98 |
98 |
99 \medskip |
99 \medskip |
100 |
100 |
101 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}. |
101 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}. |
102 |
102 |
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103 %Suppose for the moment that evaluation maps with the advertised properties exist. |
103 Let $p$ be a singular cell in $CD_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
104 Let $p$ be a singular cell in $CD_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
104 Suppose that there exists $V \sub X$ such that |
105 Suppose that there exists $V \sub X$ such that |
105 \begin{enumerate} |
106 \begin{enumerate} |
106 \item $V$ is homeomorphic to a disjoint union of balls, and |
107 \item $V$ is homeomorphic to a disjoint union of balls, and |
107 \item $\supp(p) \cup \supp(b) \sub V$. |
108 \item $\supp(p) \cup \supp(b) \sub V$. |
110 We then have a factorization |
111 We then have a factorization |
111 \[ |
112 \[ |
112 p = \gl(q, r), |
113 p = \gl(q, r), |
113 \] |
114 \] |
114 where $q \in CD_k(V, V')$ and $r' \in CD_0(W, W')$. |
115 where $q \in CD_k(V, V')$ and $r' \in CD_0(W, W')$. |
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116 We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$. |
115 According to the commutative diagram of the proposition, we must have |
117 According to the commutative diagram of the proposition, we must have |
116 \[ |
118 \[ |
117 e_X(p) = e_X(\gl(q, r)) = gl(e_{VV'}(q), e_{WW'}(r)) . |
119 e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = |
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120 gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) . |
118 \] |
121 \] |
119 \nn{need to add blob parts to above} |
122 Since $r$ is a plain, 0-parameter family of diffeomorphisms, we must have |
120 Since $r$ is a plain, 0-parameter family of diffeomorphisms, |
123 \[ |
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124 e_{WW'}(r\otimes b_W) = r(b_W), |
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125 \] |
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126 where $r(b_W)$ denotes the obvious action of diffeomorphisms on blob diagrams (in |
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127 this case a 0-blob diagram). |
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128 Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ |
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129 (by \ref{disjunion} and \ref{bcontract}). |
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130 Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$, |
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131 there is, up to homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$ |
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132 such that |
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133 \[ |
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134 \bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) . |
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135 \] |
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136 |
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137 Thus the conditions of the proposition determine (up to homotopy) the evaluation |
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138 map for generators $p\otimes b$ such that $\supp(p) \cup \supp(b)$ is contained in a disjoint |
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139 union of balls. |
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140 On the other hand, Lemma \ref{extension_lemma} allows us to homotope |
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141 \nn{is this commonly used as a verb?} arbitrary generators to sums of generators with this property. |
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142 \nn{should give a name to this property} |
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143 This (roughly) establishes the uniqueness part of the proposition. |
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144 To show existence, we must show that the various choices involved in constructing |
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145 evaluation maps in this way affect the final answer only by a homotopy. |
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146 |
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147 \nn{now for a more detailed outline of the proof...} |
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148 |
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149 |
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150 |
121 \medskip |
151 \medskip |
122 |
152 |
123 \nn{to be continued....} |
153 \nn{to be continued....} |
124 |
154 |
125 |
155 |