290 In the second approach we use a decorated colimit (as in \S \ref{ssec:spherecat}) |
290 In the second approach we use a decorated colimit (as in \S \ref{ssec:spherecat}) |
291 and various sphere modules based on $F \to E \to Y$ |
291 and various sphere modules based on $F \to E \to Y$ |
292 or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$. |
292 or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$. |
293 Information about the specific map to $Y$ has been taken out of the categories |
293 Information about the specific map to $Y$ has been taken out of the categories |
294 and put into sphere modules and decorations. |
294 and put into sphere modules and decorations. |
295 \nn{just say that one could do something along these lines} |
295 |
296 |
296 Let $F \to E \to Y$ be a fiber bundle as above. |
297 %Let $F \to E \to Y$ be a fiber bundle as above. |
297 Choose a decomposition $Y = \cup X_i$ |
298 %Choose a decomposition $Y = \cup X_i$ |
298 such that the restriction of $E$ to $X_i$ is homeomorphic to a product $F\times X_i$, |
299 %such that the restriction of $E$ to $X_i$ is a product $F\times X_i$, |
299 and choose trivializations of these products as well. |
300 %and choose trivializations of these products as well. |
300 |
301 % |
301 Let $\cF$ be the $k$-category associated to $F$. |
302 %\nn{edit marker} |
302 To each codimension-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module) for $\cF$. |
303 %To each codim-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module). |
303 More generally, to each codimension-$m$ face we have an $S^{m-1}$-module for a $(k{-}m{+}1)$-category |
304 %And more generally to each codim-$j$ face we have an $S^{j-1}$-module. |
304 associated to the (decorated) link of that face. |
305 %Decorate the decomposition with these modules and do the colimit. |
305 We can decorate the strata of the decomposition of $Y$ with these sphere modules and form a |
306 % |
306 colimit as in \S \ref{ssec:spherecat}. |
307 % |
307 This colimit computes $\bc_*(E)$. |
308 %\nn{There is a version of this last construction for arbitrary maps $E \to Y$ |
308 |
309 %(not necessarily a fibration).} |
309 There is a similar construction for general maps $M\to Y$. |
310 % |
310 |
311 % |
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312 % |
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313 %Note that Theorem \ref{thm:gluing} can be viewed as a special case of this one. |
311 %Note that Theorem \ref{thm:gluing} can be viewed as a special case of this one. |
314 %Let $X_1$ and $X_2$ be $n$-manifolds |
312 %Let $X_1$ and $X_2$ be $n$-manifolds |
315 % |
313 %\nn{...} |
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314 |
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315 |
316 |
316 |
317 |
317 |
318 \subsection{A gluing theorem} |
318 \subsection{A gluing theorem} |
319 \label{sec:gluing} |
319 \label{sec:gluing} |
320 |
320 |