54 |
54 |
55 \medskip |
55 \medskip |
56 |
56 |
57 If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted |
57 If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted |
58 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
58 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
59 For a general $k-chain$ $a\in \bc_k(X)$, define the support of $a$ to be the union |
59 For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union |
60 of the supports of the blob diagrams which appear in it. |
60 of the supports of the blob diagrams which appear in it. |
61 |
61 |
62 If $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is |
62 If $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is |
63 {\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$. |
63 {\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$. |
64 We will sometimes abuse language and talk about ``the" support of $f$, |
64 We will sometimes abuse language and talk about ``the" support of $f$, |
73 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
73 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
74 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
74 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
75 and moreover each field labeling a region cut out by the blobs is splittable |
75 and moreover each field labeling a region cut out by the blobs is splittable |
76 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
76 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
77 |
77 |
78 \begin{thm}[Small blobs] \label{thm:small-blobs-xx} |
78 \begin{lemma}[Small blobs] \label{small-blobs-b} |
79 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
79 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
80 \end{thm} |
80 \end{lemma} |
81 |
81 |
82 \begin{proof} |
82 \begin{proof} |
83 It suffices to show that for any finitely generated pair of subcomplexes |
83 It suffices to show that for any finitely generated pair of subcomplexes |
84 $(C_*, D_*) \sub (\bc_*(X), \sbc_*(X))$ |
84 $(C_*, D_*) \sub (\bc_*(X), \sbc_*(X))$ |
85 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
85 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
86 and $x + h\bd(x) + \bd h(X) \in \sbc_*(X)_*(X)$ for all $x\in C_*$. |
86 and $x + h\bd(x) + \bd h(X) \in \sbc_*(X)$ for all $x\in C_*$. |
87 |
87 |
88 For simplicity we will assume that all fields are splittable into small pieces, so that |
88 For simplicity we will assume that all fields are splittable into small pieces, so that |
89 $\sbc_0(X) = \bc_0$. |
89 $\sbc_0(X) = \bc_0$. |
90 (This is true for all of the examples presented in this paper.) |
90 (This is true for all of the examples presented in this paper.) |
91 Accordingly, we define $h_0 = 0$. |
91 Accordingly, we define $h_0 = 0$. |
295 &= x - r(x) + r(x) - r(x)\\ |
295 &= x - r(x) + r(x) - r(x)\\ |
296 &= x - r(x). |
296 &= x - r(x). |
297 \end{align*} |
297 \end{align*} |
298 \end{proof} |
298 \end{proof} |
299 |
299 |
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300 \begin{lemma} |
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301 For manifolds $X$ and $Y$, we have $\btc_*(X\du Y) \simeq \btc_*(X)\otimes\btc_*(Y)$. |
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302 \end{lemma} |
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303 \begin{proof} |
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304 This follows from the Eilenber-Zilber theorem and the fact that |
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305 \[ |
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306 \BD_k(X\du Y) \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . |
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307 \] |
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308 \end{proof} |
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309 |
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310 For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} |
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311 if there exists $S' \subeq S$, $a'\in \btc_k(S')$ |
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312 and $r\in \btc_0(X\setmin S')$ such that $a = a'\bullet r$. |
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313 |
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314 \newcommand\sbtc{\btc^{\cU}} |
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315 Let $\cU$ be an open cover of $X$. |
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316 Let $\sbtc_*(X)\sub\btc_*(X)$ be the subcomplex generated by |
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317 $a\in \btc_*(X)$ such that there is a decomposition $X = \cup_i D_i$ |
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318 such that each $D_i$ is a ball contained in some open set of $\cU$ and |
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319 $a$ is splittable along this decomposition. |
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320 In other words, $a$ can be obtained by gluing together pieces, each of which |
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321 is small with respect to $\cU$. |
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322 |
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323 \begin{lemma} \label{small-top-blobs} |
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324 For any open cover $\cU$ of $X$, the inclusion $\sbtc_*(X)\sub\btc_*(X)$ |
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325 is a homotopy equivalence. |
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326 \end{lemma} |
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327 \begin{proof} |
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328 This follows from a combination of Lemma \ref{extension_lemma_c} and the techniques of |
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329 the proof of Lemma \ref{small-blobs-b}. |
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330 |
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331 It suffices to show that we can deform a finite subcomplex $C_*$ of $\btc_*(X)$ into $\sbtc_*(X)$ |
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332 (relative to any designated subcomplex of $C_*$ already in $\sbtc_*(X)$). |
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333 The first step is to replace families of general blob diagrams with families that are |
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334 small with respect to $\cU$. |
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335 This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families. |
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336 Each such family is homotopic to a sum families which can be a ``lifted" to $\Homeo(X)$. |
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337 That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$. |
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338 (We are ignoring a complication related to twig blob labels, which might vary |
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339 independently of $g$, but this complication does not affect the conclusion we draw here.) |
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340 We now apply Lemma \ref{extension_lemma_c} to get families which are supported |
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341 on balls $D_i$ contained in open sets of $\cU$. |
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342 \end{proof} |
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343 |
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344 |
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345 \begin{proof}[Proof of \ref{lem:bt-btc}] |
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346 Armed with the above lemmas, we can now proceed similarly to the proof of \ref{small-blobs-b}. |
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347 |
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348 It suffices to show that for any finitely generated pair of subcomplexes |
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349 $(C_*, D_*) \sub (\btc_*(X), \bc_*(X))$ |
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350 we can find a homotopy $h:C_*\to \btc_*(X)$ such that $h(D_*) \sub \bc_*(X)$ |
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351 and $x + h\bd(x) + \bd h(X) \in \bc_*(X)$ for all $x\in C_*$. |
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352 |
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353 By Lemma \ref{small-top-blobs}, we may assume that $C_* \sub \btc_*^\cU(X)$ for some |
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354 cover $\cU$ of our choosing. |
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355 We choose $\cU$ fine enough so that each generator of $C_*$ is supported on a disjoint union of balls. |
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356 (This is possible since the original $C_*$ was finite and therefore had bounded dimension.) |
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357 |
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358 Since $\bc_0(X) = \btc_0(X)$, we can take $h_0 = 0$. |
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359 |
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360 |
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361 |
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362 |
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363 \nn{...} |
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364 \end{proof} |
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365 |
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366 |
300 |
367 |
301 |
368 |
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369 |
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371 |