631 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
631 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
632 The restriction of the string diagram to innermost blobs is required to be ``null" in the sense that |
632 The restriction of the string diagram to innermost blobs is required to be ``null" in the sense that |
633 it evaluates to a zero $n$-morphism of $C$. |
633 it evaluates to a zero $n$-morphism of $C$. |
634 The next few paragraphs describe this in more detail. |
634 The next few paragraphs describe this in more detail. |
635 |
635 |
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636 We will call a string diagram on a manifold a ``field". |
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637 (See \cite{1009.5025} for a more general notion of field.) |
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638 |
636 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
639 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
637 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
640 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
638 each $B_i$ appears as a connected component of one of the $M_j$. |
641 each $B_i$ appears as a connected component of one of the $M_j$. |
639 Note that this forces the balls to be pairwise either disjoint or nested. |
642 Note that this forces the balls to be pairwise either disjoint or nested. |
640 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. |
643 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. |
649 \item a linear combination $s$ of string diagrams on $W$, |
652 \item a linear combination $s$ of string diagrams on $W$, |
650 \end{itemize} |
653 \end{itemize} |
651 such that |
654 such that |
652 \begin{itemize} |
655 \begin{itemize} |
653 \item there is a permissible decomposition of $W$, compatible with the $k$ blobs, such that |
656 \item there is a permissible decomposition of $W$, compatible with the $k$ blobs, such that |
654 $s$ is the product of linear combinations of string diagrams $s_i$ on the initial pieces $X_i$ of the decomposition |
657 $s$ is the product of linear combinations of fields $s_i$ on the initial pieces $X_i$ of the decomposition |
655 (for fixed restrictions to the boundaries of the pieces), |
658 (for fixed restrictions to the boundaries of the pieces), |
656 \item the $s_i$'s corresponding to innermost blobs evaluate to zero in $\cC$, and |
659 \item the $s_i$'s corresponding to innermost blobs evaluate to zero in $\cC$, and |
657 \item the $s_i$'s corresponding to the other pieces are single string diagrams (linear combinations with only one term). |
660 \item the $s_i$'s corresponding to the other pieces are single fields (linear combinations with only one term). |
658 \end{itemize} |
661 \end{itemize} |
659 %that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
662 %that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
660 \nn{yech} |
663 \nn{yech} |
661 We call such linear combinations which evaluate to zero on a blob $B$ a ``null field on $B$". |
664 We call such linear combinations which evaluate to zero on a blob $B$ a ``null field on $B$". |
662 |
665 |
663 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
666 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
664 |
667 |
665 \nn{KW: I have not finished changng terminology from ``field" to ``string diagram"} |
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666 |
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667 We now spell this out for some small values of $k$. |
668 We now spell this out for some small values of $k$. |
668 For $k=0$, the $0$-blob group is simply linear combinations of string diagrams on $W$. |
669 For $k=0$, the $0$-blob group is simply linear combinations of fields (string diagrams) on $W$. |
669 For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. |
670 For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field. |
670 The differential simply forgets the ball. |
671 The differential simply forgets the ball. |
671 Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball. |
672 Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball. |
672 |
673 |
673 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 $B_2$. |
674 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 $B_2$. |