52 \def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}} |
52 \def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}} |
53 \applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM; |
53 \applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM; |
54 |
54 |
55 % \DeclareMathOperator{\pr}{pr} etc. |
55 % \DeclareMathOperator{\pr}{pr} etc. |
56 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} |
56 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} |
57 \applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Diff}{sign}; |
57 \applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Diff}{sign}{supp}; |
58 |
58 |
59 |
59 |
60 |
60 |
61 %%%%%% end excerpt |
61 %%%%%% end excerpt |
62 |
62 |
605 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} |
605 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} |
606 } } |
606 } } |
607 Any other map satisfying the above two properties is homotopic to $e_X$. |
607 Any other map satisfying the above two properties is homotopic to $e_X$. |
608 \end{prop} |
608 \end{prop} |
609 |
609 |
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610 \nn{Should say something stronger about uniqueness. |
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611 Something like: there is |
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612 a contractible subcomplex of the complex of chain maps |
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613 $CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), |
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614 and all choices in the construction lie in the 0-cells of this |
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615 contractible subcomplex. |
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616 Or maybe better to say any two choices are homotopic, and |
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617 any two homotopies and second order homotopic, and so on.} |
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618 |
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619 \nn{Also need to say something about associativity. |
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620 Put it in the above prop or make it a separate prop? |
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621 I lean toward the latter.} |
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622 \medskip |
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623 |
610 The proof will occupy the remainder of this section. |
624 The proof will occupy the remainder of this section. |
611 |
625 |
612 \medskip |
626 \medskip |
613 |
627 |
614 Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
628 Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
645 \end{lemma} |
659 \end{lemma} |
646 |
660 |
647 The proof will be given in Section \ref{fam_diff_sect}. |
661 The proof will be given in Section \ref{fam_diff_sect}. |
648 |
662 |
649 \medskip |
663 \medskip |
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664 |
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665 The strategy for the proof of Proposition \ref{CDprop} is as follows. |
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666 We will identify a subcomplex |
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667 \[ |
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668 G_* \sub CD_*(X) \otimes \bc_*(X) |
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669 \] |
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670 on which the evaluation map is uniquely determined (up to homotopy) by the conditions |
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671 in \ref{CDprop}. |
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672 We then show that the inclusion of $G_*$ into the full complex |
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673 is an equivalence in the appropriate sense. |
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674 \nn{need to be more specific here} |
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675 |
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676 Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
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677 Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$ |
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678 intersects at most one blob $B$ of $b$. |
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679 Since $V \cup B$ might not itself be a ball, we need a more careful and complicated definition. |
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680 Choose a metric for $X$. |
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681 We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that |
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682 $N_\epsilon(b) \cup \supp(p)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon |
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683 neighborhood of the support of $b$. |
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684 \nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.} |
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685 |
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686 \nn{need to worry about case where the intrinsic support of $p$ is not a union of balls} |
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687 |
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688 \nn{need to eventually show independence of choice of metric. maybe there's a better way than |
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689 choosing a metric. perhaps just choose a nbd of each ball, but I think I see problems |
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690 with that as well.} |
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691 |
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692 Next we define the evaluation map on $G_*$. |
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693 We'll proceed inductively on $G_i$. |
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694 The induction starts on $G_0$, where we have no choice for the evaluation map |
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695 because $G_0 \sub CD_0\otimes \bc_0$. |
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696 Assume we have defined the evaluation map up to $G_{k-1}$ and |
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697 let $p\otimes b$ be a generator of $G_k$. |
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698 Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. |
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699 |
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700 |
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701 |
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702 |
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703 |
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704 \medskip |
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705 \hrule |
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706 \medskip |
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707 \hrule |
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708 \medskip |
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709 \nn{older stuff:} |
650 |
710 |
651 Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$ |
711 Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$ |
652 (e.g.~the support of a blob diagram). |
712 (e.g.~the support of a blob diagram). |
653 We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if |
713 We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if |
654 $f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$ |
714 $f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$ |