454 %To harmonize notation with the next section, |
454 %To harmonize notation with the next section, |
455 %let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so |
455 %let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so |
456 %$\bc_0(X) = \lf(X)$. |
456 %$\bc_0(X) = \lf(X)$. |
457 \begin{defn} |
457 \begin{defn} |
458 \label{defn:TQFT-invariant} |
458 \label{defn:TQFT-invariant} |
459 The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is |
459 The TQFT invariant of $X$ associated to a system of fields $\cC$ and local relations $U$ is |
460 $$A(X) \deq \lf(X) / U(X),$$ |
460 $$A(X) \deq \lf(X) / U(X),$$ |
461 where $\cU(X) \sub \lf(X)$ is the space of local relations in $\lf(X)$: |
461 where $U(X) \sub \lf(X)$ is the space of local relations in $\lf(X)$: |
462 $\cU(X)$ is generated by fields of the form $u\bullet r$, where |
462 $U(X)$ is generated by fields of the form $u\bullet r$, where |
463 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
463 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
464 \end{defn} |
464 \end{defn} |
465 The blob complex, defined in the next section, |
465 The blob complex, defined in the next section, |
466 is in some sense the derived version of $A(X)$. |
466 is in some sense the derived version of $A(X)$. |
467 If $X$ has boundary we can similarly define $A(X; c)$ for each |
467 If $X$ has boundary we can similarly define $A(X; c)$ for each |