
771 |
\] |
|
236 | 772 |
where $p'_1 \in CH_*(V_1)$, $p''_1 \in CH_*(X\setmin V_1)$, |
84 | 773 |
$b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$, |
774 |
$f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$. |
|
88 | 775 |
Inductively, $\bd f'_1 = 0$ and $f_1'' = p_1''(b_1'')$. |
84 | 776 |
Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$. |
777 |
Define |
|
778 |
\[ |
|
779 |
h(p\ot b) \deq x'_1 \bullet p''_1(b''_1) . |
|
780 |
\] |
|
781 |
This completes the construction of the first-order homotopy when $m \ge 1$. |
|
782 |
||
783 |
The $j$-th order homotopy is constructed similarly, with $V_j$ replacing $V_1$ above. |
|
784 |
\end{proof} |
|
785 |
||
786 |
Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
|
249 | 787 |
$e_{i,m}$ and $e_{i,m+1}$. |
788 |
An easy variation on the above lemma shows that |
|
789 |
the restrictions of $e_{i,m}$ and $e_{i,m+1}$ to $G_*^{i,m+1}$ are $m$-th |
|
84 | 790 |
order homotopic. |
791 |
||
236 | 792 |
Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
85 | 793 |
$G_*^{i,m}$. |
794 |
Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
|
795 |
Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
|
345 | 796 |
Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is |
797 |
spanned by families of homeomorphisms with support compatible with $\cU_j$, |
|
798 |
as described in Lemma \ref{extension_lemma}. |
|
86 | 799 |
Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
85 | 800 |
supports. |
801 |
Define |
|
802 |
\[ |
|
803 |
g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
|
804 |
\] |
|
805 |
The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
|
806 |
$g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
|
247 | 807 |
(depending on $b$, $\deg(p)$ and $m$). |
808 |
%(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
|
85 | 809 |
|
87 | 810 |
\begin{lemma} \label{Gim_approx} |
236 | 811 |
Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$. |
85 | 812 |
Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
255 | 813 |
there exists another constant $j_{ibmn}$ such that for all $j \ge j_{ibmn}$ and all $p\in CH_n(X)$ |
85 | 814 |
we have $g_j(p)\ot b \in G_*^{i,m}$. |
815 |
\end{lemma} |
|
816 |
||
255 | 817 |
For convenience we also define $k_{bmp} = k_{bmn}$ |
818 |
and $j_{ibmp} = j_{ibmn}$ where $n=\deg(p)$. |
|
254 | 819 |
Note that we may assume that |
820 |
\[ |
|
821 |
k_{bmp} \ge k_{alq} |
|
822 |
\] |
|
823 |
for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
|
255 | 824 |
Additionally, we may assume that |
825 |
\[ |
|
826 |
j_{ibmp} \ge j_{ialq} |
|
827 |
\] |
|
828 |
for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$. |
|
829 |
||
254 | 830 |
|
85 | 831 |
\begin{proof} |
430 | 832 |
|
544
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Kevin Walker <kevin@canyon23.net>
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diff
changeset
|
833 |
There exists $\lambda > 0$ such that for every subset $c$ of the blobs of $b$ the set |
24be062a87a1
breaking lines, and one other minor comment
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parents:
541
diff
changeset
|
834 |
$\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ . |
434 | 835 |
(Here we are using the fact that the blobs are |
836 |
piecewise smooth or piecewise-linear and that $\bd c$ is collared.) |
|
90 | 837 |
We need to consider all such $c$ because all generators appearing in |
247 | 838 |
iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
85 | 839 |
|
86 | 840 |
Let $r = \deg(b)$ and |
841 |
\[ |
|
90 | 842 |
t = r+n+m+1 = \deg(p\ot b) + m + 1. |
86 | 843 |
\] |
85 | 844 |
|
845 |
Choose $k = k_{bmn}$ such that |
|
846 |
\[ |
|
248 | 847 |
t\ep_k < \lambda |
85 | 848 |
\] |
849 |
and |
|
850 |
\[ |
|
90 | 851 |
n\cdot (2 (\phi_t + 1) \delta_k) < \ep_k . |
85 | 852 |
\] |
853 |
Let $i \ge k_{bmn}$. |
|
854 |
Choose $j = j_i$ so that |
