286 In the second approach we use a decorated colimit (as in \S \ref{ssec:spherecat}) |
286 In the second approach we use a decorated colimit (as in \S \ref{ssec:spherecat}) |
287 and various sphere modules based on $F \to E \to Y$ |
287 and various sphere modules based on $F \to E \to Y$ |
288 or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$. |
288 or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$. |
289 Information about the specific map to $Y$ has been taken out of the categories |
289 Information about the specific map to $Y$ has been taken out of the categories |
290 and put into sphere modules and decorations. |
290 and put into sphere modules and decorations. |
291 |
291 \nn{...} |
292 Let $F \to E \to Y$ be a fiber bundle as above. |
292 |
293 Choose a decomposition $Y = \cup X_i$ |
293 %Let $F \to E \to Y$ be a fiber bundle as above. |
294 such that the restriction of $E$ to $X_i$ is a product $F\times X_i$. |
294 %Choose a decomposition $Y = \cup X_i$ |
295 \nn{resume revising here} |
295 %such that the restriction of $E$ to $X_i$ is a product $F\times X_i$, |
296 Choose the product structure (trivialization of the bundle restricted to $X_i$) as well. |
296 %and choose trivializations of these products as well. |
297 To each codim-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module). |
297 % |
298 And more generally to each codim-$j$ face we have an $S^{j-1}$-module. |
298 %\nn{edit marker} |
299 Decorate the decomposition with these modules and do the colimit. |
299 %To each codim-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module). |
300 |
300 %And more generally to each codim-$j$ face we have an $S^{j-1}$-module. |
301 |
301 %Decorate the decomposition with these modules and do the colimit. |
302 \nn{There is a version of this last construction for arbitrary maps $E \to Y$ |
302 % |
303 (not necessarily a fibration). |
303 % |
304 In fact, there is also a version of the first construction for non-fibrations.} |
304 %\nn{There is a version of this last construction for arbitrary maps $E \to Y$ |
305 |
305 %(not necessarily a fibration).} |
306 |
306 % |
307 |
307 % |
308 Note that Theorem \ref{thm:gluing} can be viewed as a special case of this one. |
308 % |
309 Let $X_1$ and $X_2$ be $n$-manifolds |
309 %Note that Theorem \ref{thm:gluing} can be viewed as a special case of this one. |
310 |
310 %Let $X_1$ and $X_2$ be $n$-manifolds |
|
311 % |
311 |
312 |
312 |
313 |
313 \subsection{A gluing theorem} |
314 \subsection{A gluing theorem} |
314 \label{sec:gluing} |
315 \label{sec:gluing} |
315 |
316 |