382 ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
382 ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
383 Adding $B$ as a blob to $x$ gives a contraction. |
383 Adding $B$ as a blob to $x$ gives a contraction. |
384 \nn{need to say something else in degree zero} |
384 \nn{need to say something else in degree zero} |
385 \end{proof} |
385 \end{proof} |
386 |
386 |
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387 \medskip |
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388 |
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389 For purposes of illustration, we describe an explicit chain map |
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390 $HC_*(M) \to K_*(M)$ |
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391 between the Hochschild complex and the blob complex (with bimodule point) |
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392 for degree $\le 2$. |
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393 This map can be completed to a homotopy equivalence, though we will not prove that here. |
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394 There are of course many such maps; what we describe here is one of the simpler possibilities. |
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395 Describing the extension to higher degrees is straightforward but tedious. |
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396 \nn{but probably we should include the general case in a future version of this paper} |
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397 |
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398 Recall that in low degrees $HC_*(M)$ is |
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399 \[ |
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400 \cdots \stackrel{\bd}{\to} M \otimes C\otimes C \stackrel{\bd}{\to} |
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401 M \otimes C \stackrel{\bd}{\to} M |
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402 \] |
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403 with |
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404 \eqar{ |
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405 \bd(m\otimes a) & = & ma - am \\ |
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406 \bd(m\otimes a \otimes b) & = & ma\otimes b - m\otimes ab + bm \otimes a . |
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407 } |
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408 In degree 0, we send $m\in M$ to the 0-blob diagram in Figure xx0; the base point |
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409 in $S^1$ is labeled by $m$ and there are no other labeled points. |
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410 In degree 1, we send $m\ot a$ to the sum of two 1-blob diagrams |
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411 as shown in Figure xx1. |
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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 |
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413 Figure xx2. |
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414 In Figure xx2 the 1- and 2-blob diagrams are indicated only by their support. |
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415 We leave it to the reader to determine the labels of the 1-blob diagrams. |
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416 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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417 1-blob diagrams in its boundary. |
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418 Such a 2-cell corresponds to a sum of the 2-blob diagrams obtained by adding $V$ |
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419 as an outer (non-twig) blob to each of the 1-blob diagrams in the boundary of the 2-cell. |
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420 Figure xx3 shows this explicitly for one of the 2-cells. |
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421 Note that the (blob complex) boundary of this sum of 2-blob diagrams is |
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422 precisely the sum of the 1-blob diagrams corresponding to the boundary of the 2-cell. |
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423 (Compare with the proof of \ref{bcontract}.) |
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424 |
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425 |
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426 |
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427 \medskip |
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428 \nn{old stuff; delete soon....} |
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429 |
387 We can also describe explicitly a map from the standard Hochschild |
430 We can also describe explicitly a map from the standard Hochschild |
388 complex to the blob complex on the circle. \nn{What properties does this |
431 complex to the blob complex on the circle. \nn{What properties does this |
389 map have?} |
432 map have?} |
390 |
433 |
391 \begin{figure}% |
434 \begin{figure}% |