720 |
724 |
721 Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists |
725 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 |
726 a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and |
723 this homotopy are unique up to iterated homotopy. |
727 this homotopy are unique up to iterated homotopy. |
724 |
728 |
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. |
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732 Note that if a family of diffeomorphisms $p$ is adapted to |
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733 $\cU$ and $b$ is a blob diagram occurring in $x$, then $p\otimes b \in G_*$. |
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734 \nn{maybe emphasize this more; it's one of the main ideas in the proof} |
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735 |
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736 Let $k$ be the degree of $x$ and choose a cover $\cU$ of $X$ such that $\cU$ is |
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737 $k$-compatible with each of the (finitely many) blob diagrams occurring in $x$. |
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738 We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to |
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739 construct the homotopy to $G_*$. |
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740 First we will construct a homotopy $h \in G_*$ from $\bd x$ to a cycle $z$ such that |
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741 each family of diffeomorphisms $p$ occurring in $z$ is adapted to $\cU$. |
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742 Then we will construct a homotopy (rel boundary) $r$ from $x + h$ to $y$ such that |
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743 each family of diffeomorphisms $p$ occurring in $y$ is adapted to $\cU$. |
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744 This implies that $y \in G_*$. |
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745 $r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek. |
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746 |
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747 We will define $h$ inductive on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$. |
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748 Define $h$ to be zero on bidegree $(0, k-1)$. |
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749 Let $p\otimes b$ be a generator occurring in $\bd x$ with bidegree $(1, k-2)$. |
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750 Using Lemma \ref{extension_lemma}, construct a homotopy $q$ from $p$ to $p'$ which is adapted to $\cU$. |
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751 Define $h$ at $p\otimes b$ to be $q\otimes b$. |
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752 Let $p'\otimes b'$ be a generator occurring in $\bd x$ with bidegree $(2, k-3)$. |
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753 Let $a$ be that portion of $\bd(p'\otimes b')$ which intersects the boundary of |
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754 bidegree $(1, k-2)$ stuff. |
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755 Apply Lemma \ref{extension_lemma} to $p'$ plus the diffeo part of $h(a)$ |
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756 (rel the outer boundary of said part), |
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757 yielding a family of diffeos $q'$. |
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758 \nn{definitely need to say this better} |
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759 Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$. |
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760 Continuing in this way, we define all of $h$. |
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761 |
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762 The homotopy $r$ is constructed similarly. |
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763 |
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764 \nn{need to say something about uniqueness of $r$, $h$ etc. |
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765 postpone this until second draft.} |
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766 |
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767 At this point, we have finished defining the evaluation map. |
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768 The uniqueness statement in the proposition is clear from the method of proof. |
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769 All that remains is to show that the evaluation map gets along well with cutting and gluing, |
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770 as claimed in the proposition. |
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771 This is in fact not difficult, since the myriad choices involved in defining the |
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772 evaluation map can be made in parallel for the top and bottom |
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773 arrows in the commutative diagram. |
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774 |
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775 This completes the proof of Proposition \ref{CDprop}. |
728 |
776 |
729 \medskip |
777 \medskip |
730 \hrule |
778 |
731 \medskip |
779 \nn{say something about associativity here} |
732 \hrule |
780 |
733 \medskip |
781 |
734 \nn{older stuff:} |
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735 |
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736 Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$ |
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737 (e.g.~the support of a blob diagram). |
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738 We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if |
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739 $f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$ |
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740 either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$. |
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741 A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells, |
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742 each of which is compatible. |
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743 (Note that we could strengthen the definition of compatibility to incorporate |
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744 a factorization condition, similar to the definition of ``adapted to" above. |
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745 The weaker definition given here will suffice for our needs below.) |
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746 |
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747 \begin{cor} \label{extension_lemma_2} |
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748 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is compatible with $\{B_j\}$. |
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749 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$. |
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750 \end{cor} |
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751 \begin{proof} |
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752 This will follow from Lemma \ref{extension_lemma} for |
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753 appropriate choice of cover $\cU = \{U_\alpha\}$. |
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754 Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let |
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755 $V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$. |
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756 Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$ |
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757 either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$. |
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758 |
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759 Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$, |
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760 with the (compatible) support of $f_i$ in place of $X$. |
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761 This insures that the resulting homotopy $h_i$ is compatible. |
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762 Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$. |
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763 \nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this} |
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764 \end{proof} |
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765 |
|
766 \medskip |
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767 |
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768 ((argument continues roughly as follows: up to homotopy, there is only one way to define $e_X$ |
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769 on compatible $x\otimes y \in CD_*(X)\otimes \bc_*(X)$. |
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770 This is because $x$ is the gluing of $x'$ and $x''$, where $x'$ has degree zero and is defined on |
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771 the complement of the $D_i$'s, and $x''$ is defined on the $D_i$'s. |
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772 We have no choice on $x'$, since we already know the map on 0-parameter families of diffeomorphisms. |
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773 We have no choice, up to homotopy, on $x''$, since the target chain complex is contractible.)) |
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774 |
782 |
775 |
783 |
776 \section{Families of Diffeomorphisms} \label{fam_diff_sect} |
784 \section{Families of Diffeomorphisms} \label{fam_diff_sect} |
777 |
785 |
778 |
786 |