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 %