equal
deleted
inserted
replaced
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 $\bc_*(p^*(E))$, when $\dim(D) = k$, |
270 assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, when $\dim(D) = k$, |
271 or the fields $\cE(p^*(E))$, when $\dim(D) < k$. |
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 on 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 |
276 |
277 \begin{thm} |
277 \begin{thm} |
278 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. |
279 Then |
279 Then |
289 |
289 |
290 As before, we define a map |
290 As before, we define a map |
291 \[ |
291 \[ |
292 \psi: \cl{\cF_E}(Y) \to \bc_*(E) . |
292 \psi: \cl{\cF_E}(Y) \to \bc_*(E) . |
293 \] |
293 \] |
294 0-simplices of the homotopy colimit $\cl{\cF_E}(Y)$ are glued up to give an element of $\bc_*(E)$. |
294 The 0-simplices of the homotopy colimit $\cl{\cF_E}(Y)$ are glued up to give an element of $\bc_*(E)$. |
295 Simplices of positive degree are sent to zero. |
295 Simplices of positive degree are sent to zero. |
296 |
296 |
297 Let $G_* \sub \bc_*(E)$ be the image of $\psi$. |
297 Let $G_* \sub \bc_*(E)$ be the image of $\psi$. |
298 By Lemma \ref{thm:small-blobs}, $\bc_*(Y\times F; \cE)$ |
298 By Lemma \ref{thm:small-blobs}, $\bc_*(Y\times F; \cE)$ |
299 is homotopic to a subcomplex of $G_*$. |
299 is homotopic to a subcomplex of $G_*$. |