31 |
31 |
32 [some things to cover in the intro] |
32 [some things to cover in the intro] |
33 \begin{itemize} |
33 \begin{itemize} |
34 \item explain relation between old and new blob complex definitions |
34 \item explain relation between old and new blob complex definitions |
35 \item overview of sections |
35 \item overview of sections |
36 \item ?? we have resisted the temptation |
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37 (actually, it was not a temptation) to state things in the greatest |
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38 generality possible |
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39 \item related: we are being unsophisticated from a homotopy theory point of |
36 \item related: we are being unsophisticated from a homotopy theory point of |
40 view and using chain complexes in many places where we could be by with spaces |
37 view and using chain complexes in many places where we could be by with spaces |
41 \item ? one of the points we make (far) below is that there is not really much |
38 \item ? one of the points we make (far) below is that there is not really much |
42 difference between (a) systems of fields and local relations and (b) $n$-cats; |
39 difference between (a) systems of fields and local relations and (b) $n$-cats; |
43 thus we tend to switch between talking in terms of one or the other |
40 thus we tend to switch between talking in terms of one or the other |
248 \end{property} |
245 \end{property} |
249 |
246 |
250 \begin{property}[Higher dimensional Deligne conjecture] |
247 \begin{property}[Higher dimensional Deligne conjecture] |
251 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
248 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
252 \end{property} |
249 \end{property} |
253 \begin{rem} |
250 See \S \ref{sec:deligne} for an explanation of the terms appearing here, and (in a later edition of this paper) the proof. |
254 The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
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255 of $n$-manifolds |
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256 $R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms |
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257 $f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$. |
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258 (Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to |
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259 the $n$-ball is equivalent to the little $n{+}1$-disks operad.) |
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260 If $A$ and $B$ are $n$-manifolds sharing the same boundary, we define |
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261 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be |
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262 $A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both |
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263 collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. |
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264 The ``holes" in the above |
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265 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. |
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266 \end{rem} |
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267 |
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268 |
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269 |
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270 |
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271 |
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272 |
251 |
273 |
252 |
274 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
253 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
275 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
254 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
276 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
255 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
277 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
256 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
278 and Property \ref{property:gluing} in \S \ref{sec:gluing}. |
257 and Property \ref{property:gluing} in \S \ref{sec:gluing}. |
279 \nn{need to say where the remaining properties are proved.} |
258 \nn{need to say where the remaining properties are proved.} |
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259 |
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260 \subsection{Future directions} |
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261 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds. |
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262 |
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263 Many results in Hochschild homology can be understood `topologically' via the blob complex. For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ simply corresponds to the gluing operation on $\bc_*(S^1 \times I, A)$, but haven't investigated the details. |
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264 |
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265 Most importantly, however, \nn{applications!} \nn{$n=2$ cases, contact, Kh} |
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266 |
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267 |
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268 \subsection{Thanks and acknowledgements} |