1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{Basic properties of the blob complex} |
3 \section{Basic properties of the blob complex} |
4 \label{sec:basic-properties} |
4 \label{sec:basic-properties} |
5 |
5 |
6 \begin{prop} \label{disjunion} |
6 In this section we complete the proofs of Properties 1-4. Throughout the paper, where possible, we prove results using Properties 1-4, rather than the actual definition of blob homology. This allows the possibility of future improvements to or alternatives on our definition. In fact, we hope that there may be a characterisation of blob homology in terms of Properties 1-4, but at this point we are unaware of one. |
7 There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
7 |
8 \end{prop} |
8 Recall Property \ref{property:disjoint-union}, that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
9 \begin{proof} |
9 |
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10 \begin{proof}[Proof of Property \ref{property:disjoint-union}] |
10 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
11 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
11 (putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
12 (putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
12 blob diagram $(b_1, b_2)$ on $X \du Y$. |
13 blob diagram $(b_1, b_2)$ on $X \du Y$. |
13 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
14 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
14 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
15 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
15 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
16 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
16 a pair of blob diagrams on $X$ and $Y$. |
17 a pair of blob diagrams on $X$ and $Y$. |
17 These two maps are compatible with our sign conventions. |
18 These two maps are compatible with our sign conventions. (We follow the usual convention for tensors products of complexes, as in e.g. \cite{MR1438306}: $d(a \tensor b) = da \tensor b + (-1)^{\deg(a)} a \tensor db$.) |
18 The two maps are inverses of each other. |
19 The two maps are inverses of each other. |
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} |
20 \end{proof} |
22 |
21 |
23 For the next proposition we will temporarily restore $n$-manifold boundary |
22 For the next proposition we will temporarily restore $n$-manifold boundary |
24 conditions to the notation. |
23 conditions to the notation. |
25 |
24 |
42 an $(i{+}1)$-st blob equal to all of $B^n$. |
41 an $(i{+}1)$-st blob equal to all of $B^n$. |
43 In other words, add a new outermost blob which encloses all of the others. |
42 In other words, add a new outermost blob which encloses all of the others. |
44 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
43 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
45 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
44 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
46 \end{proof} |
45 \end{proof} |
47 |
46 This proves Property \ref{property:contractibility} (the second half of the statement of this Property was immediate from the definitions). |
48 Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
47 Note that even when there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
49 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
48 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
50 |
49 |
51 For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
50 For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
52 where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
51 where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
53 |
52 |
54 \begin{cor} \label{disj-union-contract} |
53 \begin{cor} \label{disj-union-contract} |
55 If $X$ is a disjoint union of $n$-balls, then $\bc_*(X; c)$ is contractible. |
54 If $X$ is a disjoint union of $n$-balls, then $\bc_*(X; c)$ is contractible. |
56 \end{cor} |
55 \end{cor} |
57 |
56 |
58 \begin{proof} |
57 \begin{proof} |
59 This follows from \ref{disjunion} and \ref{bcontract}. |
58 This follows from Properties \ref{property:disjoint-union} and \ref{property:contractibility}. |
60 \end{proof} |
59 \end{proof} |
61 |
60 |
62 Define the {\it support} of a blob diagram to be the union of all the |
61 Define the {\it support} of a blob diagram to be the union of all the |
63 blobs of the diagram. |
62 blobs of the diagram. |
64 Define the support of a linear combination of blob diagrams to be the union of the |
63 Define the support of a linear combination of blob diagrams to be the union of the |
82 note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$. |
81 note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$. |
83 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), |
82 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), |
84 so $f$ and the identity map are homotopic. |
83 so $f$ and the identity map are homotopic. |
85 \end{proof} |
84 \end{proof} |
86 |
85 |
87 |
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88 \medskip |
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89 |
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90 \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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91 But I think it's worth saying that the Diff actions will be enhanced later. |
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92 Maybe put that in the intro too.} |
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93 |
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94 As we noted above, |
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95 \begin{prop} |
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96 There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
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97 \qed |
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98 \end{prop} |
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99 |
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100 |
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101 \begin{prop} |
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102 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
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103 of $n$-manifolds and homeomorphisms to the category of chain complexes and |
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104 (chain map) isomorphisms. |
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105 \qed |
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106 \end{prop} |
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107 |
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108 In particular, |
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109 \begin{prop} \label{diff0prop} |
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110 There is an action of $\Homeo(X)$ on $\bc_*(X)$. |
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111 \qed |
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112 \end{prop} |
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113 |
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114 The above will be greatly strengthened in Section \ref{sec:evaluation}. |
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115 |
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116 \medskip |
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117 |
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118 For the next proposition we will temporarily restore $n$-manifold boundary |
86 For the next proposition we will temporarily restore $n$-manifold boundary |
119 conditions to the notation. |
87 conditions to the notation. |
120 |
88 |
121 Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
89 Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
122 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
90 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
123 with boundary $Z\sgl$. |
91 with boundary $Z\sgl$. |
124 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
92 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
125 we have the blob complex $\bc_*(X; a, b, c)$. |
93 we have the blob complex $\bc_*(X; a, b, c)$. |
126 If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
94 If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
127 $X$ to get blob diagrams on $X\sgl$: |
95 $X$ to get blob diagrams on $X\sgl$. This proves Property \ref{property:gluing-map}, which we restate here in more detail. |
128 |
96 |
129 \begin{prop} |
97 \textbf{Property \ref{property:gluing-map}.}\emph{ |
130 There is a natural chain map |
98 There is a natural chain map |
131 \eq{ |
99 \eq{ |
132 \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
100 \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
133 } |
101 } |
134 The sum is over all fields $a$ on $Y$ compatible at their |
102 The sum is over all fields $a$ on $Y$ compatible at their |
135 ($n{-}2$-dimensional) boundaries with $c$. |
103 ($n{-}2$-dimensional) boundaries with $c$. |
136 `Natural' means natural with respect to the actions of diffeomorphisms. |
104 `Natural' means natural with respect to the actions of diffeomorphisms. |
137 \qed |
105 } |
138 \end{prop} |
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139 |
106 |
140 The above map is very far from being an isomorphism, even on homology. |
107 This map is very far from being an isomorphism, even on homology. |
141 This will be fixed in Section \ref{sec:gluing} below. |
108 We fix this deficit in Section \ref{sec:gluing} below. |
142 |
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143 %\nn{Next para not needed, since we already use bullet = gluing notation above(?)} |
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144 |
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145 %An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
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146 %and $X\sgl = X_1 \cup_Y X_2$. |
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147 %(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
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148 %For $x_i \in \bc_*(X_i)$, we introduce the notation |
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149 %\eq{ |
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150 % x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
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151 %} |
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152 %Note that we have resumed our habit of omitting boundary labels from the notation. |
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153 |
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154 |
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155 |
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