725 

   729 Given $k>0$ and a blob diagram $b$, we say that a cover of $X$ $\cU$ is $k$-compatible with

   726 

   730 $b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union 

   727 

   731 $U_1\cup\cdots\cup U_k$ is a union of balls which satisfies the condition used to define $G_*$ above.

   732 Note that if a family of diffeomorphisms $p$ is adapted to 

   733 $\cU$ and $b$ is a blob diagram occurring in $x$, then $p\otimes b \in G_*$.

   734 \nn{maybe emphasize this more; it's one of the main ideas in the proof}

   735 

   736 Let $k$ be the degree of $x$ and choose a cover $\cU$ of $X$ such that $\cU$ is

   737 $k$-compatible with each of the (finitely many) blob diagrams occurring in $x$.

   738 We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to 

   739 construct the homotopy to $G_*$.

   740 First we will construct a homotopy $h \in G_*$ from $\bd x$ to a cycle $z$ such that

   741 each family of diffeomorphisms $p$ occurring in $z$ is adapted to $\cU$.

   742 Then we will construct a homotopy (rel boundary) $r$ from $x + h$ to $y$ such that

   743 each family of diffeomorphisms $p$ occurring in $y$ is adapted to $\cU$.

   744 This implies that $y \in G_*$.

   745 $r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek.

   746 

   747 We will define $h$ inductive on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$.

   748 Define $h$ to be zero on bidegree $(0, k-1)$.

   749 Let $p\otimes b$ be a generator occurring in $\bd x$ with bidegree $(1, k-2)$.

   750 Using Lemma \ref{extension_lemma}, construct a homotopy $q$ from $p$ to $p'$ which is adapted to $\cU$.

   751 Define $h$ at $p\otimes b$ to be $q\otimes b$.

   752 Let $p'\otimes b'$ be a generator occurring in $\bd x$ with bidegree $(2, k-3)$.

   753 Let $a$ be that portion of $\bd(p'\otimes b')$ which intersects the boundary of

   754 bidegree $(1, k-2)$ stuff.

   755 Apply Lemma \ref{extension_lemma} to $p'$ plus the diffeo part of $h(a)$

   756 (rel the outer boundary of said part),

   757 yielding a family of diffeos $q'$.

   758 \nn{definitely need to say this better}

   759 Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$.

   760 Continuing in this way, we define all of $h$.

   761 

   762 The homotopy $r$ is constructed similarly.

   763 

   764 \nn{need to say something about uniqueness of $r$, $h$ etc.  

   765 postpone this until second draft.}

   766 

   767 At this point, we have finished defining the evaluation map.

   768 The uniqueness statement in the proposition is clear from the method of proof.

   769 All that remains is to show that the evaluation map gets along well with cutting and gluing,

   770 as claimed in the proposition.

   771 This is in fact not difficult, since the myriad choices involved in defining the

   772 evaluation map can be made in parallel for the top and bottom

   773 arrows in the commutative diagram.

   774 

   775 This completes the proof of Proposition \ref{CDprop}.

   730 \hrule

   778 

   731 \medskip

   779 \nn{say something about associativity here}

   732 \hrule

   780 

   733 \medskip

   781 

   734 \nn{older stuff:}

   735 

   736 Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$

   737 (e.g.~the support of a blob diagram).

   738 We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if

   739 $f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$

   740 either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$.

   741 A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells,

   742 each of which is compatible.

   743 (Note that we could strengthen the definition of compatibility to incorporate

   744 a factorization condition, similar to the definition of ``adapted to" above.

   745 The weaker definition given here will suffice for our needs below.)

   746 

   747 \begin{cor}  \label{extension_lemma_2}

   748 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is compatible with $\{B_j\}$.

   749 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$.

   750 \end{cor}

   751 \begin{proof}

   752 This will follow from Lemma \ref{extension_lemma} for

   753 appropriate choice of cover $\cU = \{U_\alpha\}$.

   754 Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let

   755 $V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$.

   756 Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$

   757 either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$.

   758 

   759 Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$,

   760 with the (compatible) support of $f_i$ in place of $X$.

   761 This insures that the resulting homotopy $h_i$ is compatible.

   762 Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$.

   763 \nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this}

   764 \end{proof}

   765 

   766 \medskip

   767 

   768 ((argument continues roughly as follows: up to homotopy, there is only one way to define $e_X$

   769 on compatible $x\otimes y \in CD_*(X)\otimes \bc_*(X)$.

   770 This is because $x$ is the gluing of $x'$ and $x''$, where $x'$ has degree zero and is defined on

   771 the complement of the $D_i$'s, and $x''$ is defined on the $D_i$'s.

   772 We have no choice on $x'$, since we already know the map on 0-parameter families of diffeomorphisms.

   773 We have no choice, up to homotopy, on $x''$, since the target chain complex is contractible.))