pnas/pnas.tex
changeset 579 0bd4aca0546b
parent 578 d2c073d785fd
child 580 99611dfed1f3
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   597 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains.
   597 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains.
   598 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad,
   598 Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad,
   599 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball.
   599 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball.
   600 \end{thm}
   600 \end{thm}
   601 
   601 
   602 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.
   602 An $n$-dimensional surgery cylinder is an alternating sequence of mapping cylinders and surgeries, modulo changing the order of distant surgeries, and conjugating a 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 
   604 \todo{Explain blob cochains}
   606 \todo{Explain blob cochains}
   605 \todo{Sketch proof}
   607 \todo{Sketch proof}
   606 
   608 
   607 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
   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