|
855 |
\[ |
|
90 | 856 |
\gamma_j < \delta_i |
857 |
\] |
|
858 |
and also so that $\phi_t \gamma_j$ is less than the constant $\rho(M)$ of Lemma \ref{xxzz11}. |
|
859 |
||
236 | 860 |
Let $j \ge j_i$ and $p\in CH_n(X)$. |
90 | 861 |
Let $q$ be a generator appearing in $g_j(p)$. |
862 |
Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$, |
|
863 |
which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$. |
|
864 |
We must show that $q\ot b \in G_*^{i,m}$, which means finding neighborhoods |
|
865 |
$V_0,\ldots,V_m \sub X$ of $|q|\cup |b|$ such that each $V_j$ |
|
866 |
is homeomorphic to a disjoint union of balls and |
|
867 |
\[ |
|
868 |
N_{i,n}(q\ot b) \subeq V_0 \subeq N_{i,n+1}(q\ot b) |
|
869 |
\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,t}(q\ot b) . |
|
870 |
\] |
|
248 | 871 |
Recall that |
872 |
\[ |
|
873 |
N_{i,a}(q\ot b) \deq \Nbd_{a\ep_i}(|b|) \cup \Nbd_{\phi_a\delta_i}(|q|). |
|
874 |
\] |
|
90 | 875 |
By repeated applications of Lemma \ref{xx2phi} we can find neighborhoods $U_0,\ldots,U_m$ |
876 |
of $|q|$, each homeomorphic to a disjoint union of balls, with |
|
877 |
\[ |
|
878 |
\Nbd_{\phi_{n+l} \delta_i}(|q|) \subeq U_l \subeq \Nbd_{\phi_{n+l+1} \delta_i}(|q|) . |
|
85 | 879 |
\] |
248 | 880 |
The inequalities above guarantee that |
881 |
for each $0\le l\le m$ we can find $u_l$ with |
|
90 | 882 |
\[ |
883 |
(n+l)\ep_i \le u_l \le (n+l+1)\ep_i |
|
884 |
\] |
|
885 |
such that each component of $U_l$ is either disjoint from $\Nbd_{u_l}(|b|)$ or contained in |
|
886 |
$\Nbd_{u_l}(|b|)$. |
|
887 |
This is because there are at most $n$ components of $U_l$, and each component |
|
888 |
has radius $\le (\phi_t + 1) \delta_i$. |
|
889 |
It follows that |
|
890 |
\[ |
|
891 |
V_l \deq \Nbd_{u_l}(|b|) \cup U_l |
|
892 |
\] |
|
893 |
is homeomorphic to a disjoint union of balls and satisfies |
|
894 |
\[ |
|
895 |
N_{i,n+l}(q\ot b) \subeq V_l \subeq N_{i,n+l+1}(q\ot b) . |
|
896 |
\] |
|
86 | 897 |
|
90 | 898 |
The same argument shows that each generator involved in iterated boundaries of $q\ot b$ |
899 |
is in $G_*^{i,m}$. |
|
86 | 900 |
\end{proof} |
901 |
||
430 | 902 |
In the next three lemmas, which provide the estimates needed above, we have made no effort to optimize the various bounds. |
86 | 903 |
(The bounds are, however, optimal in the sense of minimizing the amount of work |
904 |
we do. Equivalently, they are the first bounds we thought of.) |
|
905 |
||
906 |
We say that a subset $S$ of a metric space has radius $\le r$ if $S$ is contained in |
|
907 |
some metric ball of radius $r$. |
|
908 |
||
909 |
\begin{lemma} |
|
910 |
Let $S \sub \ebb^n$ (Euclidean $n$-space) have radius $\le r$. |
|
911 |
Then $\Nbd_a(S)$ is homeomorphic to a ball for $a \ge 2r$. |
|
912 |
\end{lemma} |
|
913 |
||
914 |
\begin{proof} \label{xxyy2} |
|
915 |
Let $S$ be contained in $B_r(y)$, $y \in \ebb^n$. |
|
89 | 916 |
Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. |
917 |
Let $z\in \Nbd_a(S) \setmin B_r(y)$. |
|
918 |
Consider the triangle |
|
494
cb76847c439e
many small fixes in ncat.tex
Scott Morrison <scott@tqft.net>
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492
diff
changeset
|
919 |
with vertices $z$, $y$ and $s$ with $s\in S$ such that $z \in B_a(s)$. |
89 | 920 |
The length of the edge $yz$ is greater than $r$ which is greater |
921 |
than the length of the edge $ys$. |
|
922 |
It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), |
|
923 |
which means that points on the edge $yz$ near $z$ are closer to $s$ than $z$ is, |
|
924 |
which implies that these points are also in $\Nbd_a(S)$. |
|
925 |
Hence $\Nbd_a(S)$ is star-shaped with respect to $y$. |
|
926 |
\end{proof} |
|
927 |
||
928 |
If we replace $\ebb^n$ above with an arbitrary compact Riemannian manifold $M$, |
|
929 |
the same result holds, so long as $a$ is not too large: |
|
430 | 930 |
\nn{replace this with a PL version} |
