127 quasi-isomorphic to the 0-step complex $C$. |
127 quasi-isomorphic to the 0-step complex $C$. |
128 \item \label{item:hochschild-coinvariants}% |
128 \item \label{item:hochschild-coinvariants}% |
129 $HH_0(M)$ is isomorphic to the coinvariants of $M$, $M/\langle cm-mc \rangle$. |
129 $HH_0(M)$ is isomorphic to the coinvariants of $M$, $M/\langle cm-mc \rangle$. |
130 \end{enumerate} |
130 \end{enumerate} |
131 (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) |
131 (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) |
132 We'll first explain why these properties are characteristic. Take some |
132 We'll first recall why these properties are characteristic. |
133 $C$-$C$ bimodule $M$. If $M$ is free, that is, a direct sum of copies of |
133 |
134 $C \tensor C$, then properties \ref{item:hochschild-additive} and |
134 Take some $C$-$C$ bimodule $M$, and choose a free resolution |
135 \ref{item:hochschild-free} determine $HC_*(M)$. Otherwise, choose some |
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136 free cover $F \onto M$, and define $K$ to be this map's kernel. Thus we |
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137 have a short exact sequence $0 \to K \into F \onto M \to 0$, and hence a |
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138 short exact sequence of complexes $0 \to HC_*(K) \into HC_*(F) \onto HC_*(M) |
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139 \to 0$. Such a sequence gives a long exact sequence on homology |
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140 \begin{equation*} |
135 \begin{equation*} |
141 %\begin{split} |
136 \cdots \to F_2 \xrightarrow{f_2} F_1 \xrightarrow{f_1} F_0. |
142 \cdots \to HH_{i+1}(F) \to HH_{i+1}(M) \to HH_i(K) \to HH_i(F) \to \cdots % \\ |
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143 %\cdots \to HH_1(F) \to HH_1(M) \to HH_0(K) \to HH_0(F) \to HH_0(M). |
|
144 %\end{split} |
|
145 \end{equation*} |
137 \end{equation*} |
146 For any $i \geq 1$, $HH_{i+1}(F) = HH_i(F) = 0$, by properties |
138 There's a quotient map $\pi: F_0 \onto M$, and by construction the cone of the chain map $\pi: F_j \to M$ is acyclic. Now construct the total complex |
147 \ref{item:hochschild-additive} and \ref{item:hochschild-free}, and so |
139 $HC_i(F_j)$, with $i,j \geq 0$, graded by $i+j$. |
148 $HH_{i+1}(M) \iso HH_i(F)$. For $i=0$, \todo{}. |
140 |
149 |
141 Observe that we have two chain maps |
150 This tells us how to |
142 \begin{align*} |
151 compute every homology group of $HC_*(M)$; we already know $HH_0(M)$ |
143 HC_i(F_j) & \xrightarrow{HC_i(\pi)} HC_i(M) \\ |
152 (it's just coinvariants, by property \ref{item:hochschild-coinvariants}), |
144 \intertext{and} |
153 and higher homology groups are determined by lower ones in $HC_*(K)$, and |
145 HC_i(F_j) & \xrightarrow{HC_0(F_j) \onto HH_0(F_j)} \operatorname{coinv}(F_j). |
154 hence recursively as coinvariants of some other bimodule. |
146 \end{align*} |
155 |
147 The cone of each chain map is acyclic. In the first case, this is because the `rows' indexed by $i$ are acyclic since $HC_i$ is exact. |
156 The proposition then follows from the following lemmas, establishing that $F_*$ has precisely these required properties. |
148 In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free. |
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149 |
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150 Because the cones are acyclic, the chain maps are quasi-isomorphisms. Composing one with the inverse of the other, we obtain the desired quasi-isomorphism |
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151 $$HC_*(M) \iso \operatorname{coinv}(F_*).$$ |
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152 |
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153 %If $M$ is free, that is, a direct sum of copies of |
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154 %$C \tensor C$, then properties \ref{item:hochschild-additive} and |
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155 %\ref{item:hochschild-free} determine $HC_*(M)$. Otherwise, choose some |
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156 %free cover $F \onto M$, and define $K$ to be this map's kernel. Thus we |
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157 %have a short exact sequence $0 \to K \into F \onto M \to 0$, and hence a |
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158 %short exact sequence of complexes $0 \to HC_*(K) \into HC_*(F) \onto HC_*(M) |
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159 %\to 0$. Such a sequence gives a long exact sequence on homology |
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160 %\begin{equation*} |
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161 %%\begin{split} |
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162 %\cdots \to HH_{i+1}(F) \to HH_{i+1}(M) \to HH_i(K) \to HH_i(F) \to \cdots % \\ |
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163 %%\cdots \to HH_1(F) \to HH_1(M) \to HH_0(K) \to HH_0(F) \to HH_0(M). |
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164 %%\end{split} |
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165 %\end{equation*} |
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166 %For any $i \geq 1$, $HH_{i+1}(F) = HH_i(F) = 0$, by properties |
