1432 Note that $\Sigma^0(M)$ is a point. |
1435 Note that $\Sigma^0(M)$ is a point. |
1433 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
1436 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
1434 |
1437 |
1435 Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$. |
1438 Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$. |
1436 |
1439 |
1437 \begin{prop} |
1440 \begin{prop} \label{sympowerprop} |
1438 $\bc_*(M^n, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
1441 $\bc_*(M^n, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
1439 \end{prop} |
1442 \end{prop} |
1440 |
1443 |
1441 \begin{proof} |
1444 \begin{proof} |
1442 To define the chain maps between the two complexes we will use the following lemma: |
1445 To define the chain maps between the two complexes we will use the following lemma: |
1531 \nn{say something about $t$-degrees also matching up?} |
1534 \nn{say something about $t$-degrees also matching up?} |
1532 |
1535 |
1533 By xxxx and \ref{ktcdprop}, |
1536 By xxxx and \ref{ktcdprop}, |
1534 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
1537 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
1535 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
1538 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 |
1539 If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree |
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1540 0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. |
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1541 The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |
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1542 degrees and 0 in odd degrees. |
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1543 This agrees with the calculation in \nn{Loday, 3.1.7}. |
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1544 |
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1545 \medskip |
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1546 |
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1547 Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. |
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1548 Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations |
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1549 of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable. |
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1550 The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers |
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1551 corresponding to the number of points of each color of a configuration. |
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1552 A proof similar to that of \ref{sympowerprop} shows that |
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1553 |
|
1554 \begin{prop} |
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1555 $\bc_*(M^n, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
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1556 \end{prop} |
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1557 |
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1558 According to \nn{Loday, 3.2.2}, |
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1559 \[ |
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1560 HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
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1561 \] |
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1562 Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
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1563 We will content ourselves with the case $k = \z$. |
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1564 One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the same color repel each other and points of different colors do not interact. |
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1565 This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent |
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1566 to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple |
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1567 corresponding to $X$. |
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1568 The homology calculation we desire follows easily from this. |
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1569 |
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1570 \nn{say something about cyclic homology in this case? probably not necessary.} |
1537 |
1571 |
1538 |
1572 |
1539 |
1573 |
1540 |
1574 |
1541 |
1575 |