171 

   172 Next we prove a gluing theorem.

   173 Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$.

   174 We will need an explicit collar on $Y$, so rewrite this as

   175 $X = X_1\cup (Y\times J) \cup X_2$.

   176 \nn{need figure}

   177 Given this data we have: \nn{need refs to above for these}

   178 \begin{itemize}

   179 \item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball

   180 $D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$

   181 (for $m+k = n$). \nn{need to explain $c$}.

   182 \item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly.

   183 \item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked

   184 $m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$)

   185 or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$).

   186 \end{itemize}

   187 

   188 \begin{thm}

   189 $\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.

   190 \end{thm}

   191 

   192 \begin{proof}

   193 The proof is similar to that of Theorem \ref{product_thm}.

   194 \nn{need to say something about dimensions less than $n$, 

   195 but for now concentrate on top dimension.}

   196 

   197 Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.

   198 Let $D$ be an $n{-}k$-ball.

   199 There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$.

   200 To get a map in the other direction, we replace $\bc_*(D\times X)$ with a subcomplex

   201 $\cS_*$ which is adapted to a fine open cover of $D\times X$.

   202 For sufficiently small $j$ (depending on the cover), we can find, for each $j$-blob diagram $b$

   203 on $D\times X$, a decomposition of $J$ such that $b$ splits on the corresponding

   204 decomposition of $D\times X$.

   205 The proof that these two maps are inverse to each other is the same as in

   206 Theorem \ref{product_thm}.

   207 \end{proof}

   208 

   209 

   210 \medskip