609 |
609 |
610 \nn{need to rewrite for self-gluing instead of gluing two pieces together} |
610 \nn{need to rewrite for self-gluing instead of gluing two pieces together} |
611 |
611 |
612 \nn{Should say something stronger about uniqueness. |
612 \nn{Should say something stronger about uniqueness. |
613 Something like: there is |
613 Something like: there is |
614 a contractible subcomplex of the complex of chain maps |
614 a contractible subcomplex of the complex of chain maps |
615 $CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), |
615 $CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), |
616 and all choices in the construction lie in the 0-cells of this |
616 and all choices in the construction lie in the 0-cells of this |
617 contractible subcomplex. |
617 contractible subcomplex. |
618 Or maybe better to say any two choices are homotopic, and |
618 Or maybe better to say any two choices are homotopic, and |
619 any two homotopies and second order homotopic, and so on.} |
619 any two homotopies and second order homotopic, and so on.} |
620 |
620 |
621 \nn{Also need to say something about associativity. |
621 \nn{Also need to say something about associativity. |
622 Put it in the above prop or make it a separate prop? |
622 Put it in the above prop or make it a separate prop? |
623 I lean toward the latter.} |
623 I lean toward the latter.} |
667 \medskip |
667 \medskip |
668 |
668 |
669 The strategy for the proof of Proposition \ref{CDprop} is as follows. |
669 The strategy for the proof of Proposition \ref{CDprop} is as follows. |
670 We will identify a subcomplex |
670 We will identify a subcomplex |
671 \[ |
671 \[ |
672 G_* \sub CD_*(X) \otimes \bc_*(X) |
672 G_* \sub CD_*(X) \otimes \bc_*(X) |
673 \] |
673 \] |
674 on which the evaluation map is uniquely determined (up to homotopy) by the conditions |
674 on which the evaluation map is uniquely determined (up to homotopy) by the conditions |
675 in \ref{CDprop}. |
675 in \ref{CDprop}. |
676 We then show that the inclusion of $G_*$ into the full complex |
676 We then show that the inclusion of $G_*$ into the full complex |
677 is an equivalence in the appropriate sense. |
677 is an equivalence in the appropriate sense. |
678 \nn{need to be more specific here} |
678 \nn{need to be more specific here} |
679 |
679 |
680 Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
680 Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
681 Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$ |
681 Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$ |
682 intersects at most one blob $B$ of $b$. |
682 intersects at most one blob $B$ of $b$. |
683 Since $V \cup B$ might not itself be a ball, we need a more careful and complicated definition. |
683 Since $V \cup B$ might not itself be a ball, we need a more careful and complicated definition. |
684 Choose a metric for $X$. |
684 Choose a metric for $X$. |
685 We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that |
685 We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that |
686 $N_\epsilon(b) \cup \supp(p)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon |
686 $\supp(p) \cup N_\epsilon(b)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon |
687 neighborhood of the support of $b$. |
687 neighborhood of the support of $b$. |
688 \nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.} |
688 \nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.} |
689 |
689 |
690 \nn{need to worry about case where the intrinsic support of $p$ is not a union of balls} |
690 \nn{need to worry about case where the intrinsic support of $p$ is not a union of balls} |
691 |
691 |
702 because $G_0 \sub CD_0\otimes \bc_0$. |
702 because $G_0 \sub CD_0\otimes \bc_0$. |
703 Assume we have defined the evaluation map up to $G_{k-1}$ and |
703 Assume we have defined the evaluation map up to $G_{k-1}$ and |
704 let $p\otimes b$ be a generator of $G_k$. |
704 let $p\otimes b$ be a generator of $G_k$. |
705 Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. |
705 Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. |
706 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 $C$. |
706 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 $C$. |
|
707 \scott{Shouldn't this be $p = g\circ p'$?} |
707 Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$. |
708 Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$. |
708 We may assume inductively that $e_X(\bd(p\otimes b))$ has the form $x\bullet g(b'')$, where |
709 We may assume inductively \scott{why? I don't get this.} that $e_X(\bd(p\otimes b))$ has the form $x\bullet g(b'')$, where |
709 $x \in \bc_*(g(C))$. |
710 $x \in \bc_*(g(C))$. |