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167 %\ref{item:hochschild-additive} and \ref{item:hochschild-free}, and so |
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168 %$HH_{i+1}(M) \iso HH_i(F)$. For $i=0$, \todo{}. |
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169 % |
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170 %This tells us how to |
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171 %compute every homology group of $HC_*(M)$; we already know $HH_0(M)$ |
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172 %(it's just coinvariants, by property \ref{item:hochschild-coinvariants}), |
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173 %and higher homology groups are determined by lower ones in $HC_*(K)$, and |
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174 %hence recursively as coinvariants of some other bimodule. |
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175 |
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176 The proposition then follows from the following lemmas, establishing that $K_*$ has precisely these required properties. |
157 \begin{lem} |
177 \begin{lem} |
158 \label{lem:hochschild-additive}% |
178 \label{lem:hochschild-additive}% |
159 Directly from the definition, $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$. |
179 Directly from the definition, $K_*(M_1 \oplus M_2) \cong K_*(M_1) \oplus K_*(M_2)$. |
160 \end{lem} |
180 \end{lem} |
161 \begin{lem} |
181 \begin{lem} |
162 \label{lem:hochschild-exact}% |
182 \label{lem:hochschild-exact}% |
163 An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an |
183 An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an |
164 exact sequence $0 \to F_*(M_1) \into F_*(M_2) \onto F_*(M_3) \to 0$. |
184 exact sequence $0 \to K_*(M_1) \into K_*(M_2) \onto K_*(M_3) \to 0$. |
165 \end{lem} |
185 \end{lem} |
166 \begin{lem} |
186 \begin{lem} |
167 \label{lem:hochschild-free}% |
187 \label{lem:hochschild-free}% |
168 $F_*(C\otimes C)$ is quasi-isomorphic to the 0-step complex $C$. |
188 $K_*(C\otimes C)$ is quasi-isomorphic to the 0-step complex $C$. |
169 \end{lem} |
189 \end{lem} |
170 \begin{lem} |
190 \begin{lem} |
171 \label{lem:hochschild-coinvariants}% |
191 \label{lem:hochschild-coinvariants}% |
172 $H_0(F_*(M))$ is isomorphic to the coinvariants of $M$, $M/\langle cm-mc \rangle$. |
192 $H_0(K_*(M))$ is isomorphic to the coinvariants of $M$, $M/\langle cm-mc \rangle$. |
173 \end{lem} |
193 \end{lem} |
174 |
194 |
175 The remainder of this section is devoted to proving Lemmas |
195 The remainder of this section is devoted to proving Lemmas |
176 \ref{lem:module-blob}, |
196 \ref{lem:module-blob}, |
177 \ref{lem:hochschild-exact}, \ref{lem:hochschild-free} and |
197 \ref{lem:hochschild-exact}, \ref{lem:hochschild-free} and |
178 \ref{lem:hochschild-coinvariants}. |
198 \ref{lem:hochschild-coinvariants}. |
179 \end{proof} |
199 \end{proof} |
180 |
200 |
181 \begin{proof}[Proof of Lemma \ref{lem:module-blob}] |
201 \begin{proof}[Proof of Lemma \ref{lem:module-blob}] |
182 We show that $F_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
202 We show that $K_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
183 $F_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
203 $K_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
184 is always a labeled point in $F_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
204 is always a labeled point in $K_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
185 In other words, there is an inclusion map $i: F_*(C) \to \bc_*(S^1)$. |
205 In other words, there is an inclusion map $i: K_*(C) \to \bc_*(S^1)$. |
186 |
206 |
187 We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows. |
207 We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to K_*(C)$ to the inclusion as follows. |
188 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
208 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
189 * is a labeled point in $y$. |
209 * is a labeled point in $y$. |
190 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
210 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
191 Let $x \in \bc_*(S^1)$. |
211 Let $x \in \bc_*(S^1)$. |
192 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
212 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
193 $x$ with $y$. |
213 $x$ with $y$. |
194 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
214 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
195 |
215 |
196 Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
216 Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
197 in a neighborhood $B_\ep$ of *, except perhaps *. |
217 in a neighborhood $B_\ep$ of *, except perhaps *. |
198 Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$. |
218 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
199 \nn{rest of argument goes similarly to above} |
219 \nn{rest of argument goes similarly to above} |
200 \end{proof} |
220 \end{proof} |
201 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
221 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
202 \todo{} |
222 \todo{} |
203 \end{proof} |
223 \end{proof} |
204 \begin{proof}[Proof of Lemma \ref{lem:hochschild-free}] |
224 \begin{proof}[Proof of Lemma \ref{lem:hochschild-free}] |
