107 and that the cyclic homology of $k[t]$ is related to the action of rotations |
107 and that the cyclic homology of $k[t]$ is related to the action of rotations |
108 on $C_*(\Sigma^\infty(S^1), k)$. |
108 on $C_*(\Sigma^\infty(S^1), k)$. |
109 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
109 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
110 Let us check this directly. |
110 Let us check this directly. |
111 |
111 |
112 According to \cite[3.2.2]{MR1600246}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$. |
112 The algebra $k[t]$ has Koszul resolution $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, which has coinvariants $k[t] \xrightarrow{0} k[t]$. This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. |
113 \nn{say something about $t$-degree? is this in Loday?} |
113 (See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one. |
114 |
114 |
115 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
115 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
116 The fixed points of this flow are the equally spaced configurations. |
116 The fixed points of this flow are the equally spaced configurations. |
117 This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
117 This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
118 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
118 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
121 |
121 |
122 In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
122 In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
123 of course $\Sigma^0(S^1)$ is a point. |
123 of course $\Sigma^0(S^1)$ is a point. |
124 Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
124 Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
125 and is zero for $i\ge 2$. |
125 and is zero for $i\ge 2$. |
126 \nn{say something about $t$-degrees also matching up?} |
126 Note that the $j$-grading here matches with the $t$-grading on the algebraic side. |
127 |
127 |
128 By xxxx and \ref{ktchprop}, |
128 By xxxx and Proposition \ref{ktchprop}, |
129 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
129 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
130 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
130 Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
131 If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree |
131 If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree |
132 0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. |
132 0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. |
133 The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |
133 The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |