86 |
86 |
87 We then show that blob homology enjoys the following properties. |
87 We then show that blob homology enjoys the following properties. |
88 |
88 |
89 \begin{property}[Functoriality] |
89 \begin{property}[Functoriality] |
90 \label{property:functoriality}% |
90 \label{property:functoriality}% |
91 Blob homology is functorial with respect to homeomorphisms. That is, |
91 The blob complex is functorial with respect to homeomorphisms. That is, |
92 for fixed $n$-category / fields $\cC$, the association |
92 for fixed $n$-category / fields $\cC$, the association |
93 \begin{equation*} |
93 \begin{equation*} |
94 X \mapsto \bc_*^{\cC}(X) |
94 X \mapsto \bc_*^{\cC}(X) |
95 \end{equation*} |
95 \end{equation*} |
96 is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. |
96 is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. |
97 \end{property} |
97 \end{property} |
98 |
98 |
99 \nn{should probably also say something about being functorial in $\cC$} |
99 The blob complex is also functorial with respect to $\cC$, although we will not address this in detail here. |
100 |
100 |
101 \begin{property}[Disjoint union] |
101 \begin{property}[Disjoint union] |
102 \label{property:disjoint-union} |
102 \label{property:disjoint-union} |
103 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
103 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
104 \begin{equation*} |
104 \begin{equation*} |
105 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
105 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
106 \end{equation*} |
106 \end{equation*} |
107 \end{property} |
107 \end{property} |
108 |
108 |
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109 If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$-submanifold of its boundary, write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. Note that this includes the case of gluing two disjoint manifolds together. |
109 \begin{property}[Gluing map] |
110 \begin{property}[Gluing map] |
110 \label{property:gluing-map}% |
111 \label{property:gluing-map}% |
111 If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, |
112 %If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
112 there is a chain map |
113 %\begin{equation*} |
113 \begin{equation*} |
114 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
114 \gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
115 %\end{equation*} |
115 \end{equation*} |
116 Given a gluing $X_\mathrm{cut} \to X_\mathrm{glued}$, there is |
116 \nn{alternate version:}Given a gluing $X_\mathrm{cut} \to X_\mathrm{gl}$, there is |
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117 a natural map |
117 a natural map |
118 \[ |
118 \[ |
119 \bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{gl}) . |
119 \bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) . |
120 \] |
120 \] |
121 (Natural with respect to homeomorphisms, and also associative with respect to iterated gluings.) |
121 (Natural with respect to homeomorphisms, and also associative with respect to iterated gluings.) |
122 \end{property} |
122 \end{property} |
123 |
123 |
124 \begin{property}[Contractibility] |
124 \begin{property}[Contractibility] |
125 \label{property:contractibility}% |
125 \label{property:contractibility}% |
126 \todo{Err, requires a splitting?} |
126 \todo{Err, requires a splitting?} |
127 The blob complex for an $n$-category on an $n$-ball is quasi-isomorphic to its $0$-th homology. |
127 The blob complex on an $n$-ball is contractible in the sense that it is quasi-isomorphic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the original $n$-category $\cC$. |
128 \begin{equation} |
128 \begin{equation} |
129 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))} |
129 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))} |
130 \end{equation} |
130 \end{equation} |
131 \todo{Say that this is just the original $n$-category?} |
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132 \end{property} |
131 \end{property} |
133 |
132 |
134 \begin{property}[Skein modules] |
133 \begin{property}[Skein modules] |
135 \label{property:skein-modules}% |
134 \label{property:skein-modules}% |
136 The $0$-th blob homology of $X$ is the usual |
135 The $0$-th blob homology of $X$ is the usual |
144 \begin{property}[Hochschild homology when $X=S^1$] |
143 \begin{property}[Hochschild homology when $X=S^1$] |
145 \label{property:hochschild}% |
144 \label{property:hochschild}% |
146 The blob complex for a $1$-category $\cC$ on the circle is |
145 The blob complex for a $1$-category $\cC$ on the circle is |
147 quasi-isomorphic to the Hochschild complex. |
146 quasi-isomorphic to the Hochschild complex. |
148 \begin{equation*} |
147 \begin{equation*} |
149 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & HC_*(\cC)} |
148 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & \HC_*(\cC)} |
150 \end{equation*} |
149 \end{equation*} |
151 \end{property} |
150 \end{property} |
152 |
151 |
153 \nn{$HC_*$ or $\rm{Hoch}_*$?} |
152 |
154 |
153 \begin{property}[$C_*(\Diff(-))$ action] |
155 \begin{property}[$C_*(\Diff(\cdot))$ action] |
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156 \label{property:evaluation}% |
154 \label{property:evaluation}% |
157 There is a chain map |
155 There is a chain map |
158 \begin{equation*} |
156 \begin{equation*} |
159 \ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
157 \ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
160 \end{equation*} |
158 \end{equation*} |
173 \end{equation*} |
171 \end{equation*} |
174 \nn{should probably say something about associativity here (or not?)} |
172 \nn{should probably say something about associativity here (or not?)} |
175 \nn{maybe do self-gluing instead of 2 pieces case} |
173 \nn{maybe do self-gluing instead of 2 pieces case} |
176 \end{property} |
174 \end{property} |
177 |
175 |
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176 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
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177 instead of a garden variety $n$-category. |
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178 |
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179 \begin{property}[Product formula] |
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180 Let $M^n = Y^{n-k}\times W^k$ and let $\cC$ be an $n$-category. |
