484 

   484 \hrule\medskip\hrule\medskip

   485 

   485 

   486 \nn{outline of what remains to be done:}

   487 

   488 \begin{itemize}

   489 \item We need to assemble the maps for the various $G^{i,m}$ into

   490 a map for all of $CD_*\ot \bc_*$.

   491 One idea: Think of the $g_j$ as a sort of homotopy (from $CD_*\ot \bc_*$ to itself) 

   492 parameterized by $[0,\infty)$.  For each $p\ot b$ in $CD_*\ot \bc_*$ choose a sufficiently

   493 large $j'$.  Use these choices to reparameterize $g_\bullet$ so that each

   494 $p\ot b$ gets pushed as far as the corresponding $j'$.

   495 \item Independence of metric, $\ep_i$, $\delta_i$:

   496 For a different metric etc. let $\hat{G}^{i,m}$ denote the alternate subcomplexes

   497 and $\hat{N}_{i,l}$ the alternate neighborhoods.

   498 Main idea is that for all $i$ there exists sufficiently large $k$ such that

   499 $\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed.

   500 \item Also need to prove associativity.

   501 \end{itemize}

   502 

   503 

   504 \nn{to be continued....}