89 | 931 |
|
932 |
\begin{lemma} \label{xxzz11} |
|
933 |
Let $M$ be a compact Riemannian manifold. |
|
934 |
Then there is a constant $\rho(M)$ such that for all |
|
935 |
subsets $S\sub M$ of radius $\le r$ and all $a$ such that $2r \le a \le \rho(M)$, |
|
936 |
$\Nbd_a(S)$ is homeomorphic to a ball. |
|
937 |
\end{lemma} |
|
938 |
||
939 |
\begin{proof} |
|
940 |
Choose $\rho = \rho(M)$ such that $3\rho/2$ is less than the radius of injectivity of $M$, |
|
941 |
and also so that for any point $y\in M$ the geodesic coordinates of radius $3\rho/2$ around |
|
942 |
$y$ distort angles by only a small amount. |
|
943 |
Now the argument of the previous lemma works. |
|
85 | 944 |
\end{proof} |
945 |
||
946 |
||
89 | 947 |
|
948 |
\begin{lemma} \label{xx2phi} |
|
949 |
Let $S \sub M$ be contained in a union (not necessarily disjoint) |
|
86 | 950 |
of $k$ metric balls of radius $r$. |
89 | 951 |
Let $\phi_1, \phi_2, \ldots$ be an increasing sequence of real numbers satisfying |
952 |
$\phi_1 \ge 2$ and $\phi_{i+1} \ge \phi_i(2\phi_i + 2) + \phi_i$. |
|
953 |
For convenience, let $\phi_0 = 0$. |
|
248 | 954 |
Assume also that $\phi_k r \le \rho(M)$, |
955 |
where $\rho(M)$ is as in Lemma \ref{xxzz11}. |
|
89 | 956 |
Then there exists a neighborhood $U$ of $S$, |
957 |
homeomorphic to a disjoint union of balls, such that |
|
86 | 958 |
\[ |
89 | 959 |
\Nbd_{\phi_{k-1} r}(S) \subeq U \subeq \Nbd_{\phi_k r}(S) . |
86 | 960 |
\] |
961 |
\end{lemma} |
|
962 |
||
963 |
\begin{proof} |
|
89 | 964 |
For $k=1$ this follows from Lemma \ref{xxzz11}. |
965 |
Assume inductively that it holds for $k-1$. |
|
86 | 966 |
Partition $S$ into $k$ disjoint subsets $S_1,\ldots,S_k$, each of radius $\le r$. |
89 | 967 |
By Lemma \ref{xxzz11}, each $\Nbd_{\phi_{k-1} r}(S_i)$ is homeomorphic to a ball. |
968 |
If these balls are disjoint, let $U$ be their union. |
|
969 |
Otherwise, assume WLOG that $S_{k-1}$ and $S_k$ are distance less than $2\phi_{k-1}r$ apart. |
|
970 |
Let $R_i = \Nbd_{\phi_{k-1} r}(S_i)$ for $i = 1,\ldots,k-2$ |
|
971 |
and $R_{k-1} = \Nbd_{\phi_{k-1} r}(S_{k-1})\cup \Nbd_{\phi_{k-1} r}(S_k)$. |
|
972 |
Each $R_i$ is contained in a metric ball of radius $r' \deq (2\phi_{k-1}+2)r$. |
|
91 | 973 |
Note that the defining inequality of the $\phi_i$ guarantees that |
974 |
\[ |
|
975 |
\phi_{k-1}r' = \phi_{k-1}(2\phi_{k-1}+2)r \le \phi_k r \le \rho(M) . |
|
976 |
\] |
|
89 | 977 |
By induction, there is a neighborhood $U$ of $R \deq \bigcup_i R_i$, |
978 |
homeomorphic to a disjoint union |
|
979 |
of balls, and such that |
|
86 | 980 |
\[ |
89 | 981 |
U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |
86 | 982 |
\] |
89 | 983 |
where $t = \phi_{k-1}(2\phi_{k-1}+2)r + \phi_{k-1} r$. |
86 | 984 |
\end{proof} |
985 |
||
430 | 986 |
|
987 |
We now return to defining the chain maps $e_X$. |
|
988 |
||
70 | 989 |
|
254 | 990 |
Let $R_*$ be the chain complex with a generating 0-chain for each non-negative |
991 |
integer and a generating 1-chain connecting each adjacent pair $(j, j+1)$. |
|
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992 |
(So $R_*$ is a simplicial version of the non-negative reals.) |
254 | 993 |
Denote the 0-chains by $j$ (for $j$ a non-negative integer) and the 1-chain connecting $j$ and $j+1$ |
994 |
by $\iota_j$. |
|
995 |
Define a map (homotopy equivalence) |
|
250 | 996 |
\[ |
254 | 997 |
\sigma: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to CH_*(X, X)\ot \bc_*(X) |
250 | 998 |
\] |
254 | 999 |
as follows. |
1000 |
On $R_0\ot CH_*(X, X) \otimes \bc_*(X)$ we define |
|
1001 |
\[ |
|
1002 |
\sigma(j\ot p\ot b) = g_j(p)\ot b . |
|
1003 |
\] |
|
255 | 1004 |
On $R_1\ot CH_*(X, X) \otimes \bc_*(X)$ we define |
1005 |
\[ |
|
1006 |
\sigma(\iota_j\ot p\ot b) = f_j(p)\ot b , |
|
1007 |
\] |
|
1008 |
where $f_j$ is the homotopy from $g_j$ to $g_{j+1}$. |
|
86 | 1009 |
|
254 | 1010 |