205 We show that $F_*(C\otimes C)$ is |
225 We show that $K_*(C\otimes C)$ is |
206 quasi-isomorphic to the 0-step complex $C$. |
226 quasi-isomorphic to the 0-step complex $C$. |
207 |
227 |
208 Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of |
228 Let $K'_* \sub K_*(C\otimes C)$ be the subcomplex where the label of |
209 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
229 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
210 We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism. |
230 We will show that the inclusion $i: K'_* \to K_*(C\otimes C)$ is a quasi-isomorphism. |
211 |
231 |
212 Fix a small $\ep > 0$. |
232 Fix a small $\ep > 0$. |
213 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
233 Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
214 Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex |
234 Let $K_*^\ep \sub K_*(C\otimes C)$ be the subcomplex |
215 generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from |
235 generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from |
216 or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$. |
236 or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$. |
217 For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
237 For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
218 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
238 labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
219 (See Figure xxxx.) |
239 (See Figure xxxx.) |
220 Note that $y - s_\ep(y) \in U(B_\ep)$. |
240 Note that $y - s_\ep(y) \in U(B_\ep)$. |
221 \nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.} |
241 \nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.} |
222 |
242 |
223 Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows. |
243 Define a degree 1 chain map $j_\ep : K_*^\ep \to K_*^\ep$ as follows. |
224 Let $x \in F^\ep_*$ be a blob diagram. |
244 Let $x \in K_*^\ep$ be a blob diagram. |
225 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
245 If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
226 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
246 $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
227 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
247 If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
228 Let $y_i$ be the restriction of $z_i$ to $B_\ep$. |
248 Let $y_i$ be the restriction of $z_i$ to $B_\ep$. |
229 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
249 Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
230 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
250 and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
231 Define $j_\ep(x) = \sum x_i$. |
251 Define $j_\ep(x) = \sum x_i$. |
232 \nn{need to check signs coming from blob complex differential} |
252 \nn{need to check signs coming from blob complex differential} |
233 |
253 |
234 Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also. |
254 Note that if $x \in K'_* \cap K_*^\ep$ then $j_\ep(x) \in K'_*$ also. |
235 |
255 |
236 The key property of $j_\ep$ is |
256 The key property of $j_\ep$ is |
237 \eq{ |
257 \eq{ |
238 \bd j_\ep + j_\ep \bd = \id - \sigma_\ep , |
258 \bd j_\ep + j_\ep \bd = \id - \sigma_\ep , |
239 } |
259 } |
240 where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field |
260 where $\sigma_\ep : K_*^\ep \to K_*^\ep$ is given by replacing the restriction $y$ of each field |
241 mentioned in $x \in F^\ep_*$ with $s_\ep(y)$. |
261 mentioned in $x \in K_*^\ep$ with $s_\ep(y)$. |
242 Note that $\sigma_\ep(x) \in F'_*$. |
262 Note that $\sigma_\ep(x) \in K'_*$. |
243 |
263 |
244 If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$ |
264 If $j_\ep$ were defined on all of $K_*(C\otimes C)$, it would show that $\sigma_\ep$ |
245 is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$. |
265 is a homotopy inverse to the inclusion $K'_* \to K_*(C\otimes C)$. |
246 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
266 One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
247 $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$. |
267 $\ep$ and show that $K'_*$ is homotopy equivalent to $K_*(C\otimes C)$. |
248 Instead, we'll be less ambitious and just show that |
268 Instead, we'll be less ambitious and just show that |
249 $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
269 $K'_*$ is quasi-isomorphic to $K_*(C\otimes C)$. |
250 |
270 |
251 If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
271 If $x$ is a cycle in $K_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
252 $x \in F_*^\ep$. |
272 $x \in K_*^\ep$. |
253 (This is true for any chain in $F_*(C\otimes C)$, since chains are sums of |
273 (This is true for any chain in $K_*(C\otimes C)$, since chains are sums of |
254 finitely many blob diagrams.) |
274 finitely many blob diagrams.) |
255 Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map |
275 Then $x$ is homologous to $s_\ep(x)$, which is in $K'_*$, so the inclusion map |
256 $F'_* \sub F_*(C\otimes C)$ is surjective on homology. |
276 $K'_* \sub K_*(C\otimes C)$ is surjective on homology. |
257 If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$ |
277 If $y \in K_*(C\otimes C)$ and $\bd y = x \in K'_*$, then $y \in K_*^\ep$ for some $\ep$ |
258 and |
278 and |
259 \eq{ |
279 \eq{ |
