93 \nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
93 \nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
94 (for this, might need a lemma that says we can assume that blob diameters are small)} |
94 (for this, might need a lemma that says we can assume that blob diameters are small)} |
95 \end{proof} |
95 \end{proof} |
96 |
96 |
97 |
97 |
98 \begin{prop} \label{ktcdprop} |
98 \begin{prop} \label{ktchprop} |
99 The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
99 The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
100 \end{prop} |
100 \end{prop} |
101 |
101 |
102 \begin{proof} |
102 \begin{proof} |
103 The actions agree in degree 0, and both are compatible with gluing. |
103 The actions agree in degree 0, and both are compatible with gluing. |
104 (cf. uniqueness statement in \ref{CDprop}.) |
104 (cf. uniqueness statement in \ref{CHprop}.) |
105 \end{proof} |
105 \end{proof} |
106 |
106 |
107 \medskip |
107 \medskip |
108 |
108 |
109 In view of \ref{hochthm}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
109 In view of \ref{hochthm}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
126 of course $\Sigma^0(S^1)$ is a point. |
126 of course $\Sigma^0(S^1)$ is a point. |
127 Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
127 Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
128 and is zero for $i\ge 2$. |
128 and is zero for $i\ge 2$. |
129 \nn{say something about $t$-degrees also matching up?} |
129 \nn{say something about $t$-degrees also matching up?} |
130 |
130 |
131 By xxxx and \ref{ktcdprop}, |
131 By xxxx and \ref{ktchprop}, |
132 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
132 the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
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 |