809 %\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
809 %\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
810 |
810 |
811 \section{Deligne conjecture for $n$-categories} |
811 \section{Deligne conjecture for $n$-categories} |
812 \label{sec:applications} |
812 \label{sec:applications} |
813 |
813 |
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814 Let $M$ and $N$ be $n$-manifolds with common boundary $E$. |
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815 Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$ |
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816 acts on $\bc_*(M)$ and $\bc_*(N)$. |
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817 Let $\hom_A(\bc_*(M), \bc_*(N))$ denote the chain complex of $A_\infty$ module maps |
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818 from $\bc_*(M)$ to $\bc_*(N)$. |
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819 Let $R$ be another $n$-manifold with boundary $-E$. |
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820 There is a chain map |
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821 \[ |
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822 \hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) . |
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823 \] |
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824 We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
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825 replaces it with $N$, yielding $N\cup_E R$. |
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826 |
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827 Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
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828 blob complex. |
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829 \nn{...} |
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830 |
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831 |
814 \begin{thm}[Higher dimensional Deligne conjecture] |
832 \begin{thm}[Higher dimensional Deligne conjecture] |
815 \label{thm:deligne} |
833 \label{thm:deligne} |
816 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
834 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
817 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
835 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
818 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
836 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
862 %% \appendix[Appendix Title] |
880 %% \appendix[Appendix Title] |
863 |
881 |
864 \begin{acknowledgments} |
882 \begin{acknowledgments} |
865 It is a pleasure to acknowledge helpful conversations with |
883 It is a pleasure to acknowledge helpful conversations with |
866 Kevin Costello, |
884 Kevin Costello, |
867 Mike Freedman, |
885 Michael Freedman, |
868 Justin Roberts, |
886 Justin Roberts, |
869 and |
887 and |
870 Peter Teichner. |
888 Peter Teichner. |
871 We also thank the Aspen Center for Physics for providing a pleasant and productive |
889 We also thank the Aspen Center for Physics for providing a pleasant and productive |
872 environment during the last stages of this project. |
890 environment during the last stages of this project. |