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1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{Commutative algebras as $n$-categories} |
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4 |
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5 \nn{this should probably not be a section by itself. i'm just trying to write down the outline |
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6 while it's still fresh in my mind.} |
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7 |
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8 If $C$ is a commutative algebra it |
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9 can (and will) also be thought of as an $n$-category whose $j$-morphisms are trivial for |
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10 $j<n$ and whose $n$-morphisms are $C$. |
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11 The goal of this \nn{subsection?} is to compute |
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12 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
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13 |
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14 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
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15 |
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16 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
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17 unlabeled points in $M$. |
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18 Note that $\Sigma^0(M)$ is a point. |
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19 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
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20 |
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21 Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$. |
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22 |
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23 \begin{prop} \label{sympowerprop} |
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24 $\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
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25 \end{prop} |
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26 |
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27 \begin{proof} |
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28 To define the chain maps between the two complexes we will use the following lemma: |
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29 |
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30 \begin{lemma} |
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31 Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
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32 a basis (e.g.\ blob diagrams or singular simplices). |
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33 For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
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34 such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$. |
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35 Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
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36 $f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
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37 \end{lemma} |
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38 |
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39 \begin{proof} |
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40 \nn{easy, but should probably write the details eventually} |
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41 \end{proof} |
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42 |
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43 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
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44 satisfying the conditions of the above lemma. |
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45 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
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46 finite unordered collection of points of $M$ with multiplicities, which is |
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47 a point in $\Sigma^\infty(M)$. |
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48 Define $R(b)_*$ to be the singular chain complex of this point. |
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49 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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50 The path components of $\Sigma^\infty(D)$ are contractible, and these components are indexed |
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51 by the numbers of points in each component of $D$. |
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52 We may assume that the blob labels $u$ have homogeneous $t$ degree in $k[t]$, and so |
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53 $u$ picks out a component $X \sub \Sigma^\infty(D)$. |
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54 The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, |
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55 and using this point we can embed $X$ in $\Sigma^\infty(M)$. |
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56 Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a |
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57 subspace of $\Sigma^\infty(M)$. |
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58 It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from the above lemma. |
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59 Thus we have defined (up to homotopy) a map from |
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60 $\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$. |
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61 |
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62 Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace |
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63 $R(c)_* \sub \bc_*(M^n, k[t])$. |
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64 If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and |
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65 $\Sigma^\infty(M)$ described above. |
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66 Now let $c$ be an $i$-simplex of $\Sigma^j(M)$. |
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67 Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. |
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68 We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ |
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69 is homotopy equivalent to the subcomplex of small simplices. |
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70 How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. |
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71 Let $T\sub M$ be the ``track" of $c$ in $M$. |
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72 \nn{do we need to define this precisely?} |
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73 Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. |
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74 \nn{need to say more precisely how small} |
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75 Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$. |
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76 This is contractible by \ref{bcontract}. |
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77 We can arrange that the boundary/inclusion condition is satisfied if we start with |
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78 low-dimensional simplices and work our way up. |
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79 \nn{need to be more precise} |
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80 |
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81 \nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
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82 (for this, might need a lemma that says we can assume that blob diameters are small)} |
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83 \end{proof} |
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84 |
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85 |
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86 \begin{prop} \label{ktcdprop} |
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87 The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
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88 \end{prop} |
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89 |
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90 \begin{proof} |
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91 The actions agree in degree 0, and both are compatible with gluing. |
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92 (cf. uniqueness statement in \ref{CDprop}.) |
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93 \end{proof} |
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94 |
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95 \medskip |
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96 |
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97 In view of \ref{hochthm}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
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98 and that the cyclic homology of $k[t]$ is related to the action of rotations |
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99 on $C_*(\Sigma^\infty(S^1), k)$. |
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100 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
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101 Let us check this directly. |
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102 |
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103 According to \nn{Loday, 3.2.2}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$. |
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104 \nn{say something about $t$-degree? is this in [Loday]?} |
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105 |
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106 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
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107 The fixed points of this flow are the equally spaced configurations. |
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108 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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109 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
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110 and the holonomy of the $\Delta^{j-1}$ bundle |
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111 over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices. |
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112 |
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113 In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
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114 of course $\Sigma^0(S^1)$ is a point. |
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115 Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
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116 and is zero for $i\ge 2$. |
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117 \nn{say something about $t$-degrees also matching up?} |
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118 |
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119 By xxxx and \ref{ktcdprop}, |
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120 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
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121 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
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122 If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree |
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123 0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. |
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124 The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |
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125 degrees and 0 in odd degrees. |
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126 This agrees with the calculation in \nn{Loday, 3.1.7}. |
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127 |
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128 \medskip |
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129 |
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130 Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. |
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131 Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations |
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132 of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable. |
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133 The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers |
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134 corresponding to the number of points of each color of a configuration. |
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135 A proof similar to that of \ref{sympowerprop} shows that |
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136 |
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137 \begin{prop} |
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138 $\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
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139 \end{prop} |
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140 |
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141 According to \nn{Loday, 3.2.2}, |
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142 \[ |
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143 HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
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144 \] |
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145 Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
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146 We will content ourselves with the case $k = \z$. |
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147 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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148 This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent |
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149 to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple |
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150 corresponding to $X$. |
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151 The homology calculation we desire follows easily from this. |
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152 |
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153 \nn{say something about cyclic homology in this case? probably not necessary.} |
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154 |
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155 \medskip |
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156 |
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157 Next we consider the case $C = k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. |
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158 Define $\Delta_l \sub \Sigma^\infty(M)$ to be those configurations of points with $l$ or |
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159 more points coinciding. |
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160 |
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161 \begin{prop} |
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162 $\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ |
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163 (relative singular chains with coefficients in $k$). |
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164 \end{prop} |
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165 |
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166 \begin{proof} |
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167 \nn{...} |
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168 \end{proof} |
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169 |
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170 \nn{...} |
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171 |