%!TEX root = ../blob1.tex

\section{Commutative algebras as \texorpdfstring{$n$}{n}-categories}
\label{sec:comm_alg}

If $C$ is a commutative algebra it
can also be thought of as an $n$-category whose $j$-morphisms are trivial for
$j<n$ and whose $n$-morphisms are $C$. 
The goal of this appendix 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.


\medskip

Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$.

Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$
unlabeled points in $M$.
Note that $\Sigma^0(M)$ is a point.
Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$.

Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$.

\begin{prop} \label{sympowerprop}
$\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$.
\end{prop}

\begin{proof}
We will use acyclic models (\S \ref{sec:moam}).
Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$
satisfying the conditions of Theorem \ref{moam-thm}.
If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a 
finite unordered collection of points of $M$ with multiplicities, which is
a point in $\Sigma^\infty(M)$.
Define $R(b)_*$ to be the singular chain complex of this point.
If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs).
The path components of $\Sigma^\infty(D)$ are contractible, and these components are indexed 
by the numbers of points in each component of $D$.
We may assume that the blob labels $u$ have homogeneous $t$ degree in $k[t]$, and so
$u$ picks out a component $X \sub \Sigma^\infty(D)$.
The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$,
and using this point we can embed $X$ in $\Sigma^\infty(M)$.
Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a 
subspace of $\Sigma^\infty(M)$.
It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from 
Theorem \ref{moam-thm}.
Thus we have defined (up to homotopy) a map from 
$\bc_*(M, k[t])$ to $C_*(\Sigma^\infty(M))$.

Next we define a map going the other direction.
First we replace $C_*(\Sigma^\infty(M))$ with a homotopy equivalent 
subcomplex $S_*$ of small simplices.
Roughly, we define $c\in C_*(\Sigma^\infty(M))$ to be small if the 
corresponding track of points in $M$
is contained in a disjoint union of balls.
Because there could be different, inequivalent choices of such balls, we must a bit more careful.
\nn{this runs into the same issues as in defining evmap.
either refer there for details, or use the simp-space-ish version of the blob complex,
which makes things easier here.}

\nn{...}


We will define, for each simplex $c$ of $S_*$, a contractible subspace
$R(c)_* \sub \bc_*(M, k[t])$.
If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and 
$\Sigma^\infty(M)$ described above.
Now let $c$ be an $i$-simplex of $S_*$.
Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$.
We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$
is homotopy equivalent to the subcomplex of small simplices.
How small?  $(2r)/3j$, where $r$ is the radius of injectivity of the metric.
Let $T\sub M$ be the ``track" of $c$ in $M$.
\nn{do we need to define this precisely?}
Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter.
\nn{need to say more precisely how small}
Define $R(c)_*$ to be $\bc_*(D; k[t]) \sub \bc_*(M; k[t])$.
This is contractible by Proposition \ref{bcontract}.
We can arrange that the boundary/inclusion condition is satisfied if we start with
low-dimensional simplices and work our way up.
\nn{need to be more precise}

\nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity
(for this, might need a lemma that says we can assume that blob diameters are small)}
\end{proof}


\begin{prop} \label{ktchprop}
The above maps are compatible with the evaluation map actions of $C_*(\Homeo(M))$.
\end{prop}

\begin{proof}
The actions agree in degree 0, and both are compatible with gluing.
(cf. uniqueness statement in Theorem \ref{thm:CH}.)
\nn{if Theorem \ref{thm:CH} is rewritten/rearranged, make sure uniqueness discussion is properly referenced from here}
\end{proof}

\medskip

In view of Theorem \ref{thm:hochschild}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$,
and that the cyclic homology of $k[t]$ is related to the action of rotations
on $C_*(\Sigma^\infty(S^1), k)$.
\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 a resolution 
$k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, 
which has coinvariants $k[t] \xrightarrow{0} k[t]$. 
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.
This defines a deformation retraction from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation).
The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, 
and the holonomy of the $\Delta^{j-1}$ bundle
over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices.

In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and
of course $\Sigma^0(S^1)$ is a point.
Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$
and is zero for $i\ge 2$.
Note that the $j$-grading here matches with the $t$-grading on the algebraic side.

By Proposition \ref{ktchprop}, 
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 $k = \z$, $\Sigma^j(S^1)$ 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 \cite[\S 3.1.7]{MR1600246}.

\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}
$\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$.
\end{prop}

According to \cite[3.2.2]{MR1600246},
\[
	HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] .
\]
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.}

\medskip

Next we consider the case $C$ is the truncated polynomial
algebra $k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$.
Define $\Delta_l \sub \Sigma^\infty(M)$ to be configurations of points in $M$ with $l$ or
more of the points coinciding.

\begin{prop}
$\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$
(relative singular chains with coefficients in $k$).
\end{prop}

\begin{proof}
\nn{...}
\end{proof}

\medskip
\hrule
\medskip

Still to do:
\begin{itemize}
\item compare the topological computation for truncated polynomial algebra with \cite{MR1600246}
\item multivariable truncated polynomial algebras (at least mention them)
\end{itemize}