190 Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and |
190 Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and |
191 $\psi$ glues those pieces back together, yielding $a$. |
191 $\psi$ glues those pieces back together, yielding $a$. |
192 We have $\psi(r) = 0$ since $\psi$ is zero on $(\ge 1)$-simplices. |
192 We have $\psi(r) = 0$ since $\psi$ is zero on $(\ge 1)$-simplices. |
193 |
193 |
194 Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models. |
194 Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models. |
195 To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above. |
195 To each generator $(b, \ol{K})$ of $\cl{\cC_F}(Y)$ we associate the acyclic subcomplex $D(b)$ defined above. |
196 Both the identity map and $\phi\circ\psi$ are compatible with this |
196 Both the identity map and $\phi\circ\psi$ are compatible with this |
197 collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps |
197 collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps |
198 are homotopic. |
198 are homotopic. |
199 |
199 |
200 This concludes the proof of Theorem \ref{thm:product}. |
200 This concludes the proof of Theorem \ref{thm:product}. |
246 (c.f. \cite{MR2079378}). |
246 (c.f. \cite{MR2079378}). |
247 Call this a $k$-category over $Y$. |
247 Call this a $k$-category over $Y$. |
248 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
248 A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$: |
249 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$, |
249 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$, |
250 or the fields $\cE(p^*(E))$, if $\dim(D) < k$. |
250 or the fields $\cE(p^*(E))$, if $\dim(D) < k$. |
251 ($p^*(E)$ denotes the pull-back bundle over $D$.) |
251 (Here $p^*(E)$ denotes the pull-back bundle over $D$.) |
252 Let $\cF_E$ denote this $k$-category over $Y$. |
252 Let $\cF_E$ denote this $k$-category over $Y$. |
253 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
253 We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to |
254 get a chain complex $\cl{\cF_E}(Y)$. |
254 get a chain complex $\cl{\cF_E}(Y)$. |
255 The proof of Theorem \ref{thm:product} goes through essentially unchanged |
255 The proof of Theorem \ref{thm:product} goes through essentially unchanged |
256 to show that |
256 to show the following result. |
257 \begin{thm} |
257 \begin{thm} |
258 Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above. |
258 Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above. |
259 Then |
259 Then |
260 \[ |
260 \[ |
261 \bc_*(E) \simeq \cl{\cF_E}(Y) . |
261 \bc_*(E) \simeq \cl{\cF_E}(Y) . |
268 Let $M\to Y$ be a map, with $\dim(M) = n$ and $\dim(Y) = k$. |
268 Let $M\to Y$ be a map, with $\dim(M) = n$ and $\dim(Y) = k$. |
269 Call a map $D^j\to Y$ ``good" with respect to $M$ if the fibered product |
269 Call a map $D^j\to Y$ ``good" with respect to $M$ if the fibered product |
270 $D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$. |
270 $D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$. |
271 (If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$ |
271 (If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$ |
272 lying above $D$.) |
272 lying above $D$.) |
273 We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which a re good with respect to $M$. |
273 We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$. |
274 We can again adapt the homotopy colimit construction to |
274 We can again adapt the homotopy colimit construction to |
275 get a chain complex $\cl{\cF_M}(Y)$. |
275 get a chain complex $\cl{\cF_M}(Y)$. |
276 The proof of Theorem \ref{thm:product} again goes through essentially unchanged |
276 The proof of Theorem \ref{thm:product} again goes through essentially unchanged |
277 to show that |
277 to show that |
278 \begin{thm} |
278 \begin{thm} |