344 This follows from a combination of Lemma \ref{extension_lemma_c} and the techniques of |
344 This follows from a combination of Lemma \ref{extension_lemma_c} and the techniques of |
345 the proof of Lemma \ref{small-blobs-b}. |
345 the proof of Lemma \ref{small-blobs-b}. |
346 |
346 |
347 It suffices to show that we can deform a finite subcomplex $C_*$ of $\btc_*(X)$ into $\sbtc_*(X)$ |
347 It suffices to show that we can deform a finite subcomplex $C_*$ of $\btc_*(X)$ into $\sbtc_*(X)$ |
348 (relative to any designated subcomplex of $C_*$ already in $\sbtc_*(X)$). |
348 (relative to any designated subcomplex of $C_*$ already in $\sbtc_*(X)$). |
349 The first step is to replace families of general blob diagrams with families that are |
349 The first step is to replace families of general blob diagrams with families |
350 small with respect to $\cU$. |
350 of blob diagrams that are small with respect to $\cU$. |
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351 (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.) |
351 This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families. |
352 This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families. |
352 Each such family is homotopic to a sum families which can be a ``lifted" to $\Homeo(X)$. |
353 Each such family is homotopic to a sum families which can be a ``lifted" to $\Homeo(X)$. |
353 That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$. |
354 That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$. |
354 (We are ignoring a complication related to twig blob labels, which might vary |
355 (We are ignoring a complication related to twig blob labels, which might vary |
355 independently of $g$, but this complication does not affect the conclusion we draw here.) |
356 independently of $g$, but this complication does not affect the conclusion we draw here.) |