685 |
685 |
686 \nn{need to worry about case where the intrinsic support of $p$ is not a union of balls} |
686 \nn{need to worry about case where the intrinsic support of $p$ is not a union of balls} |
687 |
687 |
688 \nn{need to eventually show independence of choice of metric. maybe there's a better way than |
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 |
689 choosing a metric. perhaps just choose a nbd of each ball, but I think I see problems |
690 with that as well.} |
690 with that as well. |
691 |
691 the bottom line is that we need a scheme for choosing unions of balls |
692 Next we define the evaluation map on $G_*$. |
692 which satisfies the $C$, $C'$, $C''$ claim made a few paragraphs below.} |
|
693 |
|
694 Next we define the evaluation map $e_X$ on $G_*$. |
693 We'll proceed inductively on $G_i$. |
695 We'll proceed inductively on $G_i$. |
694 The induction starts on $G_0$, where we have no choice for the evaluation map |
696 The induction starts on $G_0$, where the evaluation map is determined |
|
697 by the action of $\Diff(X)$ on $\bc_*(X)$ |
695 because $G_0 \sub CD_0\otimes \bc_0$. |
698 because $G_0 \sub CD_0\otimes \bc_0$. |
696 Assume we have defined the evaluation map up to $G_{k-1}$ and |
699 Assume we have defined the evaluation map up to $G_{k-1}$ and |
697 let $p\otimes b$ be a generator of $G_k$. |
700 let $p\otimes b$ be a generator of $G_k$. |
698 Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. |
701 Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. |
|
702 There is a factorization $p = p' \circ g$, where $g\in \Diff(X)$ and $p'$ is a family of diffeomorphisms which is the identity outside of $C$. |
|
703 Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$. |
|
704 We may assume inductively that $e_X(\bd(p\otimes b))$ has the form $x\bullet g(b'')$, where |
|
705 $x \in \bc_*(g(C))$. |
|
706 Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$. |
|
707 \nn{need to say more if degree of $x$ is 0} |
|
708 Define $e_X(p\otimes b) = y\bullet g(b'')$. |
|
709 |
|
710 We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made. |
|
711 If we make a different series of choice of the chain $y$ in the previous paragraph, |
|
712 we can inductively construct a homotopy between the two sets of choices, |
|
713 again relying on the contractibility of $\bc_*(g(G))$. |
|
714 A similar argument shows that this homotopy is unique up to second order homotopy, and so on. |
|
715 |
|
716 Given a different set of choices $\{C'\}$ of the unions of balls $\{C\}$, |
|
717 we can find a third set of choices $\{C''\}$ such that $C, C' \sub C''$. |
|
718 The argument now proceeds as in the previous paragraph. |
|
719 \nn{should maybe say more here; also need to back up claim about third set of choices} |
|
720 |
|
721 Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists |
|
722 a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and |
|
723 this homotopy are unique up to iterated homotopy. |
699 |
724 |
700 |
725 |
701 |
726 |
702 |
727 |
703 |
728 |