Next we specify subcomplexes $G^m_* \sub R_*\ot CH_*(X, X) \otimes \bc_*(X)$ on which we will eventually |
1011 |
define a version of the action map $e_X$. |
|
255 | 1012 |
A generator $j\ot p\ot b$ is defined to be in $G^m_*$ if $j\ge j_{kbmp}$, where |
254 | 1013 |
$k = k_{bmp}$ is the constant from Lemma \ref{Gim_approx}. |
255 | 1014 |
Similarly $\iota_j\ot p\ot b$ is in $G^m_*$ if $j\ge j_{kbmp}$. |
254 | 1015 |
The inequality following Lemma \ref{Gim_approx} guarantees that $G^m_*$ is indeed a subcomplex |
1016 |
and that $G^m_* \sup G^{m+1}_*$. |
|
250 | 1017 |
|
254 | 1018 |
It is easy to see that each $G^m_*$ is homotopy equivalent (via the inclusion map) |
1019 |
to $R_*\ot CH_*(X, X) \otimes \bc_*(X)$ |
|
1020 |
and hence to $CH_*(X, X) \otimes \bc_*(X)$, and furthermore that the homotopies are well-defined |
|
1021 |
up to a contractible set of choices. |
|
250 | 1022 |
|
254 | 1023 |
Next we define a map |
1024 |
\[ |
|
1025 |
e_m : G^m_* \to \bc_*(X) . |
|
1026 |
\] |
|
255 | 1027 |
Let $p\ot b$ be a generator of $G^m_*$. |
1028 |
Each $g_j(p)\ot b$ or $f_j(p)\ot b$ is a linear combination of generators $q\ot c$, |
|
1029 |
where $\supp(q)\cup\supp(c)$ is contained in a disjoint union of balls satisfying |
|
1030 |
various conditions specified above. |
|
1031 |
As in the construction of the maps $e_{i,m}$ above, |
|
1032 |
it suffices to specify for each such $q\ot c$ a disjoint union of balls |
|
1033 |
$V_{qc} \sup \supp(q)\cup\supp(c)$, such that $V_{qc} \sup V_{q'c'}$ |
|
1034 |
whenever $q'\ot c'$ appears in the boundary of $q\ot c$. |
|
1035 |
||
1036 |
Let $q\ot c$ be a summand of $g_j(p)\ot b$, as above. |
|
1037 |
Let $i$ be maximal such that $j\ge j_{ibmp}$ |
|
1038 |
(notation as in Lemma \ref{Gim_approx}). |
|
1039 |
Then $q\ot c \in G^{i,m}_*$ and we choose $V_{qc} \sup \supp(q)\cup\supp(c)$ |
|
1040 |
such that |
|
1041 |
\[ |
|
1042 |
N_{i,d}(q\ot c) \subeq V_{qc} \subeq N_{i,d+1}(q\ot c) , |
|
1043 |
\] |
|
1044 |
where $d = \deg(q\ot c)$. |
|
1045 |
Let $\tilde q = f_j(q)$. |
|
1046 |
The summands of $f_j(p)\ot b$ have the form $\tilde q \ot c$, |
|
1047 |
where $q\ot c$ is a summand of $g_j(p)\ot b$. |
|
1048 |
Since the homotopy $f_j$ does not increase supports, we also have that |
|
1049 |
\[ |
|
1050 |
V_{qc} \sup \supp(\tilde q) \cup \supp(c) . |
|
1051 |
\] |
|
1052 |
So we define $V_{\tilde qc} = V_{qc}$. |
|
1053 |
||
1054 |
It is now easy to check that we have $V_{qc} \sup V_{q'c'}$ |
|
1055 |
whenever $q'\ot c'$ appears in the boundary of $q\ot c$. |
|
1056 |
As in the construction of the maps $e_{i,m}$ above, |
|
1057 |
this allows us to construct a map |
|
1058 |
\[ |
|
1059 |
e_m : G^m_* \to \bc_*(X) |
|
1060 |
\] |
|
1061 |
which is well-defined up to homotopy. |
|
1062 |
As in the proof of Lemma \ref{m_order_hty}, we can show that the map is well-defined up |
|
1063 |
to $m$-th order homotopy. |
|
1064 |
Put another way, we have specified an $m$-connected subcomplex of the complex of |
|
1065 |
all maps $G^m_* \to \bc_*(X)$. |
|
1066 |
On $G^{m+1}_* \sub G^m_*$ we have defined two maps, $e_m$ and $e_{m+1}$. |
|
1067 |
One can similarly (to the proof of Lemma \ref{m_order_hty}) show that |
|
1068 |
these two maps agree up to $m$-th order homotopy. |
|
1069 |
More precisely, one can show that the subcomplex of maps containing the various |
|
1070 |
$e_{m+1}$ candidates is contained in the corresponding subcomplex for $e_m$. |
|
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1071 |
|
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|
1072 |
\medskip |
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|
1073 |
|
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|
1074 |
Next we show that the action maps are compatible with gluing. |
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|
1075 |
Let $G^m_*$ and $\ol{G}^m_*$ be the complexes, as above, used for defining |
