304 Next we show how to homotope chains in $CD_*(X)\ot \bc_*(X)$ to one of the 

   305 $G_*^{i,m}$.

   306 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero.

   307 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$.

   308 Let $h_j: CD_*(X)\to CD_*(X)$ be a chain map homotopic to the identity whose image is spanned by diffeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}.

   309 Recall that $h_j$ and also its homotopy back to the identity do not increase

   310 supports.

   311 Define

   312 \[

   313 	g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 .

   314 \]

   315 The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that

   316 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ 

   317 (depending on $b$, $n = \deg(p)$ and $m$).

   318 

   319 \begin{lemma}

   320 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$.

   321 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$

   322 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$ 

   323 we have $g_j(p)\ot b \in G_*^{i,m}$.

   324 \end{lemma}

   325 

   326 \begin{proof}

   327 Let $c$ be a subset of the blobs of $b$.

   328 There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$ 

   329 and all such $c$.

   330 (Here we are using a piecewise smoothness assumption for $\bd c$).

   331 

   332 Let $r = \deg(b)$.

   333 

   334 Choose $k = k_{bmn}$ such that

   335 \[

   336 	(r+n+m+1)\ep_k < l

   337 \]

   338 and

   339 \[

   340 	n\cdot (3\delta_k\cdot(r+n+m+1)) < \ep_k .

   341 \]

   342 Let $i \ge k_{bmn}$.

   343 Choose $j = j_i$ so that

   344 \[

   345 	3\cdot(r+n+m+1)\gamma_j < \ep_i

   346 \]

   347 and also so that for any subset $S\sub X$ of diameter less than or equal to 

   348 $2n\gamma_j$ we have that $\Nbd_u(S)$ is

   349 \end{proof}

   350 

   351