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313 We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$. |
313 We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$. |
314 We can again adapt the homotopy colimit construction to |
314 We can again adapt the homotopy colimit construction to |
315 get a chain complex $\cl{\cF_M}(Y)$. |
315 get a chain complex $\cl{\cF_M}(Y)$. |
316 The proof of Theorem \ref{thm:product} again goes through essentially unchanged |
316 The proof of Theorem \ref{thm:product} again goes through essentially unchanged |
317 to show that |
317 to show that |
318 \begin{thm} |
318 %\begin{thm} |
319 Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the $k$-category over $Y$ defined above. |
319 %Let $M \to Y$ be a map of manifolds and let $\cF_M$ be the $k$-category over $Y$ defined above. |
320 Then |
320 %Then |
321 \[ |
321 \[ |
322 \bc_*(M) \simeq \cl{\cF_M}(Y) . |
322 \bc_*(M) \simeq \cl{\cF_M}(Y) . |
323 \] |
323 \] |
324 \qed |
324 %\qed |
325 \end{thm} |
325 %\end{thm} |
326 |
326 |
327 |
327 |
328 \medskip |
328 \medskip |
329 |
329 |
330 In the second approach we use a decorated colimit (as in \S \ref{ssec:spherecat}) |
330 In the second approach we use a decorated colimit (as in \S \ref{ssec:spherecat}) |