
diff -r dd9487823529 -r 195a0a91e062 blob1.tex
--- a/blob1.tex	Mon Oct 27 14:32:00 2008 +0000
+++ b/blob1.tex	Tue Oct 28 01:19:24 2008 +0000
@@ -1414,6 +1414,9 @@
 \nn{need to define/recall def of (self) tensor product over an $A_\infty$ category}
 
 
+
+
+
 \section{Commutative algebras as $n$-categories}
 
 \nn{this should probably not be a section by itself.  i'm just trying to write down the outline 
@@ -1434,7 +1437,7 @@
 
 Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$.
 
-\begin{prop}
+\begin{prop} \label{sympowerprop}
 $\bc_*(M^n, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$.
 \end{prop}
 
@@ -1533,7 +1536,38 @@
 By xxxx and \ref{ktcdprop}, 
 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$.
 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$.
-If $k = \z$, we then have 
+If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree
+0, $\z/j \z$ in odd degrees, and 0 in positive even degrees.
+The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even 
+degrees and 0 in odd degrees.
+This agrees with the calculation in \nn{Loday, 3.1.7}.
+
+\medskip
+
+Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables.
+Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations
+of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable.
+The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers
+corresponding to the number of points of each color of a configuration.
+A proof similar to that of \ref{sympowerprop} shows that
+
+\begin{prop}
+$\bc_*(M^n, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$.
+\end{prop}
+
+According to \nn{Loday, 3.2.2},
+\[
+	HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] .
+\]
+Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$.
+We will content ourselves with the case $k = \z$.
+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.
+This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent
+to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple
+corresponding to $X$.
+The homology calculation we desire follows easily from this.
+
+\nn{say something about cyclic homology in this case?  probably not necessary.}
 
 
 




