author | kevin@6e1638ff-ae45-0410-89bd-df963105f760 |
Sun, 08 Nov 2009 04:08:58 +0000 | |
changeset 182 | 8bff31138721 |
parent 166 | 75f5c197a0d4 |
child 236 | 3feb6e24a518 |
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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\nn{this should probably not be a section by itself. i'm just trying to write down the outline |
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while it's still fresh in my mind.} |
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\nn{I strongly suspect that [blob complex |
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for $M^n$ based on comm alg $C$ thought of as an $n$-category] |
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is homotopy equivalent to [higher Hochschild complex for $M^n$ with coefficients in $C$]. |
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(Thomas Tradler's idea.) |
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Should prove (or at least conjecture) that here.} |
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\nn{also, this section needs a little updating to be compatible with the rest of the paper.} |
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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 \nn{subsection?} is to compute |
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$\bc_*(M^n, C)$ for various commutative algebras $C$. |
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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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\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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\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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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 the above lemma. |
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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 the above lemma. |
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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 \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{ktcdprop} |
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The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
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100 |
\end{prop} |
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101 |
|
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102 |
\begin{proof} |
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103 |
The actions agree in degree 0, and both are compatible with gluing. |
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104 |
(cf. uniqueness statement in \ref{CDprop}.) |
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105 |
\end{proof} |
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106 |
|
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107 |
\medskip |
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108 |
|
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109 |
In view of \ref{hochthm}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
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110 |
and that the cyclic homology of $k[t]$ is related to the action of rotations |
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111 |
on $C_*(\Sigma^\infty(S^1), k)$. |
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112 |
\nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
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113 |
Let us check this directly. |
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114 |
|
166 | 115 |
According to \cite[3.2.2]{MR1600246}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$. |
116 |
\nn{say something about $t$-degree? is this in Loday?} |
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100
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117 |
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118 |
We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
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119 |
The fixed points of this flow are the equally spaced configurations. |
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120 |
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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121 |
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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124 |
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In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
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126 |
of course $\Sigma^0(S^1)$ is a point. |
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127 |
Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
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128 |
and is zero for $i\ge 2$. |
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\nn{say something about $t$-degrees also matching up?} |
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130 |
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By xxxx and \ref{ktcdprop}, |
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132 |
the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
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133 |
Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
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134 |
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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136 |
The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |
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137 |
degrees and 0 in odd degrees. |
166 | 138 |
This agrees with the calculation in \cite[3.1.7]{MR1600246}. |
100
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139 |
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140 |
\medskip |
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141 |
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142 |
Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. |
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143 |
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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145 |
The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers |
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146 |
corresponding to the number of points of each color of a configuration. |
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147 |
A proof similar to that of \ref{sympowerprop} shows that |
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148 |
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149 |
\begin{prop} |
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150 |
$\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
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151 |
\end{prop} |
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152 |
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166 | 153 |
According to \cite[3.2.2]{MR1600246}, |
100
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154 |
\[ |
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HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
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156 |
\] |
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157 |
Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
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158 |
We will content ourselves with the case $k = \z$. |
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159 |
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. |
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160 |
This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent |
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161 |
to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple |
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162 |
corresponding to $X$. |
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163 |
The homology calculation we desire follows easily from this. |
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164 |
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165 |
\nn{say something about cyclic homology in this case? probably not necessary.} |
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166 |
|
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167 |
\medskip |
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168 |
|
163 | 169 |
Next we consider the case $C$ is the truncated polynomial |
170 |
algebra $k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. |
|
171 |
Define $\Delta_l \sub \Sigma^\infty(M)$ to be configurations of points in $M$ with $l$ or |
|
172 |
more of the points coinciding. |
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100
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173 |
|
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174 |
\begin{prop} |
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175 |
$\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ |
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176 |
(relative singular chains with coefficients in $k$). |
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177 |
\end{prop} |
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178 |
|
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179 |
\begin{proof} |
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180 |
\nn{...} |
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181 |
\end{proof} |
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182 |
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163 | 183 |
\medskip |
184 |
\hrule |
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185 |
\medskip |
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100
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186 |
|
163 | 187 |
Still to do: |
188 |
\begin{itemize} |
|
166 | 189 |
\item compare the topological computation for truncated polynomial algebra with \cite{MR1600246} |
163 | 190 |
\item multivariable truncated polynomial algebras (at least mention them) |
191 |
\item ideally, say something more about higher hochschild homology (maybe sketch idea for proof of equivalence) |
|
192 |
\end{itemize} |
|
193 |