349 (relative to any designated subcomplex of $C_*$ already in $\sbtc_*(X)$). |
349 (relative to any designated subcomplex of $C_*$ already in $\sbtc_*(X)$). |
350 The first step is to replace families of general blob diagrams with families |
350 The first step is to replace families of general blob diagrams with families |
351 of blob diagrams that are small with respect to $\cU$. |
351 of blob diagrams that are small with respect to $\cU$. |
352 (If $f:P \to \BD_k$ is the family then for all $p\in P$ we have that $f(p)$ is a diagram in which the blobs are small.) |
352 (If $f:P \to \BD_k$ is the family then for all $p\in P$ we have that $f(p)$ is a diagram in which the blobs are small.) |
353 This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families. |
353 This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families. |
354 Each such family is homotopic to a sum families which can be a ``lifted" to $\Homeo(X)$. |
354 Each such family is homotopic to a sum of families which can be a ``lifted" to $\Homeo(X)$. |
355 That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$. |
355 That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$. |
356 (We are ignoring a complication related to twig blob labels, which might vary |
356 (We are ignoring a complication related to twig blob labels, which might vary |
357 independently of $g$, but this complication does not affect the conclusion we draw here.) |
357 independently of $g$, but this complication does not affect the conclusion we draw here.) |
358 We now apply Lemma \ref{extension_lemma_c} to get families which are supported |
358 We now apply Lemma \ref{extension_lemma_c} to get families which are supported |
359 on balls $D_i$ contained in open sets of $\cU$. |
359 on balls $D_i$ contained in open sets of $\cU$. |