610 \nn{Should say something stronger about uniqueness.

   611 Something like: there is

   612 a contractible subcomplex of the complex of chain maps 

   613 $CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.),

   614 and all choices in the construction lie in the 0-cells of this

   615 contractible subcomplex.

   616 Or maybe better to say any two choices are homotopic, and 

   617 any two homotopies and second order homotopic, and so on.}

   618 

   619 \nn{Also need to say something about associativity.

   620 Put it in the above prop or make it a separate prop?

   621 I lean toward the latter.}

   622 \medskip

   623 

   664 

   665 The strategy for the proof of Proposition \ref{CDprop} is as follows.

   666 We will identify a subcomplex

   667 \[

   668 	G_* \sub CD_*(X) \otimes \bc_*(X)

   669 \]

   670 on which the evaluation map is uniquely determined (up to homotopy) by the conditions

   671 in \ref{CDprop}.

   672 We then show that the inclusion of $G_*$ into the full complex

   673 is an equivalence in the appropriate sense.

   674 \nn{need to be more specific here}

   675 

   676 Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$.

   677 Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$ 

   678 intersects at most one blob $B$ of $b$.

   679 Since $V \cup B$ might not itself be a ball, we need a more careful and complicated definition.

   680 Choose a metric for $X$.

   681 We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that 

   682 $N_\epsilon(b) \cup \supp(p)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon

   683 neighborhood of the support of $b$.

   684 \nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.}

   685 

   686 \nn{need to worry about case where the intrinsic support of $p$ is not a union of balls}

   687 

   688 \nn{need to eventually show independence of choice of metric.  maybe there's a better way than

   689 choosing a metric.  perhaps just choose a nbd of each ball, but I think I see problems

   690 with that as well.}

   691 

   692 Next we define the evaluation map on $G_*$.

   693 We'll proceed inductively on $G_i$.

   694 The induction starts on $G_0$, where we have no choice for the evaluation map

   695 because $G_0 \sub CD_0\otimes \bc_0$.

   696 Assume we have defined the evaluation map up to $G_{k-1}$ and

   697 let $p\otimes b$ be a generator of $G_k$.

   698 Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$.

   699 

   700 

   701 

   702 

   703 

   704 \medskip

   705 \hrule

   706 \medskip

   707 \hrule

   708 \medskip

   709 \nn{older stuff:}