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36 defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and |
36 defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and |
37 $\cC_F(X) = \bc_*(X\times F;\cE)$ if $\dim(X) = k$. |
37 $\cC_F(X) = \bc_*(X\times F;\cE)$ if $\dim(X) = k$. |
38 |
38 |
39 |
39 |
40 \begin{thm} \label{thm:product} |
40 \begin{thm} \label{thm:product} |
41 Let $Y$ be a $k$-manifold. |
41 Let $Y$ be a $k$-manifold which admits a ball decomposition |
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42 (e.g.\ any triangulable manifold). |
42 Then there is a homotopy equivalence between ``old-fashioned" (blob diagrams) |
43 Then there is a homotopy equivalence between ``old-fashioned" (blob diagrams) |
43 and ``new-fangled" (hocolimit) blob complexes |
44 and ``new-fangled" (hocolimit) blob complexes |
44 \[ |
45 \[ |
45 \cB_*(Y \times F) \htpy \cl{\cC_F}(Y) . |
46 \cB_*(Y \times F) \htpy \cl{\cC_F}(Y) . |
46 \]\end{thm} |
47 \]\end{thm} |