379 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
379 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
380 each $B_i$ appears as a connected component of one of the $M_j$. Note that this allows the balls to be pairwise either disjoint or nested. Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. These pieces need not be manifolds, but they do automatically have permissible decompositions. |
380 each $B_i$ appears as a connected component of one of the $M_j$. Note that this allows the balls to be pairwise either disjoint or nested. Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. These pieces need not be manifolds, but they do automatically have permissible decompositions. |
381 |
381 |
382 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
382 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
383 \begin{itemize} |
383 \begin{itemize} |
384 \item a permissible collection of $k$ embedded balls (called `blobs') in $W$, |
384 \item a permissible collection of $k$ embedded balls, |
385 \item an ordering of the balls, and |
385 \item an ordering of the balls, and |
386 \item for each resulting piece of $W$, a field, |
386 \item for each resulting piece of $W$, a field, |
387 \end{itemize} |
387 \end{itemize} |
388 such that for any innermost blob $B$, the field on $B$ goes to zero under the composition map from $\cC$. |
388 such that for any innermost blob $B$, the field on $B$ goes to zero under the composition map from $\cC$. We call such a field a `null field on $B$'. |
389 |
389 |
390 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with signs given by the ordering. |
390 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with signs given by the ordering. |
391 |
391 |
392 \todo{Say why this really is the homotopy colimit} |
392 \todo{Say why this really is the homotopy colimit} |
393 \todo{Spell out $k=0, 1, 2$} |
393 |
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394 We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field that that ball is a null field. The differential simply forgets the ball. Thus we see that $H_0$ of the blob complex is the quotient of fields by null fields. |
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395 |
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396 For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined. |
394 |
397 |
395 \section{Properties of the blob complex} |
398 \section{Properties of the blob complex} |
396 \subsection{Formal properties} |
399 \subsection{Formal properties} |
397 \label{sec:properties} |
400 \label{sec:properties} |
398 The blob complex enjoys the following list of formal properties. |
401 The blob complex enjoys the following list of formal properties. |
588 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
591 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
589 \end{thm} |
592 \end{thm} |
590 |
593 |
591 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
594 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
592 Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. |
595 Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. |
593 |
596 \todo{sketch proof} |
594 |
597 |
595 \begin{thm}[Higher dimensional Deligne conjecture] |
598 \begin{thm}[Higher dimensional Deligne conjecture] |
596 \label{thm:deligne} |
599 \label{thm:deligne} |
597 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
600 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, |
601 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. |
602 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
600 \end{thm} |
603 \end{thm} |
601 |
604 |
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. |
605 An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
603 |
606 |
604 |
607 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module. |
605 |
608 |
606 \todo{Explain blob cochains} |
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607 \todo{Sketch proof} |
609 \todo{Sketch proof} |
608 |
610 |
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 |
611 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 |
610 \[ |
612 \[ |
611 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
613 C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
612 \to Hoch^*(C, C), |
614 \to Hoch^*(C, C), |
613 \] |
615 \] |
614 which we now see to be a specialization of Theorem \ref{thm:deligne}. |
616 which we now see to be a specialization of Theorem \ref{thm:deligne}. |