595 |
595 |
596 |
596 |
597 \begin{thm}[Higher dimensional Deligne conjecture] |
597 \begin{thm}[Higher dimensional Deligne conjecture] |
598 \label{thm:deligne} |
598 \label{thm:deligne} |
599 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
599 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
600 Since the little $n{+}1$-balls operad is a suboperad of the $n$-dimensional surgery cylinder operad, |
600 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, |
601 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
601 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
602 \end{thm} |
602 \end{thm} |
603 \nn{Explain and sketch} |
603 |
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604 An $n$-dimensional surgery cylinder is an alternating sequence of mapping cylinders and surgeries, modulo changing the order of distant surgeries, and conjugating the submanifold not modified in a surgery by a homeomorphism. See Figure \ref{delfig2}. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
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605 |
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606 \todo{Explain blob cochains} |
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607 \todo{Sketch proof} |
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608 |
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609 The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923} gives a map |
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610 \[ |
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611 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
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612 \to Hoch^*(C, C), |
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613 \] |
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614 which we now see to be a specialization of Theorem \ref{thm:deligne}. |
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615 |
604 |
616 |
605 %% == end of paper: |
617 %% == end of paper: |
606 |
618 |
607 %% Optional Materials and Methods Section |
619 %% Optional Materials and Methods Section |
608 %% The Materials and Methods section header will be added automatically. |
620 %% The Materials and Methods section header will be added automatically. |
677 \end{equation*} |
689 \end{equation*} |
678 \caption{A small part of $\cell(W)$} |
690 \caption{A small part of $\cell(W)$} |
679 \label{partofJfig} |
691 \label{partofJfig} |
680 \end{figure} |
692 \end{figure} |
681 |
693 |
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694 \begin{figure} |
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695 $$\mathfig{.4}{deligne/manifolds}$$ |
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696 \caption{An $n$-dimensional surgery cylinder}\label{delfig2} |
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697 \end{figure} |
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698 |
682 |
699 |
683 %% For Tables, put caption above table |
700 %% For Tables, put caption above table |
684 %% |
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686 %% and not have a period at the end |
703 %% and not have a period at the end |