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307 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
307 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
308 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
308 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
309 \end{eqnarray*} |
309 \end{eqnarray*} |
310 (See Figure \ref{glue-collar}.) |
310 (See Figure \ref{glue-collar}.) |
311 \begin{figure}[!ht]\begin{equation*} |
311 \begin{figure}[!ht]\begin{equation*} |
312 \mathfig{.9}{tempkw/glue-collar} |
312 \mathfig{.9}{tempkw/blah10} |
313 \end{equation*}\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
313 \end{equation*}\caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
314 We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map. |
314 We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map. |
315 \nn{bad terminology; fix it later} |
315 \nn{bad terminology; fix it later} |
316 \nn{also need to make clear that plain old isotopic to the identity implies |
316 \nn{also need to make clear that plain old isotopic to the identity implies |
317 extended isotopic} |
317 extended isotopic} |
614 Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
614 Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$. |
615 Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
615 Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$. |
616 (The union is along $N\times \bd W$.) |
616 (The union is along $N\times \bd W$.) |
617 (If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
617 (If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
618 the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
618 the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
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619 |
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620 \begin{figure}[!ht] |
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621 $$\mathfig{.8}{tempkw/blah15}$$ |
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622 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
619 |
623 |
620 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
624 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
621 Call such a thing a {marked $k{-}1$-hemisphere}. |
625 Call such a thing a {marked $k{-}1$-hemisphere}. |
622 |
626 |
623 \xxpar{Module boundaries, part 1:} |
627 \xxpar{Module boundaries, part 1:} |