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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 |
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299 %%%%%%%%%%%%%%%%% |
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300 \noop{ |
298 Next we show that $g$ induces a surjection on homology. |
301 Next we show that $g$ induces a surjection on homology. |
299 Fix $k > 0$ and choose an open cover $\cU$ of $M$ fine enough so that the union |
302 Fix $k > 0$ and choose an open cover $\cU$ of $M$ fine enough so that the union |
300 of any $k$ open sets of $\cU$ is contained in a disjoint union of balls in $M$. |
303 of any $k$ open sets of $\cU$ is contained in a disjoint union of balls in $M$. |
301 \nn{maybe should refer to elsewhere in this paper where we made a very similar argument} |
304 \nn{maybe should refer to elsewhere in this paper where we made a very similar argument} |
302 Let $S_*$ be the subcomplex of $C_*(\Maps(M\to T))$ generated by chains adapted to $\cU$. |
305 Let $S_*$ be the subcomplex of $C_*(\Maps(M\to T))$ generated by chains adapted to $\cU$. |
312 It follow that we can choose a decomposition $K$ of $M$ so that the support of $c$ is |
315 It follow that we can choose a decomposition $K$ of $M$ so that the support of $c$ is |
313 disjoint from the $n{-}1$-skeleton of $K$. |
316 disjoint from the $n{-}1$-skeleton of $K$. |
314 It is now easy to see that $c$ is in the image of $g$. |
317 It is now easy to see that $c$ is in the image of $g$. |
315 |
318 |
316 Next we show that $g$ is injective on homology. |
319 Next we show that $g$ is injective on homology. |
317 |
320 } |
318 |
321 |
319 |
322 |
320 |
323 |
321 \nn{...} |
324 \nn{...} |
322 |
325 |