262 |
262 |
263 \begin{thm} \label{thm:map-recon} |
263 \begin{thm} \label{thm:map-recon} |
264 $\cB^\cT(M) \simeq C_*(\Maps(M\to T))$. |
264 $\cB^\cT(M) \simeq C_*(\Maps(M\to T))$. |
265 \end{thm} |
265 \end{thm} |
266 \begin{proof} |
266 \begin{proof} |
267 \nn{obvious map in one direction; use \ref{extension_lemma_b}; ...} |
267 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
268 \end{proof} |
268 We then use \ref{extension_lemma_b} to show that $g$ induces isomorphisms on homology. |
269 |
269 |
270 \nn{should also mention version where we enrich over |
270 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
271 spaces rather than chain complexes; should comment on Lurie's (and others') similar result |
271 $j$-fold mapping cylinders, $j \ge 0$. |
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272 So, as an abelian group (but not as a chain complex), |
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273 \[ |
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274 \cB^\cT(M) = \bigoplus_{j\ge 0} C^j, |
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275 \] |
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276 where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders. |
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277 |
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278 Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by |
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279 decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms |
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280 of $\cT$. |
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281 Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs |
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282 $(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous |
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283 maps from the $n{-}1$-skeleton of $K$ to $T$. |
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284 The summand indexed by $(K, \vphi)$ is |
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285 \[ |
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286 \bigotimes_b D_*(b, \vphi), |
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287 \] |
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288 where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes |
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289 chains of maps from $b$ to $T$ compatible with $\vphi$. |
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290 We can take the product of these chains of maps to get a chains of maps from |
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291 all of $M$ to $K$. |
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292 This defines $g$ on $C^0$. |
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293 |
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294 We define $g(C^j) = 0$ for $j > 0$. |
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295 It is not hard to see that this defines a chain map from |
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296 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
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297 |
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298 \nn{...} |
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299 |
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300 |
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301 |
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302 \end{proof} |
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303 |
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304 \nn{maybe should also mention version where we enrich over |
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305 spaces rather than chain complexes; should comment on Lurie's (and others') similar result |
272 for the $E_\infty$ case, and mention that our version does not require |
306 for the $E_\infty$ case, and mention that our version does not require |
273 any connectivity assumptions} |
307 any connectivity assumptions} |
274 |
308 |
275 \medskip |
309 \medskip |
276 \hrule |
310 \hrule |