181 \begin{equation*} |
181 \begin{equation*} |
182 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
182 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
183 \end{equation*} |
183 \end{equation*} |
184 \end{property} |
184 \end{property} |
185 |
185 |
186 Here $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
186 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
187 \begin{property}[$C_*(\Homeo(-))$ action] |
187 \begin{property}[$C_*(\Homeo(-))$ action] |
188 \label{property:evaluation}% |
188 \label{property:evaluation}% |
189 There is a chain map |
189 There is a chain map |
190 \begin{equation*} |
190 \begin{equation*} |
191 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
191 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
192 \end{equation*} |
192 \end{equation*} |
193 |
193 |
194 Restricted to $C_0(\Homeo(X))$ this is just the action of homeomorphisms described in Property \ref{property:functoriality}. Further, for |
194 Restricted to $C_0(\Homeo(X))$ this is just the action of homeomorphisms described in Property \ref{property:functoriality}. |
195 any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram |
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196 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
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197 \begin{equation*} |
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198 \xymatrix{ |
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199 \CH{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\ |
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200 \CH{X_1} \otimes \CH{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
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201 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
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202 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
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203 } |
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204 \end{equation*} |
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205 \nn{should probably say something about associativity here (or not?)} |
195 \nn{should probably say something about associativity here (or not?)} |
206 Further, for |
196 |
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197 For |
207 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
198 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
208 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
199 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
209 \begin{equation*} |
200 \begin{equation*} |
210 \xymatrix@C+2cm{ |
201 \xymatrix@C+2cm{ |
211 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) \\ |
202 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) \\ |
212 \CH{X} \otimes \bc_*(X) |
203 \CH{X} \otimes \bc_*(X) |
213 \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
204 \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
214 \bc_*(X) \ar[u]_{\gl_Y} |
205 \bc_*(X) \ar[u]_{\gl_Y} |
215 } |
206 } |
216 \end{equation*} |
207 \end{equation*} |
217 \end{property} |
208 |
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209 \nn{unique up to homotopy?} |
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210 \end{property} |
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211 |
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212 Since the blob complex is functorial in the manifold $X$, we can use this to build a chain map |
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213 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
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214 satisfying corresponding conditions. |
218 |
215 |
219 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields, as well as the notion of an $A_\infty$ $n$-category. |
216 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields, as well as the notion of an $A_\infty$ $n$-category. |
220 |
217 |
221 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
218 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
222 \label{property:blobs-ainfty} |
219 \label{property:blobs-ainfty} |