605 

   605 The arguments are very similar to ones given above, so we only sketch them.

   606 \nn{...}

   606 Let $g$ and $g'$ be two metrics on $X$, and let $e$ and $e'$ be the corresponding

   607 

   607 actions $CH_*(X, X) \ot \bc_*(X)\to\bc_*(X)$.

   608 

   608 We must show that $e$ and $e'$ are homotopic.

   609 \medskip\hrule\medskip\hrule\medskip

   609 As outlined in the discussion preceding this proof,

   610 

   610 this follows from the facts that both $e$ and $e'$ are compatible

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

   611 with gluing and that $\bc_*(B^n)$ is contractible.

   612 

   612 As above, we define a subcomplex $F_*\sub  CH_*(X, X) \ot \bc_*(X)$ generated

   613 \begin{itemize}

   613 by $p\ot b$ such that $|p|\cup|b|$ is contained in a disjoint union of balls.

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

   614 Using acyclic models, we can construct a homotopy from $e$ to $e'$ on $F_*$.

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

   615 We now observe that $CH_*(X, X) \ot \bc_*(X)$ retracts to $F_*$.

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

   616 Similar arguments show that this homotopy from $e$ to $e'$ is well-defined

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

   617 up to second order homotopy, and so on.

   618 $\hat{N}_{k,l} \sub N_{i,l}$, and similarly with the roles of $N$ and $\hat{N}$ reversed.

   619 \end{itemize}

   620 

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

   622