308 & = \sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k \\ |
308 & = \sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k \\ |
309 & = 0 |
309 & = 0 |
310 \end{align*} |
310 \end{align*} |
311 where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$. |
311 where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$. |
312 |
312 |
313 The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly surjective ($\ev$ surjects onto $M$); we now show that it's injective. \todo{} |
313 The map $\pi \compose \ev: H_0(K_*(M)) \to \coinv{M}$ is clearly surjective ($\ev$ surjects onto $M$); we now show that it's injective. |
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314 This is equivalent to showing that |
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315 \[ |
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316 \ev\inv(\ker(\pi)) \sub \bd K_1(M) . |
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317 \] |
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318 The above inclusion follows from |
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319 \[ |
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320 \ker(\ev) \sub \bd K_1(M) |
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321 \] |
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322 and |
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323 \[ |
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324 \ker(\pi) \sub \ev(\bd K_1(M)) . |
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325 \] |
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326 Let $x = \sum x_i$ be in the kernel of $\ev$, where each $x_i$ is a configuration of |
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327 labeled points in $S^1$. |
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328 Since the sum is finite, we can find an interval (blob) $B$ in $S^1$ |
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329 such that for each $i$ the $C$-labeled points of $x_i$ all lie to the right of the |
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330 base point *. |
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331 Let $y_i$ be the restriction of $x_i$ to $B$ and $y = \sum y_i$. |
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332 Let $r$ be the ``empty" field on $S^1 \setmin B$. |
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333 It follows that $y \in U(B)$ and |
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334 \[ |
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335 \bd(B, y, r) = x . |
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336 \] |
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337 $\ker(\pi)$ is generated by elements of the form $cm - mc$. |
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338 As shown in Figure \ref{fig:hochschild-1-chains}, $cm - mc$ lies in $\ev(\bd K_1(M))$. |
314 \end{proof} |
339 \end{proof} |
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340 |
315 \begin{proof}[Proof of Lemma \ref{lem:hochschild-free}] |
341 \begin{proof}[Proof of Lemma \ref{lem:hochschild-free}] |
316 We show that $K_*(C\otimes C)$ is |
342 We show that $K_*(C\otimes C)$ is |
317 quasi-isomorphic to the 0-step complex $C$. We'll do this in steps, establishing quasi-isomorphisms and homotopy equivalences |
343 quasi-isomorphic to the 0-step complex $C$. We'll do this in steps, establishing quasi-isomorphisms and homotopy equivalences |
318 $$K_*(C \tensor C) \quismto K'_* \htpyto K''_* \quismto C.$$ |
344 $$K_*(C \tensor C) \quismto K'_* \htpyto K''_* \quismto C.$$ |
319 |
345 |
332 (See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(N_\ep)$. |
358 (See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(N_\ep)$. |
333 Let $\sigma_\ep: K_*^\ep \to K_*^\ep$ be the chain map |
359 Let $\sigma_\ep: K_*^\ep \to K_*^\ep$ be the chain map |
334 given by replacing the restriction $y$ to $N_\ep$ of each field |
360 given by replacing the restriction $y$ to $N_\ep$ of each field |
335 appearing in an element of $K_*^\ep$ with $s_\ep(y)$. |
361 appearing in an element of $K_*^\ep$ with $s_\ep(y)$. |
336 Note that $\sigma_\ep(x) \in K'_*$. |
362 Note that $\sigma_\ep(x) \in K'_*$. |
337 \begin{figure}[!ht] |
363 \begin{figure}[t] |
338 \begin{align*} |
364 \begin{align*} |
339 y & = \mathfig{0.2}{hochschild/y} & |
365 y & = \mathfig{0.2}{hochschild/y} & |
340 s_\ep(y) & = \mathfig{0.2}{hochschild/sy} |
366 s_\ep(y) & = \mathfig{0.2}{hochschild/sy} |
341 \end{align*} |
367 \end{align*} |