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|
1076 |
the action maps $e_{X\sgl}$ and $e_X$. |
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|
1077 |
The gluing map $X\sgl\to X$ induces a map |
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|
1078 |
\[ |
430 | 1079 |
\gl: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) , |
358
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|
1080 |
\] |
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|
1081 |
and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$. |
437 | 1082 |
From this it follows that the diagram in the statement of Theorem \ref{thm:CH} commutes. |
358
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|
1083 |
|
430 | 1084 |
\todo{this paragraph isn't very convincing, or at least I don't see what's going on} |
358
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|
1085 |
Finally we show that the action maps defined above are independent of |
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|
1086 |
the choice of metric (up to iterated homotopy). |
359
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|
1087 |
The arguments are very similar to ones given above, so we only sketch them. |
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|
1088 |
Let $g$ and $g'$ be two metrics on $X$, and let $e$ and $e'$ be the corresponding |
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|
1089 |
actions $CH_*(X, X) \ot \bc_*(X)\to\bc_*(X)$. |
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|
1090 |
We must show that $e$ and $e'$ are homotopic. |
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|
1091 |
As outlined in the discussion preceding this proof, |
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|
1092 |
this follows from the facts that both $e$ and $e'$ are compatible |
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|
1093 |
with gluing and that $\bc_*(B^n)$ is contractible. |
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|
1094 |
As above, we define a subcomplex $F_*\sub CH_*(X, X) \ot \bc_*(X)$ generated |
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|
1095 |
by $p\ot b$ such that $|p|\cup|b|$ is contained in a disjoint union of balls. |
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|
1096 |
Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$. |
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|
1097 |
We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$. |
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|
1098 |
Similar arguments show that this homotopy from $e$ to $e'$ is well-defined |
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|
1099 |
up to second order homotopy, and so on. |
430 | 1100 |
|
437 | 1101 |
This completes the proof of Theorem \ref{thm:CH}. |
84 | 1102 |
\end{proof} |
1103 |
||
1104 |
||
396
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|
1105 |
\begin{rem*} |
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diff
changeset
|
1106 |
\label{rem:for-small-blobs} |
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diff
changeset
|
1107 |
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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|
1108 |
Let $b$ be a blob diagram and $p:P\times X\to X$ be a family of homeomorphisms. |
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|
1109 |
Then we may choose $e$ such that $e(p\ot b)$ is a sum of generators, each |
385
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|
1110 |
of which has support close to $p(t,|b|)$ for some $t\in P$. |
430 | 1111 |
More precisely, the support of the generators is contained in the union of a small neighborhood |
1112 |
of $p(t,|b|)$ with some small balls. |
|
385
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|
1113 |
(Here ``small" is in terms of the metric on $X$ that we chose to construct $e$.) |
396
