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 |
1500 \begin{prop} |
1500 \begin{prop} \label{ktcdprop} |
1501 The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
1501 The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
1502 \end{prop} |
1502 \end{prop} |
1503 |
1503 |
1504 \begin{proof} |
1504 \begin{proof} |
1505 The actions agree in degree 0, and both are compatible with gluing. |
1505 The actions agree in degree 0, and both are compatible with gluing. |
1512 and that the cyclic homology of $k[t]$ is related to the action of rotations |
1512 and that the cyclic homology of $k[t]$ is related to the action of rotations |
1513 on $C_*(\Sigma^\infty(S^1), k)$. |
1513 on $C_*(\Sigma^\infty(S^1), k)$. |
1514 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
1514 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
1515 Let us check this directly. |
1515 Let us check this directly. |
1516 |
1516 |
|
1517 According to \nn{Loday, 3.2.2}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$. |
|
1518 \nn{say something about $t$-degree? is this in [Loday]?} |
|
1519 |
1517 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
1520 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
1518 The fixed points of this flow are the equally spaced configurations. |
1521 The fixed points of this flow are the equally spaced configurations. |
1519 This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation.). |
1522 This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
1520 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
1523 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
1521 and the holonomy of the $\Delta^{j-1}$ bundle |
1524 and the holonomy of the $\Delta^{j-1}$ bundle |
1522 over $S^1$ is the cyclic permutation of its $j$ vertices. |
1525 over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices. |
|
1526 |
|
1527 In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
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1528 of course $\Sigma^0(S^1)$ is a point. |
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1529 Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
|
1530 and is zero for $i\ge 2$. |
|
1531 \nn{say something about $t$-degrees also matching up?} |
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1532 |
|
1533 By xxxx and \ref{ktcdprop}, |
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1534 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
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1535 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
|
1536 If $k = \z$, we then have |
1523 |
1537 |
1524 |
1538 |
1525 |
1539 |
1526 |
1540 |
1527 |
1541 |