blob: comparison text/definitions.tex

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 \section{TQFTs via fields}
 %\label{sec:definitions}
 In this section we review the construction of TQFTs from ``topological fields".
 For more details see xxxx.
+We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
+submanifold of $X$, then $X \setmin Y$ implicitly means the closure
+$\overline{X \setmin Y}$.
 \subsection{Systems of fields}
 \label{sec:fields}
 Let $\cM_k$ denote the category with objects
 Let $X$ be an $n$-manifold.
 Assume a fixed system of fields and local relations.
 In this section we will usually suppress boundary conditions on $X$ from the notation
 (e.g. write $\lf(X)$ instead of $\lf(X; c)$).
-We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
+We want to replace the quotient
-submanifold of $X$, then $X \setmin Y$ implicitly means the closure
+\[
-$\overline{X \setmin Y}$.
+	A(X) \deq \lf(X) / U(X)
+\]
+of the previous section 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 $\bc_k(X)$.
-Define $\bc_0(X) = \lf(X)$.
+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 space of all linearized fields on $X$.
 $\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$.
 \item A field $r \in \cC(X \setmin B; c)$
 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$).
 \item A local relation field $u \in U(B; c)$
 (same $c$ as previous bullet).
 \end{itemize}
+(See Figure \ref{blob1diagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{tempkw/blob1diagram}
+\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}
 In order to get the linear structure correct, we (officially) define
 \[
 	\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) .
 \]
 The first direct sum is indexed by all blobs $B\subset X$, and the second
 \item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors.
 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$
 (where $c_i \in \cC(\bd B_i)$).
 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$.
 \end{itemize}
+(See Figure \ref{blob2ddiagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{tempkw/blob2ddiagram}
+\end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure}
 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$;
 reversing the order of the blobs changes the sign.
 Define $\bd(B_0, B_1, u_0, u_1, r) =
 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$.
 In other words, the boundary of a disjoint 2-blob diagram
 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$.
 \item A field $r \in \cC(X \setmin B_0; c_0)$
 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$.
 \item A local relation field $u_0 \in U(B_0; c_0)$.
 \end{itemize}
+(See Figure \ref{blob2ndiagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{tempkw/blob2ndiagram}
+\end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure}
 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
 (for some $c_1 \in \cC(B_1)$) and
 $r' \in \cC(X \setmin B_1; c_1)$.
 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$.
 Note that the requirement that
 local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$.
 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.
 If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$.
 It is again easy to check that $\bd^2 = 0$.
-\nn{should draw figures for 1, 2 and $k$-blob diagrams}
 As with the 1-blob diagrams, in order to get the linear structure correct it is better to define
 (officially)
 \begin{eqnarray*}
 	\bc_2(X) & \deq &
 $r$ is required to be splittable along the boundaries of all blobs, twigs or not.
 \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 $u_i = u_j$.
 \end{itemize}
+(See Figure \ref{blobkdiagram}.)
+\begin{figure}[!ht]\begin{equation*}
+\mathfig{.9}{tempkw/blobkdiagram}
+\end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure}
 If two blob diagrams $D_1$ and $D_2$
 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 $D_2$.

changeset 140	e0b304e6b975
parent 139	57291331fd82
child 141	e1d24be683bb