299 Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
299 Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, |
300 call them $e_{i,m}$ and $e_{i,m+1}$. |
300 call them $e_{i,m}$ and $e_{i,m+1}$. |
301 An easy variation on the above lemma shows that $e_{i,m}$ and $e_{i,m+1}$ are $m$-th |
301 An easy variation on the above lemma shows that $e_{i,m}$ and $e_{i,m+1}$ are $m$-th |
302 order homotopic. |
302 order homotopic. |
303 |
303 |
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304 Next we show how to homotope chains in $CD_*(X)\ot \bc_*(X)$ to one of the |
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305 $G_*^{i,m}$. |
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306 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
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307 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
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308 Let $h_j: CD_*(X)\to CD_*(X)$ be a chain map homotopic to the identity whose image is spanned by diffeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}. |
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309 Recall that $h_j$ and also its homotopy back to the identity do not increase |
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310 supports. |
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311 Define |
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312 \[ |
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313 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
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314 \] |
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315 The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
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316 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
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317 (depending on $b$, $n = \deg(p)$ and $m$). |
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318 |
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319 \begin{lemma} |
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320 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$. |
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321 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
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322 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$ |
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323 we have $g_j(p)\ot b \in G_*^{i,m}$. |
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324 \end{lemma} |
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325 |
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326 \begin{proof} |
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327 Let $c$ be a subset of the blobs of $b$. |
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328 There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$ |
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329 and all such $c$. |
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330 (Here we are using a piecewise smoothness assumption for $\bd c$). |
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331 |
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332 Let $r = \deg(b)$. |
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333 |
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334 Choose $k = k_{bmn}$ such that |
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335 \[ |
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336 (r+n+m+1)\ep_k < l |
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337 \] |
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338 and |
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339 \[ |
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340 n\cdot (3\delta_k\cdot(r+n+m+1)) < \ep_k . |
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341 \] |
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342 Let $i \ge k_{bmn}$. |
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343 Choose $j = j_i$ so that |
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344 \[ |
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345 3\cdot(r+n+m+1)\gamma_j < \ep_i |
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346 \] |
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347 and also so that for any subset $S\sub X$ of diameter less than or equal to |
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348 $2n\gamma_j$ we have that $\Nbd_u(S)$ is |
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349 \end{proof} |
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350 |
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351 |
304 |
352 |
305 \medskip |
353 \medskip |
306 |
354 |
307 \noop{ |
355 \noop{ |
308 |
356 |
309 |
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310 \begin{lemma} |
357 \begin{lemma} |
311 |
358 |
312 \end{lemma} |
359 \end{lemma} |
313 \begin{proof} |
360 \begin{proof} |
314 |
361 |
315 \end{proof} |
362 \end{proof} |
316 |
363 |
317 |
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318 } |
364 } |
319 |
365 |
320 |
366 |
321 \nn{to be continued....} |
367 \nn{to be continued....} |
322 |
368 |