11 can also be thought of as an $n$-category whose $j$-morphisms are trivial for |
11 can also be thought of as an $n$-category whose $j$-morphisms are trivial for |
12 $j<n$ and whose $n$-morphisms are $C$. |
12 $j<n$ and whose $n$-morphisms are $C$. |
13 The goal of this \nn{subsection?} is to compute |
13 The goal of this \nn{subsection?} is to compute |
14 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
14 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
15 |
15 |
16 Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). This possibility was suggested to us by Thomas Tradler. |
16 Moreover, we conjecture that the blob complex $\bc_*(M^n, $C$)$, for $C$ a commutative |
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17 algebra is homotopy equivalent to the higher Hochschild complex for $M^n$ with |
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18 coefficients in $C$ (see \cite{MR0339132, MR1755114, MR2383113}). |
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19 This possibility was suggested to us by Thomas Tradler. |
17 |
20 |
18 |
21 |
19 \medskip |
22 \medskip |
20 |
23 |
21 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
24 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
106 and that the cyclic homology of $k[t]$ is related to the action of rotations |
109 and that the cyclic homology of $k[t]$ is related to the action of rotations |
107 on $C_*(\Sigma^\infty(S^1), k)$. |
110 on $C_*(\Sigma^\infty(S^1), k)$. |
108 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
111 \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
109 Let us check this directly. |
112 Let us check this directly. |
110 |
113 |
111 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 $i\ge 2$. |
114 The algebra $k[t]$ has Koszul resolution |
112 (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. |
115 $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, |
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116 which has coinvariants $k[t] \xrightarrow{0} k[t]$. |
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117 This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. |
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118 (See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: |
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119 $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one. |
113 |
120 |
114 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
121 We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
115 The fixed points of this flow are the equally spaced configurations. |
122 The fixed points of this flow are the equally spaced configurations. |
116 This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
123 This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
117 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
124 The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
150 \[ |
157 \[ |
151 HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
158 HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
152 \] |
159 \] |
153 Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
160 Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
154 We will content ourselves with the case $k = \z$. |
161 We will content ourselves with the case $k = \z$. |
155 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. |
162 One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the |
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163 same color repel each other and points of different colors do not interact. |
156 This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent |
164 This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent |
157 to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple |
165 to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple |
158 corresponding to $X$. |
166 corresponding to $X$. |
159 The homology calculation we desire follows easily from this. |
167 The homology calculation we desire follows easily from this. |
160 |
168 |