author | Kevin Walker <kevin@canyon23.net> |
Thu, 22 Jul 2010 16:16:58 -0600 | |
changeset 476 | 4d0ca2fc4f2b |
parent 470 | ec8e9c920a34 |
child 501 | fdb012a1c8fe |
permissions | -rw-r--r-- |
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%!TEX root = ../blob1.tex |
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\section{Commutative algebras as $n$-categories} |
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\label{sec:comm_alg} |
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If $C$ is a commutative algebra it |
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can also be thought of as an $n$-category whose $j$-morphisms are trivial for |
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$j<n$ and whose $n$-morphisms are $C$. |
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The goal of this appendix is to compute |
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$\bc_*(M^n, C)$ for various commutative algebras $C$. |
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Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative |
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algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with |
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coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). |
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This possibility was suggested to us by Thomas Tradler. |
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\medskip |
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Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
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Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
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unlabeled points in $M$. |
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Note that $\Sigma^0(M)$ is a point. |
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Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
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Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$. |
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\begin{prop} \label{sympowerprop} |
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$\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
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\end{prop} |
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\begin{proof} |
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%To define the chain maps between the two complexes we will use the following lemma: |
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% |
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%\begin{lemma} |
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%Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
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%a basis (e.g.\ blob diagrams or singular simplices). |
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%For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
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%such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$. |
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%Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
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%$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
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%\end{lemma} |
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% |
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%\begin{proof} |
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%\nn{easy, but should probably write the details eventually} |
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%\nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that} |
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%\end{proof} |
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We will use acyclic models \nn{need ref}. |
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Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
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satisfying the conditions of \nn{need ref}. |
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If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
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finite unordered collection of points of $M$ with multiplicities, which is |
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a point in $\Sigma^\infty(M)$. |
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Define $R(b)_*$ to be the singular chain complex of this point. |
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If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). |
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The path components of $\Sigma^\infty(D)$ are contractible, and these components are indexed |
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by the numbers of points in each component of $D$. |
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We may assume that the blob labels $u$ have homogeneous $t$ degree in $k[t]$, and so |
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$u$ picks out a component $X \sub \Sigma^\infty(D)$. |
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The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, |
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and using this point we can embed $X$ in $\Sigma^\infty(M)$. |
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Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a |
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subspace of $\Sigma^\infty(M)$. |
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It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from |
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\nn{need ref, or state condition}. |
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Thus we have defined (up to homotopy) a map from |
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$\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$. |
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Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace |
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$R(c)_* \sub \bc_*(M^n, k[t])$. |
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If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and |
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$\Sigma^\infty(M)$ described above. |
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Now let $c$ be an $i$-simplex of $\Sigma^j(M)$. |
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Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. |
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We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ |
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is homotopy equivalent to the subcomplex of small simplices. |
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How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. |
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Let $T\sub M$ be the ``track" of $c$ in $M$. |
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\nn{do we need to define this precisely?} |
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Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. |
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\nn{need to say more precisely how small} |
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Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$. |
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This is contractible by Proposition \ref{bcontract}. |
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We can arrange that the boundary/inclusion condition is satisfied if we start with |
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low-dimensional simplices and work our way up. |
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\nn{need to be more precise} |
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\nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
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(for this, might need a lemma that says we can assume that blob diameters are small)} |
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\end{proof} |
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\begin{prop} \label{ktchprop} |
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The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
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\end{prop} |
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\begin{proof} |
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The actions agree in degree 0, and both are compatible with gluing. |
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(cf. uniqueness statement in Theorem \ref{thm:CH}.) |
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\end{proof} |
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\medskip |
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In view of Theorem \ref{thm:hochschild}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
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and that the cyclic homology of $k[t]$ is related to the action of rotations |
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on $C_*(\Sigma^\infty(S^1), k)$. |
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\nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
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Let us check this directly. |
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342 | 111 |
The algebra $k[t]$ has Koszul resolution |
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$k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, |
|
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which has coinvariants $k[t] \xrightarrow{0} k[t]$. |
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This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. |
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(See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: |
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$HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one. |
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We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
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The fixed points of this flow are the equally spaced configurations. |
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This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
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The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
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and the holonomy of the $\Delta^{j-1}$ bundle |
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over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices. |
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In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
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of course $\Sigma^0(S^1)$ is a point. |
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Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
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and is zero for $i\ge 2$. |
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Note that the $j$-grading here matches with the $t$-grading on the algebraic side. |
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By xxxx and Proposition \ref{ktchprop}, |
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the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
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Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
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If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree |
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0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. |
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The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |
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degrees and 0 in odd degrees. |
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This agrees with the calculation in \cite[3.1.7]{MR1600246}. |
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\medskip |
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Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. |
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Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations |
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of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable. |
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The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers |
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corresponding to the number of points of each color of a configuration. |
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A proof similar to that of \ref{sympowerprop} shows that |
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\begin{prop} |
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$\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
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\end{prop} |
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166 | 153 |
According to \cite[3.2.2]{MR1600246}, |
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\[ |
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HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
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\] |
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Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
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We will content ourselves with the case $k = \z$. |
342 | 159 |
One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the |
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same color repel each other and points of different colors do not interact. |
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This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent |
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to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple |
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corresponding to $X$. |
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The homology calculation we desire follows easily from this. |
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\nn{say something about cyclic homology in this case? probably not necessary.} |
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\medskip |
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163 | 170 |
Next we consider the case $C$ is the truncated polynomial |
171 |
algebra $k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. |
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Define $\Delta_l \sub \Sigma^\infty(M)$ to be configurations of points in $M$ with $l$ or |
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more of the points coinciding. |
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\begin{prop} |
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$\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ |
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(relative singular chains with coefficients in $k$). |
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\end{prop} |
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\begin{proof} |
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\nn{...} |
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\end{proof} |
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163 | 184 |
\medskip |
185 |
\hrule |
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\medskip |
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163 | 188 |
Still to do: |
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\begin{itemize} |
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166 | 190 |
\item compare the topological computation for truncated polynomial algebra with \cite{MR1600246} |
163 | 191 |
\item multivariable truncated polynomial algebras (at least mention them) |
192 |
\item ideally, say something more about higher hochschild homology (maybe sketch idea for proof of equivalence) |
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431 | 193 |
\item say something about SMCs as $n$-categories, e.g. Vect and K-theory. |
163 | 194 |
\end{itemize} |
195 |