260 \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
280 \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
261 } |
281 } |
262 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
282 Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
263 This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
283 This completes the proof that $K'_*$ is quasi-isomorphic to $K_*(C\otimes C)$. |
264 |
284 |
265 Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob. |
285 Let $K''_* \sub K'_*$ be the subcomplex of $K'_*$ where $*$ is not contained in any blob. |
266 We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence. |
286 We will show that the inclusion $i: K''_* \to K'_*$ is a homotopy equivalence. |
267 |
287 |
268 First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with |
288 First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $K''_*$ and $K'_*$, except with |
269 $S^1$ replaced some (any) neighborhood of $* \in S^1$. |
289 $S^1$ replaced some (any) neighborhood of $* \in S^1$. |
270 Then $G''_*$ and $G'_*$ are both contractible |
290 Then $G''_*$ and $G'_*$ are both contractible |
271 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
291 and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
272 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
292 For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
273 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
293 $G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
276 Thus any cycle lies in the image of the normal blob complex of a disjoint union |
296 Thus any cycle lies in the image of the normal blob complex of a disjoint union |
277 of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}). |
297 of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}). |
278 Actually, we need the further (easy) result that the inclusion |
298 Actually, we need the further (easy) result that the inclusion |
279 $G''_* \to G'_*$ induces an isomorphism on $H_0$. |
299 $G''_* \to G'_*$ induces an isomorphism on $H_0$. |
280 |
300 |
281 Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that |
301 Next we construct a degree 1 map (homotopy) $h: K'_* \to K'_*$ such that |
282 for all $x \in F'_*$ we have |
302 for all $x \in K'_*$ we have |
283 \eq{ |
303 \eq{ |
284 x - \bd h(x) - h(\bd x) \in F''_* . |
304 x - \bd h(x) - h(\bd x) \in K''_* . |
285 } |
305 } |
286 Since $F'_0 = F''_0$, we can take $h_0 = 0$. |
306 Since $K'_0 = K''_0$, we can take $h_0 = 0$. |
287 Let $x \in F'_1$, with single blob $B \sub S^1$. |
307 Let $x \in K'_1$, with single blob $B \sub S^1$. |
288 If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$. |
308 If $* \notin B$, then $x \in K''_1$ and we define $h_1(x) = 0$. |
289 If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$). |
309 If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$). |
290 Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$. |
310 Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$. |
291 Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$. |
311 Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$. |
292 Define $h_1(x) = y$. |
312 Define $h_1(x) = y$. |
293 The general case is similar, except that we have to take lower order homotopies into account. |
313 The general case is similar, except that we have to take lower order homotopies into account. |
294 Let $x \in F'_k$. |
314 Let $x \in K'_k$. |
295 If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$. |
315 If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$. |
296 Otherwise, let $B$ be the outermost blob of $x$ containing $*$. |
316 Otherwise, let $B$ be the outermost blob of $x$ containing $*$. |
297 By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$. |
317 By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$. |
298 So $x' \in G'_l$ for some $l \le k$. |
318 So $x' \in G'_l$ for some $l \le k$. |
299 Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
319 Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
300 Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
320 Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
301 Define $h_k(x) = y \bullet p$. |
321 Define $h_k(x) = y \bullet p$. |
302 This completes the proof that $i: F''_* \to F'_*$ is a homotopy equivalence. |
322 This completes the proof that $i: K''_* \to K'_*$ is a homotopy equivalence. |
303 \nn{need to say above more clearly and settle on notation/terminology} |
323 \nn{need to say above more clearly and settle on notation/terminology} |
304 |
324 |
305 Finally, we show that $F''_*$ is contractible. |
325 Finally, we show that $K''_*$ is contractible. |
306 \nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now} |
326 \nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now} |
307 Let $x$ be a cycle in $F''_*$. |
327 Let $x$ be a cycle in $K''_*$. |
308 The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a |
328 The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a |
309 ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
329 ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
310 Adding $B$ as a blob to $x$ gives a contraction. |
330 Adding $B$ as a blob to $x$ gives a contraction. |
311 \nn{need to say something else in degree zero} |
331 \nn{need to say something else in degree zero} |
312 \end{proof} |
332 \end{proof} |