diff -r c5a35886cd82 -r 93ce0ba3d2d7 text/blobdef.tex --- a/text/blobdef.tex Mon Jul 12 17:29:25 2010 -0600 +++ b/text/blobdef.tex Wed Jul 14 11:06:11 2010 -0600 @@ -4,38 +4,37 @@ \label{sec:blob-definition} Let $X$ be an $n$-manifold. -Let $\cC$ be a fixed system of fields (enriched over Vect) and local relations. -(If $\cC$ is not enriched over Vect, we can make it so by allowing finite -linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.) +Let $\cC$ be a fixed system of fields and local relations. +We'll assume it is enriched over \textbf{Vect}, and if it is not we can make it so by allowing finite +linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$. -In this section we will usually suppress boundary conditions on $X$ from the notation -(e.g. write $\lf(X)$ instead of $\lf(X; c)$). +In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\lf(X)$ instead of $\lf(X; c)$. We want to replace the quotient \[ A(X) \deq \lf(X) / U(X) \] -of the previous section with a resolution +of Definition \ref{defn:TQFT-invariant} with a resolution \[ \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . \] -We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. +We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. \todo{create a numbered definition for the general case} We of course define $\bc_0(X) = \lf(X)$. (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. We'll omit this sort of detail in the rest of this section.) -In other words, $\bc_0(X)$ is just the vector space of fields on $X$. +In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. We want the vector space $\bc_1(X)$ to capture `the space of all local relations that can be imposed on $\bc_0(X)$'. -Thus we say a $1$-blob diagram consists of +Thus we say a $1$-blob diagram consists of: \begin{itemize} \item An embedded closed ball (``blob") $B \sub X$. \item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$. \item A field $r \in \cC(X \setmin B; c)$. \item A local relation field $u \in U(B; c)$. \end{itemize} -(See Figure \ref{blob1diagram}.) +(See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation. \begin{figure}[t]\begin{equation*} \mathfig{.6}{definition/single-blob} \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} @@ -56,15 +55,18 @@ just erasing the blob from the picture (but keeping the blob label $u$). -Note that the skein space $A(X)$ -is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. -This is Theorem \ref{thm:skein-modules}, and also used in the second +Note that directly from the definition we have +\begin{thm} +\label{thm:skein-modules} +The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. +\end{thm} +This also establishes the second half of Property \ref{property:contractibility}. Next, we want the vector space $\bc_2(X)$ to capture `the space of all relations (redundancies, syzygies) among the local relations encoded in $\bc_1(X)$'. -More specifically, a $2$-blob diagram, comes in one of two types, disjoint and nested. +A $2$-blob diagram, comes in one of two types, disjoint and nested. A disjoint 2-blob diagram consists of \begin{itemize} \item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors. @@ -98,12 +100,11 @@ \mathfig{.6}{definition/nested-blobs} \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$. -Note that the requirement that -local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$. As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating sum of the two ways of erasing one of the blobs. When we erase the inner blob, the outer blob inherits the label $u\bullet r'$. -It is again easy to check that $\bd^2 = 0$. +It is again easy to check that $\bd^2 = 0$. Note that the requirement that +local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$. As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is \begin{eqnarray*} @@ -117,8 +118,8 @@ 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). +For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ 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 @@ -157,7 +158,7 @@ \item A field $r \in \cC(X \setmin B^t; c^t)$, where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ is determined by the $c_i$'s. -$r$ is required to be splittable along the boundaries of all blobs, twigs or not. +$r$ is required to be splittable along the boundaries of all blobs, twigs or not. (This is equivalent to asking for a field on of the components of $X \setmin B^t$.) \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, where $c_j$ is the restriction of $c^t$ to $\bd B_j$. If $B_i = B_j$ then $u_i = u_j$. @@ -171,12 +172,12 @@ differ only by a reordering of the blobs, then we identify $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. -$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. +Roughly, then, $\bc_k(X)$ is all finite linear combinations of $k$-blob diagrams. As before, the official definition is in terms of direct sums of tensor products: \[ \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} - \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . + \left( \bigotimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . \] Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. The index $\overline{c}$ runs over all boundary conditions, again as described above and $j$ runs over all indices of twig blobs. @@ -190,9 +191,9 @@ Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. Let $E_j(b)$ denote the result of erasing the $j$-th blob. If $B_j$ is not a twig blob, this involves only decrementing -the indices of blobs $B_{j+1},\ldots,B_{k-1}$. +the indices of blobs $B_{j+1},\ldots,B_{k}$. If $B_j$ is a twig blob, we have to assign new local relation labels -if removing $B_j$ creates new twig blobs. +if removing $B_j$ creates new twig blobs. \todo{Have to say what happens when no new twig blobs are created} If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. Finally, define @@ -203,7 +204,7 @@ Thus we have a chain complex. Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. -A homeomorphism acts in an obvious on blobs and on fields. +A homeomorphism acts in an obvious way on blobs and on fields. We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, to be the union of the blobs of $b$. @@ -225,8 +226,8 @@ \end{itemize} For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while a diagram of $k$ disjoint blobs corresponds to a $k$-cube. -(This correspondence works best if we thing of each twig label $u_i$ as having the form +(This correspondence works best if we think of each twig label $u_i$ as having the form $x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map, -and $s:C \to \cC(B_i)$ is some fixed section of $e$.) +and $s:C \to \cC(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case})