%!TEX root = ../blob1.tex

\section{Commutative algebras as $n$-categories}

\label{sec:comm_alg}

\nn{this should probably not be a section by itself.  i'm just trying to write down the outline 

while it's still fresh in my mind.}

\nn{I strongly suspect that [blob complex

for $M^n$ based on comm alg $C$ thought of as an $n$-category]

is homotopy equivalent to [higher Hochschild complex for $M^n$ with coefficients in $C$].

(Thomas Tradler's idea.)

Should prove (or at least conjecture) that here.}

If $C$ is a commutative algebra it

can (and will) 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 \nn{subsection?} is to compute

$\bc_*(M^n, C)$ for various commutative algebras $C$.

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}

To define the chain maps between the two complexes we will use the following lemma:

\begin{lemma}

Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with

a basis (e.g.\ blob diagrams or singular simplices).

For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$

such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$.

Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that

$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty).

\end{lemma}

\begin{proof}

\nn{easy, but should probably write the details eventually}

\end{proof}

Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$

satisfying the conditions of the above lemma.

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 the above lemma.

Thus we have defined (up to homotopy) a map from 

$\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$.

Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace

$R(c)_* \sub \bc_*(M^n, k[t])$.

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 $\Sigma^j(M)$.

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 $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$.

This is contractible by \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{ktcdprop}

The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$.

\end{prop}

\begin{proof}

The actions agree in degree 0, and both are compatible with gluing.

(cf. uniqueness statement in \ref{CDprop}.)

\end{proof}

\medskip

In view of \ref{hochthm}, 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.

According to \nn{Loday, 3.2.2}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$.

\nn{say something about $t$-degree?  is this in [Loday]?}

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 map 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 $i\ge 2$.

\nn{say something about $t$-degrees also matching up?}

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 $k = \z$, $\Sigma^j(S^1)$ 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}

$\bc_*(M, k[t_1, \ldots, t_m])$ 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.}

\medskip

Next we consider the case $C = k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$.

Define $\Delta_l \sub \Sigma^\infty(M)$ to be those configurations of points with $l$ or

more 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}

\nn{...}