263 calculation. |
263 calculation. |
264 |
264 |
265 We can generalize the definition of a $k$-category by replacing the categories |
265 We can generalize the definition of a $k$-category by replacing the categories |
266 of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ |
266 of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$ |
267 (c.f. \cite{MR2079378}). |
267 (c.f. \cite{MR2079378}). |
268 Call this a $k$-category over $Y$. |
268 Call this a {\it $k$-category over $Y$}. |
269 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
269 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
270 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$, |
270 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, when $\dim(D) = k$, |
271 or the fields $\cE(p^*(E))$, if $\dim(D) < k$. |
271 or the fields $\cE(p^*(E))$, when $\dim(D) < k$. |
272 (Here $p^*(E)$ denotes the pull-back bundle over $D$.) |
272 (Here $p^*(E)$ denotes the pull-back bundle over $D$.) |
273 Let $\cF_E$ denote this $k$-category over $Y$. |
273 Let $\cF_E$ denote this $k$-category over $Y$. |
274 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
274 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
275 get a chain complex $\cl{\cF_E}(Y)$. |
275 get a chain complex $\cl{\cF_E}(Y)$. |
276 The proof of Theorem \ref{thm:product} goes through essentially unchanged |
276 |
277 to show the following result. |
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278 \begin{thm} |
277 \begin{thm} |
279 Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above. |
278 Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above. |
280 Then |
279 Then |
281 \[ |
280 \[ |
282 \bc_*(E) \simeq \cl{\cF_E}(Y) . |
281 \bc_*(E) \simeq \cl{\cF_E}(Y) . |
283 \] |
282 \] |
284 \qed |
283 \qed |
285 \end{thm} |
284 \end{thm} |
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285 |
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286 \begin{proof} |
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287 The proof is nearly identical to the proof of Theorem \ref{thm:product}, so we will only give a sketch which |
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288 emphasizes the few minor changes that need to be made. |
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289 |
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290 As before, we define a map |
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291 \[ |
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292 \psi: \cl{\cF_E}(Y) \to \bc_*(E) . |
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293 \] |
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294 0-simplices of the homotopy colimit $\cl{\cF_E}(Y)$ are glued up to give an element of $\bc_*(E)$. |
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295 Simplices of positive degree are sent to zero. |
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296 |
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297 Let $G_* \sub \bc_*(E)$ be the image of $\psi$. |
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298 By Lemma \ref{thm:small-blobs}, $\bc_*(Y\times F; \cE)$ |
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299 is homotopic to a subcomplex of $G_*$. |
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300 We will define a homotopy inverse of $\psi$ on $G_*$, using acyclic models. |
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301 To each generator $a$ of $G_*$ we assign an acyclic subcomplex $D(a) \sub \cl{\cF_E}(Y)$ which consists of |
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302 0-simplices which map via $\psi$ to $a$, plus higher simplices (as described in the proof of Theorem \ref{thm:product}) |
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303 which insure that $D(a)$ is acyclic. |
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304 \end{proof} |
286 |
305 |
287 We can generalize this result still further by noting that it is not really necessary |
306 We can generalize this result still further by noting that it is not really necessary |
288 for the definition of $\cF_E$ that $E\to Y$ be a fiber bundle. |
307 for the definition of $\cF_E$ that $E\to Y$ be a fiber bundle. |
289 Let $M\to Y$ be a map, with $\dim(M) = n$ and $\dim(Y) = k$. |
308 Let $M\to Y$ be a map, with $\dim(M) = n$ and $\dim(Y) = k$. |
290 Call a map $D^j\to Y$ ``good" with respect to $M$ if the fibered product |
309 Call a map $D^j\to Y$ ``good" with respect to $M$ if the fibered product |