225 |
225 |
226 \medskip |
226 \medskip |
227 |
227 |
228 Theorem \ref{thm:product} extends to the case of general fiber bundles |
228 Theorem \ref{thm:product} extends to the case of general fiber bundles |
229 \[ |
229 \[ |
230 F \to E \to Y . |
230 F \to E \to Y , |
231 \] |
231 \] |
232 We outline one approach here and a second in \S \ref{xyxyx}. |
232 an indeed even to the case of general maps |
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233 \[ |
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234 M\to Y . |
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235 \] |
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236 We outline two approaches to these generalizations. |
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237 The first is somewhat tautological, while the second is more amenable to |
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238 calculation. |
233 |
239 |
234 We can generalize the definition of a $k$-category by replacing the categories |
240 We can generalize the definition of a $k$-category by replacing the categories |
235 of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ |
241 of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ |
236 (c.f. \cite{MR2079378}). |
242 (c.f. \cite{MR2079378}). |
237 Call this a $k$-category over $Y$. |
243 Call this a $k$-category over $Y$. |
238 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
244 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
239 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$. |
245 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$, |
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246 or the fields $\cE(p^*(E))$, if $\dim(D) < k$. |
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247 ($p^*(E)$ denotes the pull-back bundle over $D$.) |
240 Let $\cF_E$ denote this $k$-category over $Y$. |
248 Let $\cF_E$ denote this $k$-category over $Y$. |
241 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
249 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
242 get a chain complex $\cl{\cF_E}(Y)$. |
250 get a chain complex $\cl{\cF_E}(Y)$. |
243 The proof of Theorem \ref{thm:product} goes through essentially unchanged |
251 The proof of Theorem \ref{thm:product} goes through essentially unchanged |
244 to show that |
252 to show that |
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253 \begin{thm} |
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254 Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above. |
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255 Then |
245 \[ |
256 \[ |
246 \bc_*(E) \simeq \cl{\cF_E}(Y) . |
257 \bc_*(E) \simeq \cl{\cF_E}(Y) . |
247 \] |
258 \] |
248 |
259 \qed |
249 \nn{remark further that this still works when the map is not even a fibration?} |
260 \end{thm} |
250 |
261 |
251 \nn{put this later} |
262 We can generalize this result still further by noting that it is not really necessary |
252 |
263 for the definition of $\cF_E$ that $E\to Y$ be a fiber bundle. |
253 \nn{The second approach: Choose a decomposition $Y = \cup X_i$ |
264 Let $M\to Y$ be a map, with $\dim(M) = n$ and $\dim(Y) = k$. |
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265 Call a map $D^j\to Y$ ``good" with respect to $M$ if the fibered product |
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266 $D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$. |
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267 (If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$ |
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268 lying above $D$.) |
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269 We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which a re good with respect to $M$. |
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270 We can again adapt the homotopy colimit construction to |
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271 get a chain complex $\cl{\cF_M}(Y)$. |
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272 The proof of Theorem \ref{thm:product} again goes through essentially unchanged |
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273 to show that |
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274 \begin{thm} |
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275 Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the $k$-category over $Y$ defined above. |
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276 Then |
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277 \[ |
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278 \bc_*(M) \simeq \cl{\cF_M}(Y) . |
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279 \] |
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280 \qed |
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281 \end{thm} |
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282 |
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283 |
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284 \medskip |
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285 |
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286 In the second approach we use a decorated colimit (as in \S \ref{ssec:spherecat}) |
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287 and various sphere modules based on $F \to E \to Y$ |
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288 or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$. |
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289 Information about the specific map to $Y$ has been taken out of the categories |
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290 and put into sphere modules and decorations. |
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291 |
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292 Let $F \to E \to Y$ be a fiber bundle as above. |
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293 Choose a decomposition $Y = \cup X_i$ |
254 such that the restriction of $E$ to $X_i$ is a product $F\times X_i$. |
294 such that the restriction of $E$ to $X_i$ is a product $F\times X_i$. |
255 Choose the product structure as well. |
295 \nn{resume revising here} |
256 To each codim-1 face $D_i\cap D_j$ we have a bimodule ($S^0$-module). |
296 Choose the product structure (trivialization of the bundle restricted to $X_i$) as well. |
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297 To each codim-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module). |
257 And more generally to each codim-$j$ face we have an $S^{j-1}$-module. |
298 And more generally to each codim-$j$ face we have an $S^{j-1}$-module. |
258 Decorate the decomposition with these modules and do the colimit. |
299 Decorate the decomposition with these modules and do the colimit. |
259 } |
300 |
260 |
301 |
261 \nn{There is a version of this last construction for arbitrary maps $E \to Y$ |
302 \nn{There is a version of this last construction for arbitrary maps $E \to Y$ |
262 (not necessarily a fibration). |
303 (not necessarily a fibration). |
263 In fact, there is also a version of the first construction for non-fibrations.} |
304 In fact, there is also a version of the first construction for non-fibrations.} |
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305 |
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306 |
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307 |
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308 Note that Theorem \ref{thm:gluing} can be viewed as a special case of this one. |
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309 Let $X_1$ and $X_2$ be $n$-manifolds |
264 |
310 |
265 |
311 |
266 |
312 |
267 \subsection{A gluing theorem} |
313 \subsection{A gluing theorem} |
268 \label{sec:gluing} |
314 \label{sec:gluing} |