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277 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
277 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
278 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
278 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
279 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
279 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
280 |
280 |
281 \begin{thm} \label{thm:map-recon} |
281 \begin{thm} \label{thm:map-recon} |
282 $\cB^\cT(M) \simeq C_*(\Maps(M\to T))$. |
282 The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ is quasi-isomorphic to singular chains on maps from $M$ to $T$. |
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283 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
283 \end{thm} |
284 \end{thm} |
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285 \begin{rem} |
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286 \nn{This just isn't true, Lurie doesn't do this! I just heard this from Ricardo...} |
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287 Lurie has shown in \cite{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in \nn{a certain $E_n$ algebra constructed from $T$} recovers the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. This extra hypothesis is not surprising, in view of the idea that an $E_n$ algebra is roughly equivalent data as an $A_\infty$ $n$-category which is trivial at all but the topmost level. |
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288 \end{rem} |
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289 |
284 \begin{proof} |
290 \begin{proof} |
285 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
291 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
286 We then use \ref{extension_lemma_b} to show that $g$ induces isomorphisms on homology. |
292 We then use \ref{extension_lemma_b} to show that $g$ induces isomorphisms on homology. |
287 |
293 |
288 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
294 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |