25 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
25 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
26 \end{itemize} |
26 \end{itemize} |
27 |
27 |
28 We want to show that $\bc_*(S^1)$ is homotopy equivalent to the |
28 We want to show that $\bc_*(S^1)$ is homotopy equivalent to the |
29 Hochschild complex of $C$. |
29 Hochschild complex of $C$. |
30 (Note that both complexes are free (and hence projective), so it suffices to show that they |
30 Note that both complexes are free (and hence projective), so it suffices to show that they |
31 are quasi-isomorphic.) |
31 are quasi-isomorphic. |
32 In order to prove this we will need to extend the blob complex to allow points to also |
32 In order to prove this we will need to extend the blob complex to allow points to also |
33 be labeled by elements of $C$-$C$-bimodules. |
33 be labeled by elements of $C$-$C$-bimodules. |
34 |
34 |
35 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
35 Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
36 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
36 We define a blob-like complex $K_*(S^1, (p_i), (M_i))$. |
56 $C$-$C$-bimodule, not a category) is quasi-isomorphic to the blob complex |
56 $C$-$C$-bimodule, not a category) is quasi-isomorphic to the blob complex |
57 $\bc_*(S^1; C)$. (Proof later.) |
57 $\bc_*(S^1; C)$. (Proof later.) |
58 \end{lem} |
58 \end{lem} |
59 |
59 |
60 Next, we show that for any $C$-$C$-bimodule $M$, |
60 Next, we show that for any $C$-$C$-bimodule $M$, |
61 \begin{prop} |
61 \begin{prop} \label{prop:hoch} |
62 The complex $K_*(M)$ is quasi-isomorphic to $HC_*(M)$, the usual |
62 The complex $K_*(M)$ is quasi-isomorphic to $HC_*(M)$, the usual |
63 Hochschild complex of $M$. |
63 Hochschild complex of $M$. |
64 \end{prop} |
64 \end{prop} |
65 \begin{proof} |
65 \begin{proof} |
66 Recall that the usual Hochschild complex of $M$ is uniquely determined, |
66 Recall that the usual Hochschild complex of $M$ is uniquely determined, |
73 exact sequence $0 \to HC_*(M_1) \into HC_*(M_2) \onto HC_*(M_3) \to 0$. |
73 exact sequence $0 \to HC_*(M_1) \into HC_*(M_2) \onto HC_*(M_3) \to 0$. |
74 \item \label{item:hochschild-coinvariants}% |
74 \item \label{item:hochschild-coinvariants}% |
75 $HH_0(M)$ is isomorphic to the coinvariants of $M$, $\coinv(M) = |
75 $HH_0(M)$ is isomorphic to the coinvariants of $M$, $\coinv(M) = |
76 M/\langle cm-mc \rangle$. |
76 M/\langle cm-mc \rangle$. |
77 \item \label{item:hochschild-free}% |
77 \item \label{item:hochschild-free}% |
78 $HC_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is contractible; that is, |
78 $HC_*(C\otimes C)$ is contractible. |
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79 (Here $C\otimes C$ denotes |
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80 the free $C$-$C$-bimodule with one generator.) |
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81 That is, $HC_*(C\otimes C)$ is |
79 quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants}, is just $C$) via the quotient map $HC_0 \onto HH_0$. |
82 quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants}, is just $C$) via the quotient map $HC_0 \onto HH_0$. |
80 \end{enumerate} |
83 \end{enumerate} |
81 (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) |
84 (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) |
82 We'll first recall why these properties are characteristic. |
85 We'll first recall why these properties are characteristic. |
83 |
86 |
126 %compute every homology group of $HC_*(M)$; we already know $HH_0(M)$ |
129 %compute every homology group of $HC_*(M)$; we already know $HH_0(M)$ |
127 %(it's just coinvariants, by property \ref{item:hochschild-coinvariants}), |
130 %(it's just coinvariants, by property \ref{item:hochschild-coinvariants}), |
128 %and higher homology groups are determined by lower ones in $HC_*(K)$, and |
131 %and higher homology groups are determined by lower ones in $HC_*(K)$, and |
129 %hence recursively as coinvariants of some other bimodule. |
132 %hence recursively as coinvariants of some other bimodule. |
130 |
133 |
131 The proposition then follows from the following lemmas, establishing that $K_*$ has precisely these required properties. |
134 Proposition \ref{prop:hoch} then follows from the following lemmas, establishing that $K_*$ has precisely these required properties. |
132 \begin{lem} |
135 \begin{lem} |
133 \label{lem:hochschild-additive}% |
136 \label{lem:hochschild-additive}% |
134 Directly from the definition, $K_*(M_1 \oplus M_2) \cong K_*(M_1) \oplus K_*(M_2)$. |
137 Directly from the definition, $K_*(M_1 \oplus M_2) \cong K_*(M_1) \oplus K_*(M_2)$. |
135 \end{lem} |
138 \end{lem} |
136 \begin{lem} |
139 \begin{lem} |
161 |
164 |
162 We define a left inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion as follows. |
165 We define a left inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion as follows. |
163 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
166 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
164 * is a labeled point in $y$. |
167 * is a labeled point in $y$. |
165 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
168 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
166 Let $x \in \bc_*(S^1)$. |
169 Extending linearly, we get the desired map $s: \bc_*(S^1) \to K_*(C)$. |
167 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
170 %Let $x \in \bc_*(S^1)$. |
168 $x$ with $s(y)$. |
171 %Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
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172 %$x$ with $s(y)$. |
169 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
173 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
170 |
174 |
171 Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
175 Let $N_\ep$ denote the ball of radius $\ep$ around *. |