710 Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$. |
711 Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$. |
711 \nn{need to say more if degree of $x$ is 0} |
712 \nn{need to say more if degree of $x$ is 0} |
712 Define $e_X(p\otimes b) = y\bullet g(b'')$. |
713 Define $e_X(p\otimes b) = y\bullet g(b'')$. |
713 |
714 |
714 We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made. |
715 We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made. |
715 If we make a different series of choice of the chain $y$ in the previous paragraph, |
716 If we make a different series of choice of the chain $y$ in the previous paragraph, |
716 we can inductively construct a homotopy between the two sets of choices, |
717 we can inductively construct a homotopy between the two sets of choices, |
717 again relying on the contractibility of $\bc_*(g(G))$. |
718 again relying on the contractibility of $\bc_*(g(G))$. |
718 A similar argument shows that this homotopy is unique up to second order homotopy, and so on. |
719 A similar argument shows that this homotopy is unique up to second order homotopy, and so on. |
719 |
720 |
720 Given a different set of choices $\{C'\}$ of the unions of balls $\{C\}$, |
721 Given a different set of choices $\{C'\}$ of the unions of balls $\{C\}$, |
724 |
725 |
725 Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists |
726 Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists |
726 a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and |
727 a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and |
727 this homotopy are unique up to iterated homotopy. |
728 this homotopy are unique up to iterated homotopy. |
728 |
729 |
729 Given $k>0$ and a blob diagram $b$, we say that a cover of $X$ $\cU$ is $k$-compatible with |
730 Given $k>0$ and a blob diagram $b$, we say that a cover $\cU$ of $X$ is $k$-compatible with |
730 $b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union |
731 $b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union |
731 $U_1\cup\cdots\cup U_k$ is a union of balls which satisfies the condition used to define $G_*$ above. |
732 $U_1\cup\cdots\cup U_k$ 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 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 $\cU$ and $b$ is a blob diagram occurring in $x$ \scott{huh, what's $x$ here?}, then $p\otimes b \in G_*$. |
734 \nn{maybe emphasize this more; it's one of the main ideas in the proof} |
735 \nn{maybe emphasize this more; it's one of the main ideas in the proof} |
735 |
736 |
736 Let $k$ be the degree of $x$ and choose a cover $\cU$ of $X$ such that $\cU$ is |
737 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 $k$-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 We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to |
739 construct the homotopy to $G_*$. |
740 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 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 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 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 each family of diffeomorphisms $p$ occurring in $y$ is adapted to $\cU$. |
744 This implies that $y \in G_*$. |
745 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 The homotopy $r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek. |
746 |
747 |
747 We will define $h$ inductive on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$. |
748 We will define $h$ inductively on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$. |
748 Define $h$ to be zero on bidegree $(0, k-1)$. |
749 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 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 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 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 $p'\otimes b'$ be a generator occurring in $\bd x$ with bidegree $(2, k-3)$. |
759 Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$. |
760 Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$. |
760 Continuing in this way, we define all of $h$. |
761 Continuing in this way, we define all of $h$. |
761 |
762 |
762 The homotopy $r$ is constructed similarly. |
763 The homotopy $r$ is constructed similarly. |
763 |
764 |
764 \nn{need to say something about uniqueness of $r$, $h$ etc. |
765 \nn{need to say something about uniqueness of $r$, $h$ etc. |
765 postpone this until second draft.} |
766 postpone this until second draft.} |
766 |
767 |
767 At this point, we have finished defining the evaluation map. |
768 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 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 All that remains is to show that the evaluation map gets along well with cutting and gluing, |