104 $a$ (or one of its iterated boundaries), filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
104 $a$ (or one of its iterated boundaries), filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
105 filtration degree 2 stuff which kills the homology created by the |
105 filtration degree 2 stuff which kills the homology created by the |
106 filtration degree 1 stuff, and so on. |
106 filtration degree 1 stuff, and so on. |
107 More formally, |
107 More formally, |
108 |
108 |
109 \begin{lemma} |
109 \begin{lemma} \label{lem:d-a-acyclic} |
110 $D(a)$ is acyclic. |
110 $D(a)$ is acyclic. |
111 \end{lemma} |
111 \end{lemma} |
112 |
112 |
113 \begin{proof} |
113 \begin{proof} |
114 We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least} |
114 We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least} |
370 that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which |
370 that an $E_n$ algebra is roughly equivalent data to an $A_\infty$ $n$-category which |
371 is trivial at all but the topmost level. |
371 is trivial at all but the topmost level. |
372 Ricardo Andrade also told us about a similar result. |
372 Ricardo Andrade also told us about a similar result. |
373 \end{rem} |
373 \end{rem} |
374 |
374 |
375 \nn{proof is again similar to that of Theorem \ref{product_thm}. should probably say that explicitly} |
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376 |
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377 \begin{proof} |
375 \begin{proof} |
378 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
376 The proof is again similar to that of Theorem \ref{product_thm}. |
379 We then use Lemma \ref{extension_lemma_c} to show that $g$ induces isomorphisms on homology. |
377 |
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378 We begin by constructing chain map $\psi: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
380 |
379 |
381 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
380 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
382 $j$-fold mapping cylinders, $j \ge 0$. |
381 $j$-fold mapping cylinders, $j \ge 0$. |
383 So, as an abelian group (but not as a chain complex), |
382 So, as an abelian group (but not as a chain complex), |
384 \[ |
383 \[ |
398 \] |
397 \] |
399 where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes |
398 where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes |
400 chains of maps from $b$ to $T$ compatible with $\vphi$. |
399 chains of maps from $b$ to $T$ compatible with $\vphi$. |
401 We can take the product of these chains of maps to get a chains of maps from |
400 We can take the product of these chains of maps to get a chains of maps from |
402 all of $M$ to $K$. |
401 all of $M$ to $K$. |
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402 This defines $\psi$ on $C^0$. |
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403 |
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404 We define $\psi(C^j) = 0$ for $j > 0$. |
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405 It is not hard to see that this defines a chain map from |
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406 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
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407 |
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408 The image of $\psi$ is the subcomplex $G_*\sub C_*(\Maps(M\to T))$ generated by |
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409 families of maps whose support is contained in a disjoint union of balls. |
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410 It follows from Lemma \ref{extension_lemma_c} |
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411 that $C_*(\Maps(M\to T))$ is homotopic to a subcomplex of $G_*$. |
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412 |
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413 We will define a map $\phi:G_*\to \cB^\cT(M)$ via acyclic models. |
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414 Let $a$ be a generator of $G_*$. |
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415 Define $D(a)$ to be the subcomplex of $\cB^\cT(M)$ generated by all |
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416 pairs $(b, \ol{K})$, where $b$ is a generator appearing in an iterated boundary of $a$ |
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417 and $\ol{K}$ is an index of the homotopy colimit $\cB^\cT(M)$. |
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418 (See the proof of Theorem \ref{product_thm} for more details.) |
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419 The same proof as of Lemma \ref{lem:d-a-acyclic} shows that $D(a)$ is acyclic. |
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420 By the usual acyclic models nonsense, there is a (unique up to homotopy) |
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421 map $\phi:G_*\to \cB^\cT(M)$ such that $\phi(a)\in D(a)$. |
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422 Furthermore, we may choose $\phi$ such that for all $a$ |
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423 \[ |
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424 \phi(a) = (a, K) + r |
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425 \] |
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426 where $(a, K) \in C^0$ and $r\in \bigoplus_{j\ge 1} C^j$. |
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427 |
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428 It is now easy to see that $\psi\circ\phi$ is the identity on the nose. |
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429 Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity. |
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430 (See the proof of Theorem \ref{product_thm} for more details.) |
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431 \end{proof} |
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432 |
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433 \noop{ |
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434 % old proof (just start): |
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435 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$. |
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436 We then use Lemma \ref{extension_lemma_c} to show that $g$ induces isomorphisms on homology. |
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437 |
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438 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of |
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439 $j$-fold mapping cylinders, $j \ge 0$. |
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440 So, as an abelian group (but not as a chain complex), |
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441 \[ |
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442 \cB^\cT(M) = \bigoplus_{j\ge 0} C^j, |
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443 \] |
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444 where $C^j$ denotes the new chains introduced by the $j$-fold mapping cylinders. |
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445 |
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446 Recall that $C^0$ is a direct sum of chain complexes with the summands indexed by |
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447 decompositions of $M$ which have their $n{-}1$-skeletons labeled by $n{-}1$-morphisms |
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448 of $\cT$. |
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449 Since $\cT = \pi^\infty_{\leq n}(T)$, this means that the summands are indexed by pairs |
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450 $(K, \vphi)$, where $K$ is a decomposition of $M$ and $\vphi$ is a continuous |
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451 maps from the $n{-}1$-skeleton of $K$ to $T$. |
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452 The summand indexed by $(K, \vphi)$ is |
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453 \[ |
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454 \bigotimes_b D_*(b, \vphi), |
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455 \] |
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456 where $b$ runs through the $n$-cells of $K$ and $D_*(b, \vphi)$ denotes |
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457 chains of maps from $b$ to $T$ compatible with $\vphi$. |
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458 We can take the product of these chains of maps to get a chains of maps from |
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459 all of $M$ to $K$. |
403 This defines $g$ on $C^0$. |
460 This defines $g$ on $C^0$. |
404 |
461 |
405 We define $g(C^j) = 0$ for $j > 0$. |
462 We define $g(C^j) = 0$ for $j > 0$. |
406 It is not hard to see that this defines a chain map from |
463 It is not hard to see that this defines a chain map from |
407 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
464 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$. |
408 |
465 |
409 \nn{...} |
466 \nn{...} |
410 |
467 } |
411 \end{proof} |
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412 |
468 |
413 \nn{maybe should also mention version where we enrich over |
469 \nn{maybe should also mention version where we enrich over |
414 spaces rather than chain complexes;} |
470 spaces rather than chain complexes;} |
415 |
471 |
416 \medskip |
472 \medskip |