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1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{Higher-dimensional Deligne conjecture} |
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4 \label{sec:deligne} |
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5 In this section we discuss Property \ref{property:deligne}, |
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6 \begin{prop}[Higher dimensional Deligne conjecture] |
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7 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
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8 \end{prop} |
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9 |
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10 The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
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11 of $n$-manifolds |
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12 $R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms |
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13 $f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$. |
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14 (Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to |
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15 the $n$-ball is equivalent to the little $n{+}1$-disks operad.) |
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16 |
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17 If $A$ and $B$ are $n$-manifolds sharing the same boundary, we define |
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18 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be |
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19 $A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both |
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20 collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. |
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21 The ``holes" in the above |
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22 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. |