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inserted
replaced
1430 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
1430 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
1431 unlabeled points in $M$. |
1431 unlabeled points in $M$. |
1432 Note that $\Sigma^0(M)$ is a point. |
1432 Note that $\Sigma^0(M)$ is a point. |
1433 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
1433 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
1434 |
1434 |
1435 Let $C_*(X)$ denote the singular chain complex of the space $X$. |
1435 Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$. |
1436 |
1436 |
1437 \begin{prop} |
1437 \begin{prop} |
1438 $\bc_*(M^n, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M))$. |
1438 $\bc_*(M^n, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
1439 \end{prop} |
1439 \end{prop} |
1440 |
1440 |
1441 \begin{proof} |
1441 \begin{proof} |
1442 To define the chain maps between the two complexes we will use the following lemma: |
1442 To define the chain maps between the two complexes we will use the following lemma: |
1443 |
1443 |
1495 \nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
1495 \nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
1496 (for this, might need a lemma that says we can assume that blob diameters are small)} |
1496 (for this, might need a lemma that says we can assume that blob diameters are small)} |
1497 \end{proof} |
1497 \end{proof} |
1498 |
1498 |
1499 |
1499 |
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1500 \begin{prop} |
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1501 The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
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1502 \end{prop} |
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1503 |
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1504 \begin{proof} |
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1505 The actions agree in degree 0, and both are compatible with gluing. |
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1506 (cf. uniqueness statement in \ref{CDprop}.) |
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1507 \end{proof} |
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1508 |
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1509 \medskip |
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1510 |
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1511 In view of \ref{hochthm}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
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1512 and that the cyclic homology of $k[t]$ is related to the action of rotations |
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1513 on $C_*(\Sigma^\infty(S^1), k)$. |
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1514 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
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1515 Let us check this directly. |
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1516 |
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1517 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
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1518 The fixed points of this flow are the equally spaced configurations. |
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1519 This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation.). |
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1520 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
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1521 and the holonomy of the $\Delta^{j-1}$ bundle |
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1522 over $S^1$ is the cyclic permutation of its $j$ vertices. |
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1523 |
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1524 |
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1525 |
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1526 |
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1527 |
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1528 \nn{...} |
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1529 |
1500 |
1530 |
1501 |
1531 |
1502 |
1532 |
1503 \appendix |
1533 \appendix |
1504 |
1534 |