110 and that the cyclic homology of $k[t]$ is related to the action of rotations |
110 and that the cyclic homology of $k[t]$ is related to the action of rotations |
111 on $C_*(\Sigma^\infty(S^1), k)$. |
111 on $C_*(\Sigma^\infty(S^1), k)$. |
112 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
112 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
113 Let us check this directly. |
113 Let us check this directly. |
114 |
114 |
115 According to \nn{Loday, 3.2.2}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$. |
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]?} |
116 \nn{say something about $t$-degree? is this in Loday?} |
117 |
117 |
118 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
118 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
119 The fixed points of this flow are the equally spaced configurations. |
119 The fixed points of this flow are the equally spaced configurations. |
120 This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
120 This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
121 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
121 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
133 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
133 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
134 If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree |
134 If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree |
135 0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. |
135 0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. |
136 The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |
136 The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |
137 degrees and 0 in odd degrees. |
137 degrees and 0 in odd degrees. |
138 This agrees with the calculation in \nn{Loday, 3.1.7}. |
138 This agrees with the calculation in \cite[3.1.7]{MR1600246}. |
139 |
139 |
140 \medskip |
140 \medskip |
141 |
141 |
142 Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. |
142 Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. |
143 Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations |
143 Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations |
148 |
148 |
149 \begin{prop} |
149 \begin{prop} |
150 $\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
150 $\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
151 \end{prop} |
151 \end{prop} |
152 |
152 |
153 According to \nn{Loday, 3.2.2}, |
153 According to \cite[3.2.2]{MR1600246}, |
154 \[ |
154 \[ |
155 HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
155 HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
156 \] |
156 \] |
157 Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
157 Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
158 We will content ourselves with the case $k = \z$. |
158 We will content ourselves with the case $k = \z$. |