equal
deleted
inserted
replaced
1436 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
1436 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
1437 |
1437 |
1438 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 $k$. |
1439 |
1439 |
1440 \begin{prop} \label{sympowerprop} |
1440 \begin{prop} \label{sympowerprop} |
1441 $\bc_*(M^n, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
1441 $\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
1442 \end{prop} |
1442 \end{prop} |
1443 |
1443 |
1444 \begin{proof} |
1444 \begin{proof} |
1445 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: |
1446 |
1446 |
1550 The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers |
1550 The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers |
1551 corresponding to the number of points of each color of a configuration. |
1551 corresponding to the number of points of each color of a configuration. |
1552 A proof similar to that of \ref{sympowerprop} shows that |
1552 A proof similar to that of \ref{sympowerprop} shows that |
1553 |
1553 |
1554 \begin{prop} |
1554 \begin{prop} |
1555 $\bc_*(M^n, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
1555 $\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
1556 \end{prop} |
1556 \end{prop} |
1557 |
1557 |
1558 According to \nn{Loday, 3.2.2}, |
1558 According to \nn{Loday, 3.2.2}, |
1559 \[ |
1559 \[ |
1560 HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
1560 HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
1567 corresponding to $X$. |
1567 corresponding to $X$. |
1568 The homology calculation we desire follows easily from this. |
1568 The homology calculation we desire follows easily from this. |
1569 |
1569 |
1570 \nn{say something about cyclic homology in this case? probably not necessary.} |
1570 \nn{say something about cyclic homology in this case? probably not necessary.} |
1571 |
1571 |
1572 |
1572 \medskip |
1573 |
1573 |
1574 |
1574 Next we consider the case $C = k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. |
|
1575 Define $\Delta_l \sub \Sigma^\infty(M)$ to be those configurations of points with $l$ or |
|
1576 more points coinciding. |
|
1577 |
|
1578 \begin{prop} |
|
1579 $\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ |
|
1580 (relative singular chains with coefficients in $k$). |
|
1581 \end{prop} |
|
1582 |
|
1583 \begin{proof} |
|
1584 \nn{...} |
|
1585 \end{proof} |
1575 |
1586 |
1576 \nn{...} |
1587 \nn{...} |
1577 |
1588 |
1578 |
1589 |
1579 |
1590 |