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249 Theorem \ref{product_thm}. |
249 Theorem \ref{product_thm}. |
250 \end{proof} |
250 \end{proof} |
251 |
251 |
252 |
252 |
253 \medskip |
253 \medskip |
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254 |
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255 The next theorem shows how to reconstruct a mapping space from local data. |
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256 Let $T$ be a topological space, let $M$ be an $n$-manifold, |
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257 and recall the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$ |
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258 of Example \ref{ex:chains-of-maps-to-a-space}. |
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259 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
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260 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
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261 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
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262 |
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263 \begin{thm} \label{thm:map-recon} |
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264 $\cB^\cT(M) \simeq C_*(\Maps(M\to T))$. |
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265 \end{thm} |
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266 \begin{proof} |
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267 \nn{obvious map in one direction; use \ref{extension_lemma_b}; ...} |
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268 \end{proof} |
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269 |
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270 \nn{should also mention version where we enrich over |
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271 spaces rather than chain complexes; should comment on Lurie's (and others') similar result |
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272 for the $E_\infty$ case, and mention that our version does not require |
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273 any connectivity assumptions} |
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274 |
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275 \medskip |
254 \hrule |
276 \hrule |
255 \medskip |
277 \medskip |
256 |
278 |
257 \nn{to be continued...} |
279 \nn{to be continued...} |
258 \medskip |
280 \medskip |