14 new-fangled and old-fashioned blob complex. |
14 new-fangled and old-fashioned blob complex. |
15 |
15 |
16 \medskip |
16 \medskip |
17 |
17 |
18 An important technical tool in the proofs of this section is provided by the idea of `small blobs'. |
18 An important technical tool in the proofs of this section is provided by the idea of `small blobs'. |
19 Fix $\cU$, an open cover of $M$. Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$. |
19 Fix $\cU$, an open cover of $M$. |
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20 Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$. |
20 \nn{KW: We need something a little stronger: Every blob diagram (even a 0-blob diagram) is splittable into pieces which are small w.r.t.\ $\cU$. |
21 \nn{KW: We need something a little stronger: Every blob diagram (even a 0-blob diagram) is splittable into pieces which are small w.r.t.\ $\cU$. |
21 If field have potentially large coupons/boxes, then this is a non-trivial constraint. |
22 If field have potentially large coupons/boxes, then this is a non-trivial constraint. |
22 On the other hand, we could probably get away with ignoring this point. |
23 On the other hand, we could probably get away with ignoring this point. |
23 Maybe the exposition will be better if we sweep this technical detail under the rug?} |
24 Maybe the exposition will be better if we sweep this technical detail under the rug?} |
24 |
25 |
44 } |
45 } |
45 |
46 |
46 \nn{need to settle on notation; proof and statement are inconsistent} |
47 \nn{need to settle on notation; proof and statement are inconsistent} |
47 |
48 |
48 \begin{thm} \label{product_thm} |
49 \begin{thm} \label{product_thm} |
49 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by |
50 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from |
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51 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by |
50 \begin{equation*} |
52 \begin{equation*} |
51 C^{\times F}(B) = \cB_*(B \times F, C). |
53 C^{\times F}(B) = \cB_*(B \times F, C). |
52 \end{equation*} |
54 \end{equation*} |
53 Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$: |
55 Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' |
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56 blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' |
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57 (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$: |
54 \begin{align*} |
58 \begin{align*} |
55 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) |
59 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) |
56 \end{align*} |
60 \end{align*} |
57 \end{thm} |
61 \end{thm} |
58 |
62 |
303 the proof of Theorem \ref{product_thm}. |
307 the proof of Theorem \ref{product_thm}. |
304 |
308 |
305 Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
309 Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$. |
306 Recall that this is a homotopy colimit based on decompositions of the interval $J$. |
310 Recall that this is a homotopy colimit based on decompositions of the interval $J$. |
307 |
311 |
308 We define a map $\psi:\cT\to \bc_*(X)$. On filtration degree zero summands it is given |
312 We define a map $\psi:\cT\to \bc_*(X)$. |
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313 On filtration degree zero summands it is given |
309 by gluing the pieces together to get a blob diagram on $X$. |
314 by gluing the pieces together to get a blob diagram on $X$. |
310 On filtration degree 1 and greater $\psi$ is zero. |
315 On filtration degree 1 and greater $\psi$ is zero. |
311 |
316 |
312 The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split |
317 The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split |
313 over some decomposition of $J$. |
318 over some decomposition of $J$. |
351 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
356 Think of $\pi^\infty_{\leq n}(T)$ as encoding everything you would ever |
352 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
357 want to know about spaces of maps of $k$-balls into $T$ ($k\le n$). |
353 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
358 To simplify notation, let $\cT = \pi^\infty_{\leq n}(T)$. |
354 |
359 |
355 \begin{thm} \label{thm:map-recon} |
360 \begin{thm} \label{thm:map-recon} |
356 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 $T$. |
361 The blob complex for $M$ with coefficients in the fundamental $A_\infty$ $n$-category for $T$ |
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362 is quasi-isomorphic to singular chains on maps from $M$ to $T$. |
357 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
363 $$\cB^\cT(M) \simeq C_*(\Maps(M\to T)).$$ |
358 \end{thm} |
364 \end{thm} |
359 \begin{rem} |
365 \begin{rem} |
360 Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in 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 described in Example \ref{ex:e-n-alg} that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which is trivial at all but the topmost level. Ricardo Andrade also told us about a similar result. |
366 Lurie has shown in \cite[Theorem 3.8.6]{0911.0018} that the topological chiral homology |
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367 of an $n$-manifold $M$ with coefficients in a certain $E_n$ algebra constructed from $T$ recovers |
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368 the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. |
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369 This extra hypothesis is not surprising, in view of the idea described in Example \ref{ex:e-n-alg} |
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370 that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which |
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371 is trivial at all but the topmost level. |
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372 Ricardo Andrade also told us about a similar result. |
361 \end{rem} |
373 \end{rem} |
362 |
374 |
363 \nn{proof is again similar to that of Theorem \ref{product_thm}. should probably say that explicitly} |
375 \nn{proof is again similar to that of Theorem \ref{product_thm}. should probably say that explicitly} |
364 |
376 |
365 \begin{proof} |
377 \begin{proof} |