773 |
773 |
774 \medskip |
774 \medskip |
775 |
775 |
776 Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
776 Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
777 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
777 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
778 $x \notin S$ and $p, q \in P$. Equivalently \todo{really?}, $f$ is supported on $S$ if there is a family of diffeomorphisms $f' : P \times S \to S$ and a `background' |
778 $x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if there is a family of diffeomorphisms $f' : P \times S \to S$ and a `background' |
779 diffeomorphism $f_0 : X \to X$ so that |
779 diffeomorphism $f_0 : X \to X$ so that |
780 \begin{align} |
780 \begin{align} |
781 \restrict{f}{P \times S}(p,s) & = f_0(f'(p,s)) \\ |
781 f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
782 \intertext{and} |
782 \intertext{and} |
783 \restrict{f}{P \times (X \setmin S)}(p,x) & = f_0(x). |
783 f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
784 \end{align} |
784 \end{align} |
785 Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
785 Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
786 |
786 |
787 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
787 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
788 A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
788 A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
795 \eq{ |
795 \eq{ |
796 f_i : P_i \times X \to X |
796 f_i : P_i \times X \to X |
797 } |
797 } |
798 such that |
798 such that |
799 \begin{itemize} |
799 \begin{itemize} |
800 \item each $f_i(p, \cdot): X \to X$\scott{This should just read ``each $f_i$ is supported''} is supported on some connected $V_i \sub X$; |
800 \item each $f_i$ is supported on some connected $V_i \sub X$; |
801 \item the sets $V_i$ are mutually disjoint; |
801 \item the sets $V_i$ are mutually disjoint; |
802 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
802 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
803 where $k_i = \dim(P_i)$; and |
803 where $k_i = \dim(P_i)$; and |
804 \item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot) \circ g$ |
804 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
805 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$.\scott{hmm, can we do $g$ last, instead?} |
805 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
806 \end{itemize} |
806 \end{itemize} |
807 A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum |
807 A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum |
808 of singular cells, each of which is adapted to $\cU$. |
808 of singular cells, each of which is adapted to $\cU$. |
809 |
809 |
810 \begin{lemma} \label{extension_lemma} |
810 \begin{lemma} \label{extension_lemma} |
836 We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that |
836 We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that |
837 $\supp(p) \cup N_\epsilon(b)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon |
837 $\supp(p) \cup N_\epsilon(b)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon |
838 neighborhood of the support of $b$. |
838 neighborhood of the support of $b$. |
839 \nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.} |
839 \nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.} |
840 |
840 |
841 \nn{need to worry about case where the intrinsic support of $p$ is not a union of balls} |
841 \nn{need to worry about case where the intrinsic support of $p$ is not a union of balls. |
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842 probably we can just stipulate that it is (i.e. only consider families of diffeos with this property). |
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843 maybe we should build into the definition of ``adapted" that support takes up all of $U_i$.} |
842 |
844 |
843 \nn{need to eventually show independence of choice of metric. maybe there's a better way than |
845 \nn{need to eventually show independence of choice of metric. maybe there's a better way than |
844 choosing a metric. perhaps just choose a nbd of each ball, but I think I see problems |
846 choosing a metric. perhaps just choose a nbd of each ball, but I think I see problems |
845 with that as well. |
847 with that as well. |
846 the bottom line is that we need a scheme for choosing unions of balls |
848 the bottom line is that we need a scheme for choosing unions of balls |
852 by the action of $\Diff(X)$ on $\bc_*(X)$ |
854 by the action of $\Diff(X)$ on $\bc_*(X)$ |
853 because $G_0 \sub CD_0\otimes \bc_0$. |
855 because $G_0 \sub CD_0\otimes \bc_0$. |
854 Assume we have defined the evaluation map up to $G_{k-1}$ and |
856 Assume we have defined the evaluation map up to $G_{k-1}$ and |
