50 It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from |
50 It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from |
51 Theorem \ref{moam-thm}. |
51 Theorem \ref{moam-thm}. |
52 Thus we have defined (up to homotopy) a map from |
52 Thus we have defined (up to homotopy) a map from |
53 $\bc_*(M, k[t])$ to $C_*(\Sigma^\infty(M))$. |
53 $\bc_*(M, k[t])$ to $C_*(\Sigma^\infty(M))$. |
54 |
54 |
55 Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace |
55 Next we define a map going the other direction. |
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56 First we replace $C_*(\Sigma^\infty(M))$ with a homotopy equivalent |
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57 subcomplex $S_*$ of small simplices. |
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58 Roughly, we define $c\in C_*(\Sigma^\infty(M))$ to be small if the |
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59 corresponding track of points in $M$ |
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60 is contained in a disjoint union of balls. |
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61 Because there could be different, inequivalent choices of such balls, we must a bit more careful. |
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62 \nn{this runs into the same issues as in defining evmap. |
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63 either refer there for details, or use the simp-space-ish version of the blob complex, |
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64 which makes things easier here.} |
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65 |
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66 \nn{...} |
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67 |
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68 |
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69 We will define, for each simplex $c$ of $S_*$, a contractible subspace |
56 $R(c)_* \sub \bc_*(M, k[t])$. |
70 $R(c)_* \sub \bc_*(M, k[t])$. |
57 If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and |
71 If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and |
58 $\Sigma^\infty(M)$ described above. |
72 $\Sigma^\infty(M)$ described above. |
59 Now let $c$ be an $i$-simplex of $\Sigma^j(M)$. |
73 Now let $c$ be an $i$-simplex of $S_*$. |
60 Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. |
74 Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. |
61 We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ |
75 We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ |
62 is homotopy equivalent to the subcomplex of small simplices. |
76 is homotopy equivalent to the subcomplex of small simplices. |
63 How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. |
77 How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. |
64 Let $T\sub M$ be the ``track" of $c$ in $M$. |
78 Let $T\sub M$ be the ``track" of $c$ in $M$. |
75 (for this, might need a lemma that says we can assume that blob diameters are small)} |
89 (for this, might need a lemma that says we can assume that blob diameters are small)} |
76 \end{proof} |
90 \end{proof} |
77 |
91 |
78 |
92 |
79 \begin{prop} \label{ktchprop} |
93 \begin{prop} \label{ktchprop} |
80 The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
94 The above maps are compatible with the evaluation map actions of $C_*(\Homeo(M))$. |
81 \end{prop} |
95 \end{prop} |
82 |
96 |
83 \begin{proof} |
97 \begin{proof} |
84 The actions agree in degree 0, and both are compatible with gluing. |
98 The actions agree in degree 0, and both are compatible with gluing. |
85 (cf. uniqueness statement in Theorem \ref{thm:CH}.) |
99 (cf. uniqueness statement in Theorem \ref{thm:CH}.) |
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100 \nn{if Theorem \ref{thm:CH} is rewritten/rearranged, make sure uniqueness discussion is properly referenced from here} |
86 \end{proof} |
101 \end{proof} |
87 |
102 |
88 \medskip |
103 \medskip |
89 |
104 |
90 In view of Theorem \ref{thm:hochschild}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
105 In view of Theorem \ref{thm:hochschild}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
91 and that the cyclic homology of $k[t]$ is related to the action of rotations |
106 and that the cyclic homology of $k[t]$ is related to the action of rotations |
92 on $C_*(\Sigma^\infty(S^1), k)$. |
107 on $C_*(\Sigma^\infty(S^1), k)$. |
93 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
108 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
94 Let us check this directly. |
109 Let us check this directly. |
95 |
110 |
96 The algebra $k[t]$ has Koszul resolution |
111 The algebra $k[t]$ has a resolution |
97 $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, |
112 $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, |
98 which has coinvariants $k[t] \xrightarrow{0} k[t]$. |
113 which has coinvariants $k[t] \xrightarrow{0} k[t]$. |
99 This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. |
114 So we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. |
100 (See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: |
115 (See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: |
101 $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one. |
116 $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one. |
102 |
117 |
103 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. |
104 The fixed points of this flow are the equally spaced configurations. |
119 The fixed points of this flow are the equally spaced configurations. |
105 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 deformation retraction from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
106 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, |
107 and the holonomy of the $\Delta^{j-1}$ bundle |
122 and the holonomy of the $\Delta^{j-1}$ bundle |
108 over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices. |
123 over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices. |
109 |
124 |
110 In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
125 In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
111 of course $\Sigma^0(S^1)$ is a point. |
126 of course $\Sigma^0(S^1)$ is a point. |
112 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 $i=0,1$ |
113 and is zero for $i\ge 2$. |
128 and is zero for $i\ge 2$. |
114 Note that the $j$-grading here matches with the $t$-grading on the algebraic side. |
129 Note that the $j$-grading here matches with the $t$-grading on the algebraic side. |
115 |
130 |
116 By xxxx and Proposition \ref{ktchprop}, |
131 By Proposition \ref{ktchprop}, |
117 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 $\Sigma^\infty(S^1)$. |
118 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 $\Sigma^j(S^1) \simeq S^1/j$. |
119 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 |
120 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. |
121 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 |