113 are transverse to $Y$ or splittable along $Y$. |
113 are transverse to $Y$ or splittable along $Y$. |
114 \item Gluing with corners. |
114 \item Gluing with corners. |
115 Let $\bd X = (Y \du Y) \cup W$, where the two copies of $Y$ |
115 Let $\bd X = (Y \du Y) \cup W$, where the two copies of $Y$ |
116 are disjoint from each other and $\bd(Y\du Y) = \bd W$. |
116 are disjoint from each other and $\bd(Y\du Y) = \bd W$. |
117 Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$ |
117 Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$ |
118 (Figure \ref{fig:???}). |
118 (Figure \ref{fig:gluing-with-corners}). |
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119 \begin{figure}[t] |
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120 \begin{center} |
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121 \begin{tikzpicture} |
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122 |
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123 \node(A) at (-4,0) { |
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124 \begin{tikzpicture}[scale=.8, fill=blue!15!white] |
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125 \filldraw[line width=1.5pt] (-.4,1) .. controls +(-1,-.1) and +(-1,0) .. (0,-1) |
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126 .. controls +(1,0) and +(1,-.1) .. (.4,1) -- (.4,3) |
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127 .. controls +(3,-.4) and +(3,0) .. (0,-3) |
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128 .. controls +(-3,0) and +(-3,-.1) .. (-.4,3) -- cycle; |
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129 \node at (0,-2) {$X$}; |
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130 \node (W) at (-2.7,-2) {$W$}; |
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131 \node (Y1) at (-1.2,3.5) {$Y$}; |
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132 \node (Y2) at (1.4,3.5) {$Y$}; |
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133 \node[outer sep=2.3] (y1e) at (-.4,2) {}; |
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134 \node[outer sep=2.3] (y2e) at (.4,2) {}; |
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135 \node (we1) at (-2.2,-1.1) {}; |
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136 \node (we2) at (-.6,-.7) {}; |
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137 \draw[->] (Y1) -- (y1e); |
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138 \draw[->] (Y2) -- (y2e); |
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139 \draw[->] (W) .. controls +(0,.5) and +(-.5,-.2) .. (we1); |
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140 \draw[->] (W) .. controls +(.5,0) and +(-.2,-.5) .. (we2); |
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141 \end{tikzpicture} |
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142 }; |
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143 |
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144 \node(B) at (4,0) { |
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145 \begin{tikzpicture}[scale=.8, fill=blue!15!white] |
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146 \fill (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1) |
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147 .. controls +(1,0) and +(1,0) .. (0,1) -- (0,3) |
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148 .. controls +(3,0) and +(3,0) .. (0,-3) |
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149 .. controls +(-3,0) and +(-3,0) .. (0,3) -- cycle; |
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150 \draw[line width=1.5pt] (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1) |
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151 .. controls +(1,0) and +(1,0) .. (0,1); |
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152 \draw[line width=1.5pt] (0,3) .. controls +(3,0) and +(3,0) .. (0,-3) |
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153 .. controls +(-3,0) and +(-3,0) .. (0,3); |
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154 \draw[line width=.5pt, black!65!white] (0,1) -- (0,3); |
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155 \node at (0,-2) {$X\sgl$}; |
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156 \node (W) at (2.7,-2) {$W\sgl$}; |
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157 \node (we1) at (2.2,-1.1) {}; |
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158 \node (we2) at (.6,-.7) {}; |
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159 \draw[->] (W) .. controls +(0,.5) and +(.5,-.2) .. (we1); |
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160 \draw[->] (W) .. controls +(-.5,0) and +(.2,-.5) .. (we2); |
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161 \end{tikzpicture} |
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162 }; |
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163 |
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164 |
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165 \draw[->, red!50!green, line width=2pt] (A) -- node[above, black] {glue} (B); |
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166 |
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167 \end{tikzpicture} |
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168 \end{center} |
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169 \caption{Gluing with corners} |
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170 \label{fig:gluing-with-corners} |
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171 \end{figure} |
119 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
172 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
120 (without corners) along two copies of $\bd Y$. |
173 (without corners) along two copies of $\bd Y$. |
121 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let |
174 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let |
122 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
175 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
123 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
176 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |