diff -r 87b1507ebc56 -r d62402fc028e text/blobdef.tex --- a/text/blobdef.tex Sat Jul 03 15:14:24 2010 -0600 +++ b/text/blobdef.tex Sat Jul 03 19:57:58 2010 -0600 @@ -87,7 +87,7 @@ A nested 2-blob diagram consists of \begin{itemize} -\item A pair of nested balls (blobs) $B_1 \sub B_2 \sub X$. +\item A pair of nested balls (blobs) $B_1 \subseteq B_2 \subseteq X$. \item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ (for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$). \item A field $r \in \cC(X \setminus B_2; c_2)$. @@ -109,19 +109,35 @@ \begin{eqnarray*} \bc_2(X) & \deq & \left( - \bigoplus_{B_1, B_2 \text{disjoint}} \bigoplus_{c_1, c_2} + \bigoplus_{B_1, B_2\; \text{disjoint}} \bigoplus_{c_1, c_2} U(B_1; c_1) \otimes U(B_2; c_2) \otimes \lf(X\setmin (B_1\cup B_2); c_1, c_2) - \right) \\ - && \bigoplus \left( + \right) \bigoplus \\ + && \quad\quad \left( \bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2} U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) \right) . \end{eqnarray*} For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign (rather than a new, linearly independent 2-blob diagram). +\noop{ \nn{Hmm, I think we should be doing this for nested blobs too -- we shouldn't force the linear indexing of the blobs to have anything to do with the partial ordering by inclusion -- this is what happens below} +\nn{KW: I think adding that detail would only add distracting clutter, and the statement is true as written (in the sense that it yields a vector space isomorphic to the general def below} +} + +Before describing the general case we should say more precisely what we mean by +disjoint and nested blobs. +Disjoint will mean disjoint interiors. +Nested blobs are allowed to coincide, or to have overlapping boundaries. +Blob are allowed to intersect $\bd X$. +However, we require of any collection of blobs $B_1,\ldots,B_k \subseteq X$ that +$X$ is decomposable along the union of the boundaries of the blobs. +\nn{need to say more here. we want to be able to glue blob diagrams, but avoid pathological +behavior} +\nn{need to allow the case where $B\to X$ is not an embedding +on $\bd B$. this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$ +and blobs are allowed to meet $\bd X$.} Now for the general case. A $k$-blob diagram consists of @@ -132,9 +148,6 @@ (The case $B_i = B_j$ is allowed. If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) If a blob has no other blobs strictly contained in it, we call it a twig blob. -\nn{need to allow the case where $B\to X$ is not an embedding -on $\bd B$. this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$ -and blobs are allowed to meet $\bd X$.} \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. (These are implied by the data in the next bullets, so we usually suppress them from the notation.)