355 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 $T$. |
355 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 $T$. |
356 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
356 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
357 \end{thm} |
357 \end{thm} |
358 \begin{rem} |
358 \begin{rem} |
359 \nn{This just isn't true, Lurie doesn't do this! I just heard this from Ricardo...} |
359 \nn{This just isn't true, Lurie doesn't do this! I just heard this from Ricardo...} |
|
360 \nn{KW: Are you sure about that?} |
360 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. |
361 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. |
361 \end{rem} |
362 \end{rem} |
362 |
363 |
|
364 \nn{proof is again similar to that of Theorem \ref{product_thm}. should probably say that explicitly} |
|
365 |
363 \begin{proof} |
366 \begin{proof} |
364 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
367 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
365 We then use \ref{extension_lemma_b} to show that $g$ induces isomorphisms on homology. |
368 We then use Lemma \ref{extension_lemma_c} to show that $g$ induces isomorphisms on homology. |
366 |
369 |
367 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
370 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
368 $j$-fold mapping cylinders, $j \ge 0$. |
371 $j$-fold mapping cylinders, $j \ge 0$. |
369 So, as an abelian group (but not as a chain complex), |
372 So, as an abelian group (but not as a chain complex), |
370 \[ |
373 \[ |
390 |
393 |
391 We define $g(C^j) = 0$ for $j > 0$. |
394 We define $g(C^j) = 0$ for $j > 0$. |
392 It is not hard to see that this defines a chain map from |
395 It is not hard to see that this defines a chain map from |
393 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
396 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
394 |
397 |
395 |
|
396 %%%%%%%%%%%%%%%%% |
|
397 \noop{ |
|
398 Next we show that $g$ induces a surjection on homology. |
|
399 Fix $k > 0$ and choose an open cover $\cU$ of $M$ fine enough so that the union |
|
400 of any $k$ open sets of $\cU$ is contained in a disjoint union of balls in $M$. |
|
401 \nn{maybe should refer to elsewhere in this paper where we made a very similar argument} |
|
402 Let $S_*$ be the subcomplex of $C_*(\Maps(M\to T))$ generated by chains adapted to $\cU$. |
|
403 It follows from Lemma \ref{extension_lemma_b} that $C_*(\Maps(M\to T))$ |
|
404 retracts onto $S_*$. |
|
405 |
|
406 Let $S_{\le k}$ denote the chains of $S_*$ of degree less than or equal to $k$. |
|
407 We claim that $S_{\le k}$ lies in the image of $g$. |
|
408 Let $c$ be a generator of $S_{\le k}$ --- that is, a $j$-parameter family of maps $M\to T$, |
|
409 $j \le k$. |
|
410 We chose $\cU$ fine enough so that the support of $c$ is contained in a disjoint union of balls |
|
411 in $M$. |
|
412 It follow that we can choose a decomposition $K$ of $M$ so that the support of $c$ is |
|
413 disjoint from the $n{-}1$-skeleton of $K$. |
|
414 It is now easy to see that $c$ is in the image of $g$. |
|
415 |
|
416 Next we show that $g$ is injective on homology. |
|
417 } |
|
418 |
|
419 |
|
420 |
|
421 \nn{...} |
398 \nn{...} |
422 |
399 |
423 |
|
424 |
|
425 \end{proof} |
400 \end{proof} |
426 |
401 |
427 \nn{maybe should also mention version where we enrich over |
402 \nn{maybe should also mention version where we enrich over |
428 spaces rather than chain complexes; should comment on Lurie's \cite{0911.0018} (and others') similar result |
403 spaces rather than chain complexes;} |
429 for the $E_\infty$ case, and mention that our version does not require |
|
430 any connectivity assumptions} |
|
431 |
404 |
432 \medskip |
405 \medskip |
433 \hrule |
406 \hrule |
434 \medskip |
407 \medskip |
435 |
408 |
436 \nn{to be continued...} |
409 \nn{to be continued...} |
437 \medskip |
410 \medskip |
438 \nn{still to do: fiber bundles, general maps} |
411 \nn{still to do: general maps} |
439 |
412 |
440 \todo{} |
413 \todo{} |
441 Various citations we might want to make: |
414 Various citations we might want to make: |
442 \begin{itemize} |
415 \begin{itemize} |
443 \item \cite{MR2061854} McClure and Smith's review article |
416 \item \cite{MR2061854} McClure and Smith's review article |
444 \item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) |
417 \item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) |
445 \item \cite{MR0236922,MR0420609} Boardman and Vogt |
418 \item \cite{MR0236922,MR0420609} Boardman and Vogt |
446 \item \cite{MR1256989} definition of framed little-discs operad |
419 \item \cite{MR1256989} definition of framed little-discs operad |
447 \end{itemize} |
420 \end{itemize} |
448 |
421 |
449 We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction |
422 |
450 \begin{itemize} |
|
451 %\mbox{}% <-- gets the indenting right |
|
452 \item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
|
453 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
|
454 |
|
455 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
|
456 $A_\infty$ module for $\bc_*(Y \times I)$. |
|
457 |
|
458 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
|
459 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
|
460 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
|
461 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
|
462 \begin{equation*} |
|
463 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
|
464 \end{equation*} |
|
465 \end{itemize} |
|
466 |
|