327 generated by blob diagrams $b$ such that $N_\ep$ is either disjoint from |
327 generated by blob diagrams $b$ such that $N_\ep$ is either disjoint from |
328 or contained in each blob of $b$, and the only labeled point inside $N_\ep$ is $*$. |
328 or contained in each blob of $b$, and the only labeled point inside $N_\ep$ is $*$. |
329 %and the two boundary points of $N_\ep$ are not labeled points of $b$. |
329 %and the two boundary points of $N_\ep$ are not labeled points of $b$. |
330 For a field $y$ on $N_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
330 For a field $y$ on $N_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
331 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
331 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
332 (See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(N_\ep)$. We can think of |
332 (See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(N_\ep)$. |
333 $\sigma_\ep$ as a chain map $K_*^\ep \to K_*^\ep$ given by replacing the restriction $y$ to $N_\ep$ of each field |
333 Let $\sigma_\ep: K_*^\ep \to K_*^\ep$ be the chain map |
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334 given by replacing the restriction $y$ to $N_\ep$ of each field |
334 appearing in an element of $K_*^\ep$ with $s_\ep(y)$. |
335 appearing in an element of $K_*^\ep$ with $s_\ep(y)$. |
335 Note that $\sigma_\ep(x) \in K'_*$. |
336 Note that $\sigma_\ep(x) \in K'_*$. |
336 \begin{figure}[!ht] |
337 \begin{figure}[!ht] |
337 \begin{align*} |
338 \begin{align*} |
338 y & = \mathfig{0.2}{hochschild/y} & |
339 y & = \mathfig{0.2}{hochschild/y} & |
367 |
368 |
368 If $x$ is a cycle in $K_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
369 If $x$ is a cycle in $K_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
369 $x \in K_*^\ep$. |
370 $x \in K_*^\ep$. |
370 (This is true for any chain in $K_*(C\otimes C)$, since chains are sums of |
371 (This is true for any chain in $K_*(C\otimes C)$, since chains are sums of |
371 finitely many blob diagrams.) |
372 finitely many blob diagrams.) |
372 Then $x$ is homologous to $s_\ep(x)$, which is in $K'_*$, so the inclusion map |
373 Then $x$ is homologous to $\sigma_\ep(x)$, which is in $K'_*$, so the inclusion map |
373 $K'_* \sub K_*(C\otimes C)$ is surjective on homology. |
374 $K'_* \sub K_*(C\otimes C)$ is surjective on homology. |
374 If $y \in K_*(C\otimes C)$ and $\bd y = x \in K'_*$, then $y \in K_*^\ep$ for some $\ep$ |
375 If $y \in K_*(C\otimes C)$ and $\bd y = x \in K_*(C\otimes C)$, then $y \in K_*^\ep$ for some $\ep$ |
375 and |
376 and |
376 \eq{ |
377 \eq{ |
377 \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
378 \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
378 } |
379 } |
379 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
380 Since $\sigma_\ep(y) + j_\ep(x) \in K'_*$, it follows that the inclusion map is injective on homology. |
380 This completes the proof that $K'_*$ is quasi-isomorphic to $K_*(C\otimes C)$. |
381 This completes the proof that $K'_*$ is quasi-isomorphic to $K_*(C\otimes C)$. |
381 |
382 |
382 Let $K''_* \sub K'_*$ be the subcomplex of $K'_*$ where $*$ is not contained in any blob. |
383 Let $K''_* \sub K'_*$ be the subcomplex of $K'_*$ where $*$ is not contained in any blob. |
383 We will show that the inclusion $i: K''_* \to K'_*$ is a homotopy equivalence. |
384 We will show that the inclusion $i: K''_* \to K'_*$ is a homotopy equivalence. |
384 |
385 |
385 First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $K''_*$ and $K'_*$, except with |
386 First, a lemma: Let $G''_*$ and $G'_*$ be defined similarly to $K''_*$ and $K'_*$, except with |
386 $S^1$ replaced some (any) neighborhood of $* \in S^1$. |
387 $S^1$ replaced by some neighborhood $N$ of $* \in S^1$. |
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388 ($G''_*$ and $G'_*$ depend on $N$, but that is not reflected in the notation.) |
387 Then $G''_*$ and $G'_*$ are both contractible |
389 Then $G''_*$ and $G'_*$ are both contractible |
388 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
390 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
389 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
391 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
390 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
392 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
391 For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe |
393 For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe |
392 in ``basic properties" section above} away from $*$. |
394 in ``basic properties" section above} away from $*$. |
393 Thus any cycle lies in the image of the normal blob complex of a disjoint union |
395 Thus any cycle lies in the image of the normal blob complex of a disjoint union |
394 of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}). |
396 of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disj-union-contract}). |
395 Actually, we need the further (easy) result that the inclusion |
397 Finally, it is easy to see that the inclusion |
396 $G''_* \to G'_*$ induces an isomorphism on $H_0$. |
398 $G''_* \to G'_*$ induces an isomorphism on $H_0$. |
397 |
399 |
398 Next we construct a degree 1 map (homotopy) $h: K'_* \to K'_*$ such that |
400 Next we construct a degree 1 map (homotopy) $h: K'_* \to K'_*$ such that |
399 for all $x \in K'_*$ we have |
401 for all $x \in K'_*$ we have |
400 \eq{ |
402 \eq{ |
463 \end{align*} |
465 \end{align*} |
464 \caption{The image of $m \tensor a$ in the blob complex.} |
466 \caption{The image of $m \tensor a$ in the blob complex.} |
465 \label{fig:hochschild-1-chains} |
467 \label{fig:hochschild-1-chains} |
466 \end{figure} |
468 \end{figure} |
467 |
469 |
468 In degree 2, we send $m\ot a \ot b$ to the sum of 24 (=6*4) 2-blob diagrams as shown in |
470 In degree 2, we send $m\ot a \ot b$ to the sum of 24 ($=6\cdot4$) 2-blob diagrams as shown in |
469 Figure \ref{fig:hochschild-2-chains}. In Figure \ref{fig:hochschild-2-chains} the 1- and 2-blob diagrams are indicated only by their support. |
471 Figure \ref{fig:hochschild-2-chains}. In Figure \ref{fig:hochschild-2-chains} the 1- and 2-blob diagrams are indicated only by their support. |
470 We leave it to the reader to determine the labels of the 1-blob diagrams. |
472 We leave it to the reader to determine the labels of the 1-blob diagrams. |
471 \begin{figure}[!ht] |
473 \begin{figure}[!ht] |
472 \begin{equation*} |
474 \begin{equation*} |
473 \mathfig{0.6}{hochschild/2-chains-0} |
475 \mathfig{0.6}{hochschild/2-chains-0} |
480 \end{figure} |
482 \end{figure} |
481 Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all |
483 Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all |
482 1-blob diagrams in its boundary. |
484 1-blob diagrams in its boundary. |
483 Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ |
485 Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ |
484 as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. |
486 as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. |
485 Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for one of the 2-cells. |
487 Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for the 2-cell |
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488 labeled $A$ in Figure \ref{fig:hochschild-2-chains}. |
486 Note that the (blob complex) boundary of this sum of 2-blob diagrams is |
489 Note that the (blob complex) boundary of this sum of 2-blob diagrams is |
487 precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. |
490 precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. |
488 (Compare with the proof of \ref{bcontract}.) |
491 (Compare with the proof of \ref{bcontract}.) |
489 |
492 |
490 \begin{figure}[!ht] |
493 \begin{figure}[!ht] |