637 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
637 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
638 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
638 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
639 each $B_i$ appears as a connected component of one of the $M_j$. |
639 each $B_i$ appears as a connected component of one of the $M_j$. |
640 Note that this forces the balls to be pairwise either disjoint or nested. |
640 Note that this forces the balls to be pairwise either disjoint or nested. |
641 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. |
641 Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. |
642 These pieces need not be manifolds, but they do automatically have permissible decompositions. |
642 These pieces need not be manifolds, |
|
643 but they can be further subdivided into pieces which are manifolds |
|
644 and which fit into a permissible decomposition of $W$. |
643 |
645 |
644 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. |
646 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. |
645 A $k$-blob diagram consists of |
647 A $k$-blob diagram consists of |
646 \begin{itemize} |
648 \begin{itemize} |
647 \item a permissible collection of $k$ embedded balls, and |
649 \item a permissible collection of $k$ embedded balls, and |
648 \item for each resulting piece of $W$, a field, |
650 \item a linear combination $s$ of string diagrams on $W$, |
649 \end{itemize} |
651 \end{itemize} |
650 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
652 such that |
651 We call such a field a ``null field on $B$". |
653 \begin{itemize} |
|
654 \item there is a permissible decomposition of $W$, compatible with the $k$ blobs, such that |
|
655 $s$ is the product of linear combinations of string diagrams $s_i$ on the initial pieces $X_i$ of the decomposition |
|
656 (for fixed restrictions to the boundaries of the pieces), |
|
657 \item the $s_i$'s corresponding to innermost blobs evaluate to zero in $\cC$, and |
|
658 \item the $s_i$'s corresponding to the other pieces are single string diagrams (linear combinations with only one term). |
|
659 \end{itemize} |
|
660 %that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$. |
|
661 \nn{yech} |
|
662 We call such linear combinations which evaluate to zero on a blob $B$ a ``null field on $B$". |
652 |
663 |
653 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
664 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
654 |
665 |
|
666 \nn{KW: I have not finished changng terminology from ``field" to ``string diagram"} |
|
667 |
655 We now spell this out for some small values of $k$. |
668 We now spell this out for some small values of $k$. |
656 For $k=0$, the $0$-blob group is simply fields on $W$. |
669 For $k=0$, the $0$-blob group is simply linear combinations of string diagrams on $W$. |
657 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. |
658 The differential simply forgets the ball. |
671 The differential simply forgets the ball. |
659 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. |
660 |
673 |
661 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 $B_2$. |