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293 |
293 |
294 We define $g(C^j) = 0$ for $j > 0$. |
294 We define $g(C^j) = 0$ for $j > 0$. |
295 It is not hard to see that this defines a chain map from |
295 It is not hard to see that this defines a chain map from |
296 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
296 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
297 |
297 |
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298 Next we show that $g$ induces a surjection on homology. |
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299 Fix $k > 0$ and choose an open cover $\cU$ of $M$ fine enough so that the union |
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300 of any $k$ open sets of $\cU$ is contained in a disjoint union of balls in $M$. |
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301 \nn{maybe should refer to elsewhere in this paper where we made a very similar argument} |
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302 Let $S_*$ be the subcomplex of $C_*(\Maps(M\to T))$ generated by chains adapted to $\cU$. |
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303 It follows from Lemma \ref{extension_lemma_b} that $C_*(\Maps(M\to T))$ |
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304 retracts onto $S_*$. |
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305 |
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306 Let $S_{\le k}$ denote the chains of $S_*$ of degree less than or equal to $k$. |
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307 We claim that $S_{\le k}$ lies in the image of $g$. |
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308 Let $c$ be a generator of $S_{\le k}$ --- that is, a $j$-parameter family of maps $M\to T$, |
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309 $j \le k$. |
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310 We chose $\cU$ fine enough so that the support of $c$ is contained in a disjoint union of balls |
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311 in $M$. |
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312 It follow that we can choose a decomposition $K$ of $M$ so that the support of $c$ is |
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313 disjoint from the $n{-}1$-skeleton of $K$. |
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314 It is now easy to see that $c$ is in the image of $g$. |
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315 |
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316 Next we show that $g$ is injective on homology. |
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317 |
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318 |
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319 |
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320 |
298 \nn{...} |
321 \nn{...} |
299 |
322 |
300 |
323 |
301 |
324 |
302 \end{proof} |
325 \end{proof} |