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1 %!TEX root = ../blob1.tex |
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2 |
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3 Let $CD_*(X, Y)$ denote $C_*(\Diff(X \to Y))$, the singular chain complex of |
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4 the space of diffeomorphisms |
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5 \nn{or homeomorphisms} |
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6 between the $n$-manifolds $X$ and $Y$ (extending a fixed diffeomorphism $\bd X \to \bd Y$). |
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7 For convenience, we will permit the singular cells generating $CD_*(X, Y)$ to be more general |
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8 than simplices --- they can be based on any linear polyhedron. |
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9 \nn{be more restrictive here? does more need to be said?} |
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10 We also will use the abbreviated notation $CD_*(X) \deq CD_*(X, X)$. |
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11 |
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12 \begin{prop} \label{CDprop} |
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13 For $n$-manifolds $X$ and $Y$ there is a chain map |
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14 \eq{ |
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15 e_{XY} : CD_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) . |
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16 } |
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17 On $CD_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X, Y)$ on $\bc_*(X)$ |
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18 (Proposition (\ref{diff0prop})). |
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19 For any splittings $X = X_1 \cup X_2$ and $Y = Y_1 \cup Y_2$, |
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20 the following diagram commutes up to homotopy |
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21 \eq{ \xymatrix{ |
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22 CD_*(X, Y) \otimes \bc_*(X) \ar[r]^{e_{XY}} & \bc_*(Y) \\ |
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23 CD_*(X_1, Y_1) \otimes CD_*(X_2, Y_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
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24 \ar@/_4ex/[r]_{e_{X_1Y_1} \otimes e_{X_2Y_2}} \ar[u]^{\gl \otimes \gl} & |
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25 \bc_*(Y_1) \otimes \bc_*(Y_2) \ar[u]_{\gl} |
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26 } } |
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27 Any other map satisfying the above two properties is homotopic to $e_X$. |
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28 \end{prop} |
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29 |
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30 \nn{need to rewrite for self-gluing instead of gluing two pieces together} |
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31 |
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32 \nn{Should say something stronger about uniqueness. |
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33 Something like: there is |
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34 a contractible subcomplex of the complex of chain maps |
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35 $CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), |
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36 and all choices in the construction lie in the 0-cells of this |
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37 contractible subcomplex. |
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38 Or maybe better to say any two choices are homotopic, and |
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39 any two homotopies and second order homotopic, and so on.} |
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40 |
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41 \nn{Also need to say something about associativity. |
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42 Put it in the above prop or make it a separate prop? |
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43 I lean toward the latter.} |
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44 \medskip |
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45 |
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46 The proof will occupy the remainder of this section. |
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47 \nn{unless we put associativity prop at end} |
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48 |
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49 Without loss of generality, we will assume $X = Y$. |
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50 |
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51 \medskip |
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52 |
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53 Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
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54 We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
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55 $x \notin S$ 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' |
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56 diffeomorphism $f_0 : X \to X$ so that |
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57 \begin{align} |
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58 f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
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59 \intertext{and} |
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60 f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
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61 \end{align} |
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62 Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
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63 |
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64 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
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65 A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
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66 {\it adapted to $\cU$} if there is a factorization |
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67 \eq{ |
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68 P = P_1 \times \cdots \times P_m |
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69 } |
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70 (for some $m \le k$) |
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71 and families of diffeomorphisms |
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72 \eq{ |
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73 f_i : P_i \times X \to X |
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74 } |
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75 such that |
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76 \begin{itemize} |
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77 \item each $f_i$ is supported on some connected $V_i \sub X$; |
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78 \item the sets $V_i$ are mutually disjoint; |
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79 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
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80 where $k_i = \dim(P_i)$; and |
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81 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
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82 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
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83 \end{itemize} |
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84 A chain $x \in CD_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
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85 of singular cells, each of which is adapted to $\cU$. |
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86 |
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87 (Actually, in this section we will only need families of diffeomorphisms to be |
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88 {\it weakly adapted} to $\cU$, meaning that the support of $f$ is contained in the union |
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89 of at most $k$ of the $U_\alpha$'s.) |
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90 |
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91 \begin{lemma} \label{extension_lemma} |
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92 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
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93 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
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94 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
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95 \end{lemma} |
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96 |
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97 The proof will be given in Section \ref{sec:localising}. |
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98 |
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99 \medskip |
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100 |
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101 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}. |
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102 |
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103 Let $p$ be a singular cell in $CD_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
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104 Suppose that there exists $V \sub X$ such that |
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105 \begin{enumerate} |
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106 \item $V$ is homeomorphic to a disjoint union of balls, and |
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107 \item $\supp(p) \cup \supp(b) \sub V$. |
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108 \end{enumerate} |
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109 Let $W = X \setmin V$, and let $V' = p(V)$ and $W' = p(W)$. |
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110 We then have a factorization |
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111 \[ |
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112 p = \gl(q, r), |
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113 \] |
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114 where $q \in CD_k(V, V')$ and $r' \in CD_0(W, W')$. |
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115 According to the commutative diagram of the proposition, we must have |
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116 \[ |
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117 e_X(p) = e_X(\gl(q, r)) = gl(e_{VV'}(q), e_{WW'}(r)) . |
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118 \] |
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119 \nn{need to add blob parts to above} |
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120 Since $r$ is a plain, 0-parameter family of diffeomorphisms, |
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121 \medskip |
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122 |
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123 \nn{to be continued....} |
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124 |
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125 |
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126 %\nn{say something about associativity here} |
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127 |
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128 |
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129 |
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130 |
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131 |