172 in a neighborhood $B_\ep$ of $*$, except perhaps $*$, and $B_\ep$ is either disjoint from or contained in every blob. |
176 Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex |
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177 spanned by blob diagrams |
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178 where there are no labeled points |
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179 in $N_\ep$, except perhaps $*$, and $N_\ep$ is either disjoint from or contained in |
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180 every blob in the diagram. |
173 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
181 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
174 |
182 |
175 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. |
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. |
176 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding $B_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
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 |
177 of $x$ to $B_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
185 of $x$ to $N_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
178 write $y_i$ for the restriction of $z_i$ to $B_\ep$, and let |
186 write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let |
179 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin B_\ep$, |
187 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$, |
180 and have an additional blob $B_\ep$ with label $y_i - s(y_i)$. |
188 and have an additional blob $N_\ep$ with label $y_i - s(y_i)$. |
181 Define $j_\ep(x) = \sum x_i$. |
189 Define $j_\ep(x) = \sum x_i$. |
182 \todo{need to check signs coming from blob complex differential} |
190 \todo{need to check signs coming from blob complex differential} |
183 \todo{finish this} |
191 \todo{finish this} |
184 \end{proof} |
192 \end{proof} |
185 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
193 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
254 Let $K'_* \sub K_*(C\otimes C)$ be the subcomplex where the label of |
262 Let $K'_* \sub K_*(C\otimes C)$ be the subcomplex where the label of |
255 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
263 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
256 We will show that the inclusion $i: K'_* \to K_*(C\otimes C)$ is a quasi-isomorphism. |
264 We will show that the inclusion $i: K'_* \to K_*(C\otimes C)$ is a quasi-isomorphism. |
257 |
265 |
258 Fix a small $\ep > 0$. |
266 Fix a small $\ep > 0$. |
259 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
267 Let $N_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
260 Let $K_*^\ep \sub K_*(C\otimes C)$ be the subcomplex |
268 Let $K_*^\ep \sub K_*(C\otimes C)$ be the subcomplex |
261 generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from |
269 generated by blob diagrams $b$ such that $N_\ep$ is either disjoint from |
262 or contained in each blob of $b$, and the only labeled point inside $B_\ep$ is $*$. |
270 or contained in each blob of $b$, and the only labeled point inside $N_\ep$ is $*$. |
263 %and the two boundary points of $B_\ep$ are not labeled points of $b$. |
271 %and the two boundary points of $N_\ep$ are not labeled points of $b$. |
264 For a field $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
272 For a field $y$ on $N_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
265 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
273 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
266 (See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(B_\ep)$. We can think of |
274 (See Figure \ref{fig:sy}.) Note that $y - s_\ep(y) \in U(N_\ep)$. We can think of |
267 $\sigma_\ep$ as a chain map $K_*^\ep \to K_*^\ep$ given by replacing the restriction $y$ to $B_\ep$ of each field |
275 $\sigma_\ep$ as a chain map $K_*^\ep \to K_*^\ep$ given by replacing the restriction $y$ to $N_\ep$ of each field |
268 appearing in an element of $K_*^\ep$ with $s_\ep(y)$. |
276 appearing in an element of $K_*^\ep$ with $s_\ep(y)$. |
269 Note that $\sigma_\ep(x) \in K'_*$. |
277 Note that $\sigma_\ep(x) \in K'_*$. |
270 \begin{figure}[!ht] |
278 \begin{figure}[!ht] |
271 \begin{align*} |
279 \begin{align*} |
272 y & = \mathfig{0.2}{hochschild/y} & |
280 y & = \mathfig{0.2}{hochschild/y} & |
276 \label{fig:sy} |
284 \label{fig:sy} |
277 \end{figure} |
285 \end{figure} |
278 |
286 |
279 Define a degree 1 chain map $j_\ep : K_*^\ep \to K_*^\ep$ as follows. |
287 Define a degree 1 chain map $j_\ep : K_*^\ep \to K_*^\ep$ as follows. |
280 Let $x \in K_*^\ep$ be a blob diagram. |
288 Let $x \in K_*^\ep$ be a blob diagram. |
281 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
289 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $N_\ep$ to |
282 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
290 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $N_\ep$. |
283 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
291 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
284 Let $y_i$ be the restriction of $z_i$ to $B_\ep$. |
292 Let $y_i$ be the restriction of $z_i$ to $N_\ep$. |
285 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
293 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin N_\ep$, |
286 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
294 and have an additional blob $N_\ep$ with label $y_i - s_\ep(y_i)$. |
287 Define $j_\ep(x) = \sum x_i$. |
295 Define $j_\ep(x) = \sum x_i$. |
288 \nn{need to check signs coming from blob complex differential} |
296 \nn{need to check signs coming from blob complex differential} |
289 Note that if $x \in K'_* \cap K_*^\ep$ then $j_\ep(x) \in K'_*$ also. |
297 Note that if $x \in K'_* \cap K_*^\ep$ then $j_\ep(x) \in K'_*$ also. |
290 |
298 |
291 The key property of $j_\ep$ is |
299 The key property of $j_\ep$ is |