429 |
429 |
430 Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}. |
430 Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}. |
431 |
431 |
432 \subsection{Applications} |
432 \subsection{Applications} |
433 \label{sec:applications} |
433 \label{sec:applications} |
434 Finally, we give two theorems which we consider applications. % or "think of as" |
434 Finally, we give two applications of the above machinery. |
435 |
435 |
436 \newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}} |
436 \newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}} |
437 |
437 |
438 \begin{thm:map-recon}[Mapping spaces] |
438 \begin{thm:map-recon}[Mapping spaces] |
439 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
439 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
442 Then |
442 Then |
443 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
443 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
444 \end{thm:map-recon} |
444 \end{thm:map-recon} |
445 |
445 |
446 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
446 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
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447 Note that there is no restriction on the connectivity of $T$. |
447 The proof appears in \S \ref{sec:map-recon}. |
448 The proof appears in \S \ref{sec:map-recon}. |
448 |
449 |
449 \newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}} |
450 \newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}} |
450 |
451 |
451 \begin{thm:deligne}[Higher dimensional Deligne conjecture] |
452 \begin{thm:deligne}[Higher dimensional Deligne conjecture] |
452 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
453 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
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454 Since the little $n{+}1$-balls operad is a suboperad of the $n$-dimensional surgery cylinder operad, |
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455 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
453 \end{thm:deligne} |
456 \end{thm:deligne} |
454 See \S \ref{sec:deligne} for a full explanation of the statement, and the proof. |
457 See \S \ref{sec:deligne} for a full explanation of the statement, and the proof. |
455 |
458 |
456 |
459 |
457 |
460 |
458 \noop{ |
461 \noop{ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
459 \subsection{Future directions} |
462 \subsection{Future directions} |
460 \label{sec:future} |
463 \label{sec:future} |
461 \nn{KW: Perhaps we should delete this subsection and salvage only the first few sentences.} |
464 \nn{KW: Perhaps we should delete this subsection and salvage only the first few sentences.} |
462 Throughout, we have resisted the temptation to work in the greatest generality possible. |
465 Throughout, we have resisted the temptation to work in the greatest generality possible. |
463 (Don't worry, it wasn't that hard.) |
466 (Don't worry, it wasn't that hard.) |
480 For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
483 For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
481 (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1]; A)$, |
484 (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1]; A)$, |
482 but haven't investigated the details. |
485 but haven't investigated the details. |
483 |
486 |
484 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} \nn{stable categories, generalized cohomology theories} |
487 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} \nn{stable categories, generalized cohomology theories} |
485 } %%% end \noop |
488 } %%% end \noop %%%%%%%%%%%%%%%%%%%%% |
486 |
489 |
487 \subsection{Thanks and acknowledgements} |
490 \subsection{Thanks and acknowledgements} |
488 % attempting to make this chronological rather than alphabetical |
491 % attempting to make this chronological rather than alphabetical |
489 We'd like to thank |
492 We'd like to thank |
490 Justin Roberts, |
493 Justin Roberts, |