465 \end{align*} |
465 \end{align*} |
466 \caption{The image of $m \tensor a$ in the blob complex.} |
466 \caption{The image of $m \tensor a$ in the blob complex.} |
467 \label{fig:hochschild-1-chains} |
467 \label{fig:hochschild-1-chains} |
468 \end{figure} |
468 \end{figure} |
469 |
469 |
470 In degree 2, we send $m\ot a \ot b$ to the sum of 24 ($=6\cdot4$) 2-blob diagrams as shown in |
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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. |
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472 We leave it to the reader to determine the labels of the 1-blob diagrams. |
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473 \begin{figure}[!ht] |
470 \begin{figure}[!ht] |
474 \begin{equation*} |
471 \begin{equation*} |
475 \mathfig{0.6}{hochschild/2-chains-0} |
472 \mathfig{0.6}{hochschild/2-chains-0} |
476 \end{equation*} |
473 \end{equation*} |
477 \begin{equation*} |
474 \begin{equation*} |
478 \mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2} |
475 \mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2} |
479 \end{equation*} |
476 \end{equation*} |
480 \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.} |
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.} |
481 \label{fig:hochschild-2-chains} |
478 \label{fig:hochschild-2-chains} |
482 \end{figure} |
479 \end{figure} |
483 Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all |
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484 1-blob diagrams in its boundary. |
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485 Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ |
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486 as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. |
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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}. |
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489 Note that the (blob complex) boundary of this sum of 2-blob diagrams is |
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490 precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. |
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491 (Compare with the proof of \ref{bcontract}.) |
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492 |
480 |
493 \begin{figure}[!ht] |
481 \begin{figure}[!ht] |
494 \begin{equation*} |
482 \begin{equation*} |
495 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} |
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} |
496 \end{equation*} |
484 \end{equation*} |
499 v_3 & = \mathfig{0.05}{hochschild/v_3-1} - \mathfig{0.05}{hochschild/v_3-2} & v_4 & = \mathfig{0.05}{hochschild/v_4-1} - \mathfig{0.05}{hochschild/v_4-2} |
487 v_3 & = \mathfig{0.05}{hochschild/v_3-1} - \mathfig{0.05}{hochschild/v_3-2} & v_4 & = \mathfig{0.05}{hochschild/v_4-1} - \mathfig{0.05}{hochschild/v_4-2} |
500 \end{align*} |
488 \end{align*} |
501 \caption{One of the 2-cells from Figure \ref{fig:hochschild-2-chains}.} |
489 \caption{One of the 2-cells from Figure \ref{fig:hochschild-2-chains}.} |
502 \label{fig:hochschild-example-2-cell} |
490 \label{fig:hochschild-example-2-cell} |
503 \end{figure} |
491 \end{figure} |
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492 |
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493 In degree 2, we send $m\ot a \ot b$ to the sum of 24 ($=6\cdot4$) 2-blob diagrams as shown in |
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494 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. |
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495 We leave it to the reader to determine the labels of the 1-blob diagrams. |
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496 Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all |
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497 1-blob diagrams in its boundary. |
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498 Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ |
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499 as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. |
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500 Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for the 2-cell |
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501 labeled $A$ in Figure \ref{fig:hochschild-2-chains}. |
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502 Note that the (blob complex) boundary of this sum of 2-blob diagrams is |
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503 precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. |
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504 (Compare with the proof of \ref{bcontract}.) |