1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 In this section we analyze the blob complex in dimension $n=1$ |
3 In this section we analyze the blob complex in dimension $n=1$ |
4 and find that for $S^1$ the blob complex is homotopy equivalent to the |
4 and find that for $S^1$ the blob complex is homotopy equivalent to the |
5 Hochschild complex of the category (algebroid) that we started with. |
5 Hochschild complex of the category (algebroid) that we started with. |
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6 |
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7 \nn{need to be consistent about quasi-isomorphic versus homotopy equivalent |
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8 in this section. |
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9 since the various complexes are free, q.i. implies h.e.} |
6 |
10 |
7 Let $C$ be a *-1-category. |
11 Let $C$ be a *-1-category. |
8 Then specializing the definitions from above to the case $n=1$ we have: |
12 Then specializing the definitions from above to the case $n=1$ we have: |
9 \begin{itemize} |
13 \begin{itemize} |
10 \item $\cC(pt) = \ob(C)$ . |
14 \item $\cC(pt) = \ob(C)$ . |
51 |
55 |
52 This follows from two results. First, we see that |
56 This follows from two results. First, we see that |
53 \begin{lem} |
57 \begin{lem} |
54 \label{lem:module-blob}% |
58 \label{lem:module-blob}% |
55 The complex $K_*(C)$ (here $C$ is being thought of as a |
59 The complex $K_*(C)$ (here $C$ is being thought of as a |
56 $C$-$C$-bimodule, not a category) is quasi-isomorphic to the blob complex |
60 $C$-$C$-bimodule, not a category) is homotopy equivalent to the blob complex |
57 $\bc_*(S^1; C)$. (Proof later.) |
61 $\bc_*(S^1; C)$. (Proof later.) |
58 \end{lem} |
62 \end{lem} |
59 |
63 |
60 Next, we show that for any $C$-$C$-bimodule $M$, |
64 Next, we show that for any $C$-$C$-bimodule $M$, |
61 \begin{prop} \label{prop:hoch} |
65 \begin{prop} \label{prop:hoch} |
177 spanned by blob diagrams |
181 spanned by blob diagrams |
178 where there are no labeled points |
182 where there are no labeled points |
179 in $N_\ep$, except perhaps $*$, and $N_\ep$ is either disjoint from or contained in |
183 in $N_\ep$, except perhaps $*$, and $N_\ep$ is either disjoint from or contained in |
180 every blob in the diagram. |
184 every blob in the diagram. |
181 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
185 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
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186 \nn{what if * is on boundary of a blob? need preliminary homotopy to prevent this.} |
182 |
187 |
183 We define a degree $1$ chain map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. Let $x \in L_*^\ep$ be a blob diagram. |
188 We define a degree $1$ chain map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. Let $x \in L_*^\ep$ be a blob diagram. |
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189 \nn{maybe add figures illustrating $j_\ep$?} |
184 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
190 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
185 of $x$ to $N_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
191 of $x$ to $N_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
186 write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let |
192 write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let |
187 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$, |
193 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$, |
188 and have an additional blob $N_\ep$ with label $y_i - s(y_i)$. |
194 and have an additional blob $N_\ep$ with label $y_i - s(y_i)$. |
189 Define $j_\ep(x) = \sum x_i$. |
195 Define $j_\ep(x) = \sum x_i$. |
190 \todo{need to check signs coming from blob complex differential} |
196 |
191 \todo{finish this} |
197 It is not hard to show that on $L_*^\ep$ |
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198 \[ |
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199 \bd j_\ep + j_\ep \bd = \id - i \circ s . |
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200 \] |
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201 \nn{need to check signs coming from blob complex differential} |
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202 Since for $\ep$ small enough $L_*^\ep$ captures all of the |
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203 homology of $\bc_*(S^1)$, |
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204 it follows that the mapping cone of $i \circ s$ is acyclic and therefore (using the fact that |
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205 these complexes are free) $i \circ s$ is homotopic to the identity. |
192 \end{proof} |
206 \end{proof} |
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207 |
193 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
208 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
194 We now prove that $K_*$ is an exact functor. |
209 We now prove that $K_*$ is an exact functor. |
195 |
210 |
196 %\todo{p. 1478 of scott's notes} |
211 %\todo{p. 1478 of scott's notes} |
197 Essentially, this comes down to the unsurprising fact that the functor on $C$-$C$ bimodules |
212 Essentially, this comes down to the unsurprising fact that the functor on $C$-$C$ bimodules |