--- a/text/comm_alg.tex Fri Jun 04 17:00:18 2010 -0700
+++ b/text/comm_alg.tex Fri Jun 04 17:15:53 2010 -0700
@@ -13,7 +13,10 @@
The goal of this \nn{subsection?} is to compute
$\bc_*(M^n, C)$ for various commutative algebras $C$.
-Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). This possibility was suggested to us by Thomas Tradler.
+Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative
+algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with
+coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}).
+This possibility was suggested to us by Thomas Tradler.
\medskip
@@ -108,8 +111,12 @@
\nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section}
Let us check this directly.
-The algebra $k[t]$ has Koszul resolution $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, which has coinvariants $k[t] \xrightarrow{0} k[t]$. This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$.
-(See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one.
+The algebra $k[t]$ has Koszul resolution
+$k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$,
+which has coinvariants $k[t] \xrightarrow{0} k[t]$.
+This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$.
+(See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings:
+$HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one.
We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other.
The fixed points of this flow are the equally spaced configurations.
@@ -152,7 +159,8 @@
\]
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.
+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 $X$.