416 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
416 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
417 \end{thm} |
417 \end{thm} |
418 \begin{rem} |
418 \begin{rem} |
419 Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology |
419 Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology |
420 of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers |
420 of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers |
421 the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. |
421 the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n{-}1$-connected. |
422 This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} |
422 This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} |
423 that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which |
423 that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which |
424 is trivial at levels 0 through $n-1$. |
424 is trivial at levels 0 through $n-1$. |
425 Ricardo Andrade also told us about a similar result. |
425 Ricardo Andrade also told us about a similar result. |
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426 |
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427 Specializing still further, Theorem \ref{thm:map-recon} is related to the classical result that for connected spaces $T$ |
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428 we have $HH_*(C_*(\Omega T)) \cong H_*(LT)$, that is, the Hochschild homology of based loops in $T$ is isomorphic |
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429 to the homology of the free loop space of $T$ (see \cite{MR793184} and \cite{MR842427}). |
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430 Theorem \ref{thm:map-recon} says that for any space $T$ (connected or not) we have |
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431 $\bc_*(S^1; C_*(\pi^\infty_{\le 1}(T))) \simeq C_*(LT)$. |
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432 Here $C_*(\pi^\infty_{\le 1}(T))$ denotes the singular chain version of the fundamental infinity-groupoid of $T$, |
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433 whose objects are points in $T$ and morphism chain complexes are $C_*(\paths(t_1 \to t_2))$ for $t_1, t_2 \in T$. |
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434 If $T$ is connected then the $A_\infty$ 1-category $C_*(\pi^\infty_{\le 1}(T))$ is Morita equivalent to the |
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435 $A_\infty$ algebra $C_*(\Omega T)$; |
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436 the bimodule for the equivalence is the singular chains of the space of paths which start at the base point of $T$. |
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437 Theorem \ref{thm:hochschild} holds for $A_\infty$ 1-categories (though we do not prove that in this paper), |
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438 which then implies that |
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439 \[ |
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440 Hoch_*(C_*(\Omega T)) \simeq Hoch_*(C_*(\pi^\infty_{\le 1}(T))) |
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441 \simeq \bc_*(S^1; C_*(\pi^\infty_{\le 1}(T))) \simeq C_*(LT) . |
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442 \] |
426 \end{rem} |
443 \end{rem} |
427 |
444 |
428 \begin{proof} |
445 \begin{proof}[Proof of Theorem \ref{thm:map-recon}] |
429 The proof is again similar to that of Theorem \ref{thm:product}. |
446 The proof is again similar to that of Theorem \ref{thm:product}. |
430 |
447 |
431 We begin by constructing a chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
448 We begin by constructing a chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
432 |
449 |
433 Recall that |
450 Recall that |