403 with |
403 with |
404 \eqar{ |
404 \eqar{ |
405 \bd(m\otimes a) & = & ma - am \\ |
405 \bd(m\otimes a) & = & ma - am \\ |
406 \bd(m\otimes a \otimes b) & = & ma\otimes b - m\otimes ab + bm \otimes a . |
406 \bd(m\otimes a \otimes b) & = & ma\otimes b - m\otimes ab + bm \otimes a . |
407 } |
407 } |
408 In degree 0, we send $m\in M$ to the 0-blob diagram in Figure xx0; the base point |
408 In degree 0, we send $m\in M$ to the 0-blob diagram $\mathfig{0.05}{hochschild/0-chains}$; the base point |
409 in $S^1$ is labeled by $m$ and there are no other labeled points. |
409 in $S^1$ is labeled by $m$ and there are no other labeled points. |
410 In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams |
410 In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams |
411 as shown in Figure xx1. |
411 as shown in Figure \ref{fig:hochschild-1-chains}. |
412 In degree 2, we send $m\ot a \ot b$ to the sum of 22 (=4+4+4+4+3+3) 2-blob diagrams as shown in |
412 |
413 Figure xx2. |
413 \begin{figure}[!ht] |
414 In Figure xx2 the 1- and 2-blob diagrams are indicated only by their support. |
414 \begin{equation*} |
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415 \mathfig{0.4}{hochschild/1-chains} |
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416 \end{equation*} |
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417 \begin{align*} |
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418 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} |
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419 \end{align*} |
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420 \caption{The image of $m \tensor a$ in the blob complex.} |
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421 \label{fig:hochschild-1-chains} |
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422 \end{figure} |
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423 |
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424 In degree 2, we send $m\ot a \ot b$ to the sum of 24 (=6*4) 2-blob diagrams as shown in |
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425 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. |
415 We leave it to the reader to determine the labels of the 1-blob diagrams. |
426 We leave it to the reader to determine the labels of the 1-blob diagrams. |
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427 \begin{figure}[!ht] |
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428 \begin{equation*} |
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429 \mathfig{0.6}{hochschild/2-chains-0} |
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430 \end{equation*} |
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431 \begin{equation*} |
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432 \mathfig{0.4}{hochschild/2-chains-1} \qquad \mathfig{0.4}{hochschild/2-chains-2} |
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433 \end{equation*} |
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434 \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.} |
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435 \label{fig:hochschild-2-chains} |
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436 \end{figure} |
416 Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all |
437 Each 2-cell in the figure is labeled by a ball $V$ in $S^1$ which contains the support of all |
417 1-blob diagrams in its boundary. |
438 1-blob diagrams in its boundary. |
418 Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ |
439 Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ |
419 as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. |
440 as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. |
420 Figure xx3 shows this explicitly for one of the 2-cells. |
441 Figure \ref{fig:hochschild-example-2-cell} shows this explicitly for one of the 2-cells. |
421 Note that the (blob complex) boundary of this sum of 2-blob diagrams is |
442 Note that the (blob complex) boundary of this sum of 2-blob diagrams is |
422 precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. |
443 precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. |
423 (Compare with the proof of \ref{bcontract}.) |
444 (Compare with the proof of \ref{bcontract}.) |
424 |
445 |
425 |
446 \begin{figure}[!ht] |
426 |
447 \begin{equation*} |
427 \medskip |
448 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} |
428 \nn{old stuff; delete soon....} |
449 \end{equation*} |
429 |
450 \begin{align*} |
430 We can also describe explicitly a map from the standard Hochschild |
451 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} \\ |
431 complex to the blob complex on the circle. \nn{What properties does this |
452 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} |
432 map have?} |
453 \end{align*} |
433 |
454 \caption{One of the 2-cells from Figure \ref{fig:hochschild-2-chains}.} |
434 \begin{figure}% |
455 \label{fig:hochschild-example-2-cell} |
435 $$\mathfig{0.6}{barycentric/barycentric}$$ |
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436 \caption{The Hochschild chain $a \tensor b \tensor c$ is sent to |
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437 the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.} |
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438 \label{fig:Hochschild-example}% |
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439 \end{figure} |
456 \end{figure} |
440 |
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441 As an example, Figure \ref{fig:Hochschild-example} shows the image of the Hochschild chain $a \tensor b \tensor c$. Only the $0$-cells are shown explicitly. |
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442 The edges marked $x, y$ and $z$ carry the $1$-chains |
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443 \begin{align*} |
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444 x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\ |
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445 y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\ |
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446 z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab} |
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447 \end{align*} |
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448 and the $2$-chain labelled $A$ is |
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449 \begin{equation*} |
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450 A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}. |
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451 \end{equation*} |
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452 Note that we then have |
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453 \begin{equation*} |
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454 \bdy A = x+y+z. |
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455 \end{equation*} |
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456 |
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457 In general, the Hochschild chain $\Tensor_{i=1}^n a_i$ is sent to the sum of $n!$ blob $(n-1)$-chains, indexed by permutations, |
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458 $$\phi\left(\Tensor_{i=1}^n a_i\right) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$ |
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459 with ... (hmmm, problems making this precise; you need to decide where to put the labels, but then it's hard to make an honest chain map!) |
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