10 for $M^n$ based on comm alg $C$ thought of as an $n$-category] |
10 for $M^n$ based on comm alg $C$ thought of as an $n$-category] |
11 is homotopy equivalent to [higher Hochschild complex for $M^n$ with coefficients in $C$]. |
11 is homotopy equivalent to [higher Hochschild complex for $M^n$ with coefficients in $C$]. |
12 (Thomas Tradler's idea.) |
12 (Thomas Tradler's idea.) |
13 Should prove (or at least conjecture) that here.} |
13 Should prove (or at least conjecture) that here.} |
14 |
14 |
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15 \nn{also, this section needs a little updating to be compatible with the rest of the paper.} |
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16 |
15 If $C$ is a commutative algebra it |
17 If $C$ is a commutative algebra it |
16 can (and will) also be thought of as an $n$-category whose $j$-morphisms are trivial for |
18 can also be thought of as an $n$-category whose $j$-morphisms are trivial for |
17 $j<n$ and whose $n$-morphisms are $C$. |
19 $j<n$ and whose $n$-morphisms are $C$. |
18 The goal of this \nn{subsection?} is to compute |
20 The goal of this \nn{subsection?} is to compute |
19 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
21 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
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22 |
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23 \medskip |
20 |
24 |
21 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
25 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
22 |
26 |
23 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
27 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
24 unlabeled points in $M$. |
28 unlabeled points in $M$. |
43 $f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
47 $f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
44 \end{lemma} |
48 \end{lemma} |
45 |
49 |
46 \begin{proof} |
50 \begin{proof} |
47 \nn{easy, but should probably write the details eventually} |
51 \nn{easy, but should probably write the details eventually} |
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52 \nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that} |
48 \end{proof} |
53 \end{proof} |
49 |
54 |
50 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
55 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
51 satisfying the conditions of the above lemma. |
56 satisfying the conditions of the above lemma. |
52 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
57 If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
159 |
164 |
160 \nn{say something about cyclic homology in this case? probably not necessary.} |
165 \nn{say something about cyclic homology in this case? probably not necessary.} |
161 |
166 |
162 \medskip |
167 \medskip |
163 |
168 |
164 Next we consider the case $C = k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. |
169 Next we consider the case $C$ is the truncated polynomial |
165 Define $\Delta_l \sub \Sigma^\infty(M)$ to be those configurations of points with $l$ or |
170 algebra $k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. |
166 more points coinciding. |
171 Define $\Delta_l \sub \Sigma^\infty(M)$ to be configurations of points in $M$ with $l$ or |
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172 more of the points coinciding. |
167 |
173 |
168 \begin{prop} |
174 \begin{prop} |
169 $\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ |
175 $\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ |
170 (relative singular chains with coefficients in $k$). |
176 (relative singular chains with coefficients in $k$). |
171 \end{prop} |
177 \end{prop} |
172 |
178 |
173 \begin{proof} |
179 \begin{proof} |
174 \nn{...} |
180 \nn{...} |
175 \end{proof} |
181 \end{proof} |
176 |
182 |
177 \nn{...} |
183 \medskip |
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184 \hrule |
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185 \medskip |
178 |
186 |
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187 Still to do: |
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188 \begin{itemize} |
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189 \item compare the topological computation for truncated polynomial algebra with [Loday] |
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190 \item multivariable truncated polynomial algebras (at least mention them) |
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191 \item ideally, say something more about higher hochschild homology (maybe sketch idea for proof of equivalence) |
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192 \end{itemize} |
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193 |