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181 Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology, which associates to each $k$-ball $D$ the complex $A_*(Y)(D) = \bc_*(Y \times D, \cC)$. |
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182 Then |
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183 \[ |
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184 \bc_*(Y^{n-k}\times W^k, \cC) \simeq \bc_*(W, A_*(Y)) . |
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185 \] |
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186 Note on the right here we have the version of the blob complex for $A_\infty$ $n$-categories. |
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187 \nn{say something about general fiber bundles?} |
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188 \end{property} |
178 |
189 |
179 \begin{property}[Gluing formula] |
190 \begin{property}[Gluing formula] |
180 \label{property:gluing}% |
191 \label{property:gluing}% |
181 \mbox{}% <-- gets the indenting right |
192 \mbox{}% <-- gets the indenting right |
182 \begin{itemize} |
193 \begin{itemize} |
184 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
195 naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
185 |
196 |
186 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
197 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
187 $A_\infty$ module for $\bc_*(Y \times I)$. |
198 $A_\infty$ module for $\bc_*(Y \times I)$. |
188 |
199 |
189 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
200 \item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of |
190 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
201 $\bc_*(X_\text{cut})$ as an $\bc_*(Y \times I)$-bimodule. |
191 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
202 \begin{equation*} |
192 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
203 \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow |
193 \begin{equation*} |
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194 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
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195 \end{equation*} |
204 \end{equation*} |
196 \end{itemize} |
205 \end{itemize} |
197 \end{property} |
206 \end{property} |
198 |
207 |
199 |
208 |
200 |
209 |
201 \begin{property}[Relation to mapping spaces] |
210 \begin{property}[Relation to mapping spaces] |
202 There is a version of the blob complex for $C$ an $A_\infty$ $n$-category |
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203 instead of a garden variety $n$-category. |
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204 |
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205 Let $\pi^\infty_{\le n}(W)$ denote the $A_\infty$ $n$-category based on maps |
211 Let $\pi^\infty_{\le n}(W)$ denote the $A_\infty$ $n$-category based on maps |
206 $B^n \to W$. |
212 $B^n \to W$. |
207 (The case $n=1$ is the usual $A_\infty$ category of paths in $W$.) |
213 (The case $n=1$ is the usual $A_\infty$ category of paths in $W$.) |
208 Then $\bc_*(M, \pi^\infty_{\le n}(W))$ is |
214 Then |
209 homotopy equivalent to $C_*(\{\text{maps}\; M \to W\})$. |
215 $$\bc_*(M, \pi^\infty_{\le n}(W) \simeq \CM{M}{W}.$$ |
210 \end{property} |
216 \end{property} |
211 |
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212 |
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213 |
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214 |
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215 \begin{property}[Product formula] |
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216 Let $M^n = Y^{n-k}\times W^k$ and let $C$ be an $n$-category. |
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217 Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology. |
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218 Then |
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219 \[ |
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220 \bc_*(Y^{n-k}\times W^k, C) \simeq \bc_*(W, A_*(Y)) . |
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221 \] |
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222 \nn{say something about general fiber bundles?} |
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223 \end{property} |
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224 |
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225 |
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226 |
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227 |
217 |
228 \begin{property}[Higher dimensional Deligne conjecture] |
218 \begin{property}[Higher dimensional Deligne conjecture] |
229 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
219 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
230 |
220 \end{property} |
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221 \begin{rem} |
231 The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
222 The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
232 of $n$-manifolds |
223 of $n$-manifolds |
233 $R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms |
224 $R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms |
234 $f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$. |
225 $f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$. |
235 (Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to |
226 (Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to |
236 the $n$-ball is equivalent to the little $n{+}1$-disks operad.) |
227 the $n$-ball is equivalent to the little $n{+}1$-disks operad.) |
237 |
228 If $A$ and $B$ are $n$-manifolds sharing the same boundary, we define |
238 If $A$ and $B$ are $n$-manifolds sharing the same boundary, define |
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239 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be |
229 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be |
240 $A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both |
230 $A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both |
241 (collections of) complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. |
231 collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. |
242 The ``holes" in the above |
232 The ``holes" in the above |
243 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. |
233 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. |
244 \end{property} |
234 \end{rem} |
245 |
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246 |
235 |
247 |
236 |
248 |
237 |
249 |
238 |
250 |
239 |