855 let $p\otimes b$ be a generator of $G_k$. |
857 let $p\otimes b$ be a generator of $G_k$. |
856 Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. |
858 Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. |
857 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 $C$. |
859 There is a factorization $p = g \circ p'$, where $g\in \Diff(X)$ and $p'$ is a family of diffeomorphisms which is the identity outside of $C$. |
858 \scott{Shouldn't this be $p = g\circ p'$?} |
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859 Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$. |
860 Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$. |
860 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 |
861 We may assume inductively |
861 $x \in \bc_*(g(C))$. |
862 (cf the end of this paragraph) |
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863 that $e_X(\bd(p\otimes b))$ has a similar factorization $x\bullet g(b'')$, where |
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864 $x \in \bc_*(g(C))$ and $\bd x = 0$. |
862 Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$. |
865 Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$. |
863 \nn{need to say more if degree of $x$ is 0} |
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864 Define $e_X(p\otimes b) = y\bullet g(b'')$. |
866 Define $e_X(p\otimes b) = y\bullet g(b'')$. |
865 |
867 |
866 We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made. |
868 We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made. |
867 If we make a different series of choice of the chain $y$ in the previous paragraph, |
869 If we make a different series of choice of the chain $y$ in the previous paragraph, |
868 we can inductively construct a homotopy between the two sets of choices, |
870 we can inductively construct a homotopy between the two sets of choices, |
871 |
873 |
872 Given a different set of choices $\{C'\}$ of the unions of balls $\{C\}$, |
874 Given a different set of choices $\{C'\}$ of the unions of balls $\{C\}$, |
873 we can find a third set of choices $\{C''\}$ such that $C, C' \sub C''$. |
875 we can find a third set of choices $\{C''\}$ such that $C, C' \sub C''$. |
874 The argument now proceeds as in the previous paragraph. |
876 The argument now proceeds as in the previous paragraph. |
875 \nn{should maybe say more here; also need to back up claim about third set of choices} |
877 \nn{should maybe say more here; also need to back up claim about third set of choices} |
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878 \nn{this definitely needs reworking} |
876 |
879 |
877 Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists |
880 Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists |
878 a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and |
881 a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and |
879 this homotopy are unique up to iterated homotopy. |
882 this homotopy are unique up to iterated homotopy. |
880 |
883 |
881 Given $k>0$ and a blob diagram $b$, we say that a cover $\cU$ of $X$ is $k$-compatible with |
884 Given $k>0$ and a blob diagram $b$, we say that a cover $\cU$ of $X$ is $k$-compatible with |
882 $b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union |
885 $b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union |
883 $U_1\cup\cdots\cup U_k$ is a union of balls which satisfies the condition used to define $G_*$ above. |
886 $U_1\cup\cdots\cup U_k$ is a union of balls which satisfies the condition used to define $G_*$ above. |
884 Note that if a family of diffeomorphisms $p$ is adapted to |
887 It follows from Lemma \ref{extension_lemma} |
885 $\cU$ and $b$ is a blob diagram occurring in $x$ \scott{huh, what's $x$ here?}, then $p\otimes b \in G_*$. |
888 that if $\cU$ is $k$-compatible with $b$ and |
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889 $p$ is a $k$-parameter family of diffeomorphisms which is adapted to $\cU$, then |
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890 $p\otimes b \in G_*$. |
886 \nn{maybe emphasize this more; it's one of the main ideas in the proof} |
891 \nn{maybe emphasize this more; it's one of the main ideas in the proof} |
887 |
892 |
888 Let $k$ be the degree of $x$ and choose a cover $\cU$ of $X$ such that $\cU$ is |
893 Let $k$ be the degree of $x$ and choose a cover $\cU$ of $X$ such that $\cU$ is |
889 $k$-compatible with each of the (finitely many) blob diagrams occurring in $x$. |
894 $k$-compatible with each of the (finitely many) blob diagrams occurring in $x$. |
890 We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to |
895 We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to |