160 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
160 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
161 \end{equation*} |
161 \end{equation*} |
162 \end{itemize} |
162 \end{itemize} |
163 \end{property} |
163 \end{property} |
164 |
164 |
165 \nn{add product formula? $n$-dimensional fat graph operad stuff?} |
165 |
|
166 |
|
167 \begin{property}[Relation to mapping spaces] |
|
168 There is a version of the blob complex for $C$ an $A_\infty$ $n$-category |
|
169 instead of a garden variety $n$-category. |
|
170 |
|
171 Let $\pi^\infty_{\le n}(W)$ denote the $A_\infty$ $n$-category based on maps |
|
172 $B^n \to W$. |
|
173 (The case $n=1$ is the usual $A_\infty$ category of paths in $W$.) |
|
174 Then $\bc_*(M, \pi^\infty_{\le n}(W))$ is |
|
175 homotopy equivalent to $C_*(\{\text{maps}\; M \to W\})$. |
|
176 \end{property} |
|
177 |
|
178 |
|
179 |
|
180 |
|
181 \begin{property}[Product formula] |
|
182 Let $M^n = Y^{n-k}\times W^k$ and let $C$ be an $n$-category. |
|
183 Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology. |
|
184 Then |
|
185 \[ |
|
186 \bc_*(Y^{n-k}\times W^k, C) \simeq \bc_*(W, A_*(Y)) . |
|
187 \] |
|
188 \nn{say something about general fiber bundles?} |
|
189 \end{property} |
|
190 |
|
191 |
|
192 |
|
193 |
|
194 \begin{property}[Higher dimensional Deligne conjecture] |
|
195 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
|
196 |
|
197 The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
|
198 of $n$-manifolds |
|
199 $R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms |
|
200 $f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$. |
|
201 (Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to |
|
202 the $n$-ball is equivalent to the little $n{+}1$-disks operad.) |
|
203 |
|
204 If $A$ and $B$ are $n$-manifolds sharing the same boundary, define |
|
205 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be |
|
206 $A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both |
|
207 (collections of) complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. |
|
208 The ``holes" in the above |
|
209 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. |
|
210 \end{property} |
|
211 |
|
212 |
|
213 |
|
214 |
|
215 |
|
216 |
|
217 |
166 |
218 |
167 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
219 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
168 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
220 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
169 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
221 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
170 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
222 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
171 and Property \ref{property:gluing} in \S \ref{sec:gluing}. |
223 and Property \ref{property:gluing} in \S \ref{sec:gluing}. |
|
224 \nn{need to say where the remaining properties are proved.} |