342 \caption{Defining $s_\ep$.} |
368 \caption{Defining $s_\ep$.} |
411 Define $h_1(x) = y$. |
437 Define $h_1(x) = y$. |
412 The general case is similar, except that we have to take lower order homotopies into account. |
438 The general case is similar, except that we have to take lower order homotopies into account. |
413 Let $x \in K'_k$. |
439 Let $x \in K'_k$. |
414 If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$. |
440 If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$. |
415 Otherwise, let $B$ be the outermost blob of $x$ containing $*$. |
441 Otherwise, let $B$ be the outermost blob of $x$ containing $*$. |
416 By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$. |
442 We can decompose $x = x' \bullet p$, |
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443 where $x'$ is supported on $B$ and $p$ is supported away from $B$. |
417 So $x' \in G'_l$ for some $l \le k$. |
444 So $x' \in G'_l$ for some $l \le k$. |
418 Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
445 Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
419 Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
446 Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
420 Define $h_k(x) = y \bullet p$. |
447 Define $h_k(x) = y \bullet p$. |
421 This completes the proof that $i: K''_* \to K'_*$ is a homotopy equivalence. |
448 This completes the proof that $i: K''_* \to K'_*$ is a homotopy equivalence. |
454 In degree 0, we send $m\in M$ to the 0-blob diagram $\mathfig{0.05}{hochschild/0-chains}$; the base point |
481 In degree 0, we send $m\in M$ to the 0-blob diagram $\mathfig{0.05}{hochschild/0-chains}$; the base point |
455 in $S^1$ is labeled by $m$ and there are no other labeled points. |
482 in $S^1$ is labeled by $m$ and there are no other labeled points. |
456 In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams |
483 In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams |
457 as shown in Figure \ref{fig:hochschild-1-chains}. |
484 as shown in Figure \ref{fig:hochschild-1-chains}. |
458 |
485 |
459 \begin{figure}[!ht] |
486 \begin{figure}[t] |
460 \begin{equation*} |
487 \begin{equation*} |
461 \mathfig{0.4}{hochschild/1-chains} |
488 \mathfig{0.4}{hochschild/1-chains} |
462 \end{equation*} |
489 \end{equation*} |
463 \begin{align*} |
490 \begin{align*} |
464 u_1 & = \mathfig{0.05}{hochschild/u_1-1} - \mathfig{0.05}{hochschild/u_1-2} & u_2 & = \mathfig{0.05}{hochschild/u_2-1} - \mathfig{0.05}{hochschild/u_2-2} |
491 u_1 & = \mathfig{0.05}{hochschild/u_1-1} - \mathfig{0.05}{hochschild/u_1-2} & u_2 & = \mathfig{0.05}{hochschild/u_2-1} - \mathfig{0.05}{hochschild/u_2-2} |
465 \end{align*} |
492 \end{align*} |
466 \caption{The image of $m \tensor a$ in the blob complex.} |
493 \caption{The image of $m \tensor a$ in the blob complex.} |
467 \label{fig:hochschild-1-chains} |
494 \label{fig:hochschild-1-chains} |
468 \end{figure} |
495 \end{figure} |
469 |
496 |
470 \begin{figure}[!ht] |
497 \begin{figure}[t] |
471 \begin{equation*} |
498 \begin{equation*} |
472 \mathfig{0.6}{hochschild/2-chains-0} |
499 \mathfig{0.6}{hochschild/2-chains-0} |
473 \end{equation*} |
500 \end{equation*} |
474 \begin{equation*} |
501 \begin{equation*} |
475 \mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2} |
502 \mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2} |
476 \end{equation*} |
503 \end{equation*} |
477 \caption{The 0-, 1- and 2-chains in the image of $m \tensor a \tensor b$. Only the supports of the 1- and 2-blobs are shown.} |
504 \caption{The 0-, 1- and 2-chains in the image of $m \tensor a \tensor b$. Only the supports of the 1- and 2-blobs are shown.} |
478 \label{fig:hochschild-2-chains} |
505 \label{fig:hochschild-2-chains} |
479 \end{figure} |
506 \end{figure} |
480 |
507 |
481 \begin{figure}[!ht] |
508 \begin{figure}[t] |
482 \begin{equation*} |
509 \begin{equation*} |
483 A = \mathfig{0.1}{hochschild/v_1} + \mathfig{0.1}{hochschild/v_2} + \mathfig{0.1}{hochschild/v_3} + \mathfig{0.1}{hochschild/v_4} |
510 A = \mathfig{0.1}{hochschild/v_1} + \mathfig{0.1}{hochschild/v_2} + \mathfig{0.1}{hochschild/v_3} + \mathfig{0.1}{hochschild/v_4} |
484 \end{equation*} |
511 \end{equation*} |
485 \begin{align*} |
512 \begin{align*} |
486 v_1 & = \mathfig{0.05}{hochschild/v_1-1} - \mathfig{0.05}{hochschild/v_1-2} & v_2 & = \mathfig{0.05}{hochschild/v_2-1} - \mathfig{0.05}{hochschild/v_2-2} \\ |
513 v_1 & = \mathfig{0.05}{hochschild/v_1-1} - \mathfig{0.05}{hochschild/v_1-2} & v_2 & = \mathfig{0.05}{hochschild/v_2-1} - \mathfig{0.05}{hochschild/v_2-2} \\ |