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1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{Basic properties of the blob complex} |
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4 \label{sec:basic-properties} |
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5 |
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6 \begin{prop} \label{disjunion} |
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7 There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
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8 \end{prop} |
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9 \begin{proof} |
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10 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
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11 (putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
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12 blob diagram $(b_1, b_2)$ on $X \du Y$. |
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13 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
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14 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
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15 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
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16 a pair of blob diagrams on $X$ and $Y$. |
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17 These two maps are compatible with our sign conventions. |
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18 The two maps are inverses of each other. |
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19 \nn{should probably say something about sign conventions for the differential |
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20 in a tensor product of chain complexes; ask Scott} |
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21 \end{proof} |
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22 |
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23 For the next proposition we will temporarily restore $n$-manifold boundary |
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24 conditions to the notation. |
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25 |
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26 Suppose that for all $c \in \cC(\bd B^n)$ |
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27 we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
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28 of the quotient map |
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29 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
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30 For example, this is always the case if you coefficient ring is a field. |
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31 Then |
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32 \begin{prop} \label{bcontract} |
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33 For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
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34 is a chain homotopy equivalence |
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35 with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
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36 Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0. |
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37 \end{prop} |
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38 \begin{proof} |
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39 By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map |
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40 $h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$. |
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41 For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
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42 an $(i{+}1)$-st blob equal to all of $B^n$. |
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43 In other words, add a new outermost blob which encloses all of the others. |
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44 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
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45 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
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46 \end{proof} |
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47 |
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48 Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
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49 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
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50 |
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51 For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
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52 where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
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53 |
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54 \medskip |
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55 |
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56 \nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction. |
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57 But I think it's worth saying that the Diff actions will be enhanced later. |
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58 Maybe put that in the intro too.} |
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59 |
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60 As we noted above, |
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61 \begin{prop} |
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62 There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
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63 \qed |
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64 \end{prop} |
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65 |
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66 |
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67 \begin{prop} |
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68 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
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69 of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
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70 (chain map) isomorphisms. |
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71 \qed |
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72 \end{prop} |
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73 |
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74 In particular, |
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75 \begin{prop} \label{diff0prop} |
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76 There is an action of $\Diff(X)$ on $\bc_*(X)$. |
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77 \qed |
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78 \end{prop} |
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79 |
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80 The above will be greatly strengthened in Section \ref{sec:evaluation}. |
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81 |
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82 \medskip |
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83 |
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84 For the next proposition we will temporarily restore $n$-manifold boundary |
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85 conditions to the notation. |
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86 |
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87 Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
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88 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
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89 with boundary $Z\sgl$. |
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90 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
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91 we have the blob complex $\bc_*(X; a, b, c)$. |
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92 If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
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93 $X$ to get blob diagrams on $X\sgl$: |
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94 |
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95 \begin{prop} |
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96 There is a natural chain map |
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97 \eq{ |
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98 \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
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99 } |
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100 The sum is over all fields $a$ on $Y$ compatible at their |
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101 ($n{-}2$-dimensional) boundaries with $c$. |
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102 `Natural' means natural with respect to the actions of diffeomorphisms. |
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103 \qed |
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104 \end{prop} |
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105 |
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106 The above map is very far from being an isomorphism, even on homology. |
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107 This will be fixed in Section \ref{sec:gluing} below. |
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108 |
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109 \nn{Next para not need, since we already use bullet = gluing notation above(?)} |
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110 |
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111 An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
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112 and $X\sgl = X_1 \cup_Y X_2$. |
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113 (Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
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114 For $x_i \in \bc_*(X_i)$, we introduce the notation |
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115 \eq{ |
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116 x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
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117 } |
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118 Note that we have resumed our habit of omitting boundary labels from the notation. |
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119 |
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120 |
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121 |