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|
1114 |
\end{rem*} |
385
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|
1115 |
|
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|
1116 |
|
437 | 1117 |
\begin{thm} |
1118 |
\label{thm:CH-associativity} |
|
357
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|
1119 |
The $CH_*(X, Y)$ actions defined above are associative. |
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|
1120 |
That is, the following diagram commutes up to homotopy: |
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|
1121 |
\[ \xymatrix{ |
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|
1122 |
& CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\ |
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|
1123 |
CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\ |
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|
1124 |
& CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} & |
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|
1125 |
} \] |
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|
1126 |
Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition |
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|
1127 |
of homeomorphisms. |
437 | 1128 |
\end{thm} |
70 | 1129 |
|
357
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|
1130 |
\begin{proof} |
437 | 1131 |
The strategy of the proof is similar to that of Theorem \ref{thm:CH}. |
357
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|
1132 |
We will identify a subcomplex |
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|
1133 |
\[ |
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|
1134 |
G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) |
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|
1135 |
\] |
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|
1136 |
where it is easy to see that the two sides of the diagram are homotopic, then |
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|
1137 |
show that there is a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. |
70 | 1138 |
|
357
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|
1139 |
Let $p\ot q\ot b$ be a generator of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$. |
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|
1140 |
By definition, $p\ot q\ot b\in G_*$ if there is a disjoint union of balls in $X$ which |
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|
1141 |
contains $|p| \cup p\inv(|q|) \cup |b|$. |
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|
1142 |
(If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of |
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|
1143 |
$p(x, \cdot)\inv(|q|)$.) |
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|
1144 |
|
437 | 1145 |
As in the proof of Theorem \ref{thm:CH}, we can construct a homotopy |
357
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|
1146 |
between the upper and lower maps restricted to $G_*$. |
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|
1147 |
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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|
1148 |
that they are compatible with gluing, and the contractibility of $\bc_*(X)$. |
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|
1149 |
|
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|
1150 |
We can now apply Lemma \ref{extension_lemma_c}, using a series of increasingly fine covers, |
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|
1151 |
to construct a deformation retraction of $CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)$ into $G_*$. |
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|
1152 |
\end{proof} |
524
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523
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|
1153 |
|
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|
1154 |
} % end \noop |