2112 \cS(A\cup B\cup \ol{B}) \stackrel{\id\ot\psi}{\longrightarrow} |
2112 \cS(A\cup B\cup \ol{B}) \stackrel{\id\ot\psi}{\longrightarrow} |
2113 \cS(A\cup(D\times I)) \stackrel{\cong}{\longrightarrow} \cS(A) . |
2113 \cS(A\cup(D\times I)) \stackrel{\cong}{\longrightarrow} \cS(A) . |
2114 \] |
2114 \] |
2115 (See Figure \ref{jun23b}.) |
2115 (See Figure \ref{jun23b}.) |
2116 \begin{figure}[t] |
2116 \begin{figure}[t] |
2117 \begin{equation*} |
2117 $$ |
2118 \mathfig{.5}{tempkw/jun23b} |
2118 \begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=1.5cm] |
2119 \end{equation*} |
2119 \draw (0,0) node(R) {} |
|
2120 -- (0.75,0) node[below] {$\bar{B}$} |
|
2121 --(1.5,0) node[circle,fill=black,inner sep=2pt] {} |
|
2122 arc (0:80:1.5) node[above] {$D \times I$} |
|
2123 arc (80:180:1.5); |
|
2124 \foreach \r in {0.3, 0.6, 0.9, 1.2} { |
|
2125 \draw[blue!50, line width = 0.5pt] (\r,0) arc (0:180:\r); |
|
2126 } |
|
2127 \draw[fill=white] |
|
2128 (R) node[circle,fill=black,inner sep=2pt] {} |
|
2129 arc (45:65:3) node[below] {$B$} |
|
2130 arc (65:90:3) node[below] {$A$} |
|
2131 arc (90:135:3) node[circle,fill=black,inner sep=2pt] {} |
|
2132 arc (-135:-90:3) node[below] {$C$} |
|
2133 arc (-90:-45:3); |
|
2134 \draw[fill] (150:1.5) circle (2pt) node[above=4pt] {$D$}; |
|
2135 \node[green!50!brown] at (-2,0) {\scalebox{2.0}{$\uparrow f$}}; |
|
2136 \node[green!50!brown] at (0.2,0.8) {\scalebox{2.0}{$\uparrow \psi$}}; |
|
2137 \end{tikzpicture} |
|
2138 $$ |
2120 \caption{Moving $B$ from top to bottom} |
2139 \caption{Moving $B$ from top to bottom} |
2121 \label{jun23b} |
2140 \label{jun23b} |
2122 \end{figure} |
2141 \end{figure} |
2123 Let $D' = B\cap C$. |
2142 Let $D' = B\cap C$. |
2124 Using the inner products there is an adjoint map |
2143 Using the inner products there is an adjoint map |
2134 \cS(A\cup B) . |
2153 \cS(A\cup B) . |
2135 \] |
2154 \] |
2136 (See Figure \ref{jun23c}.) |
2155 (See Figure \ref{jun23c}.) |
2137 \begin{figure}[t] |
2156 \begin{figure}[t] |
2138 \begin{equation*} |
2157 \begin{equation*} |
2139 \mathfig{.5}{tempkw/jun23c} |
2158 \begin{tikzpicture}[baseline,line width = 1pt,x=1.5cm,y=-1.5cm] |
|
2159 \draw (0,0) node(R) {} |
|
2160 -- (0.75,0) node[above] {$B$} |
|
2161 --(1.5,0) node[circle,fill=black,inner sep=2pt] {} |
|
2162 arc (0:80:1.5) node[below] {$D' \times I$} |
|
2163 arc (80:180:1.5); |
|
2164 \foreach \r in {0.3, 0.6, 0.9, 1.2} { |
|
2165 \draw[blue!50, line width = 0.5pt] (\r,0) arc (0:180:\r); |
|
2166 } |
|
2167 \draw[fill=white] |
|
2168 (R) node[circle,fill=black,inner sep=2pt] {} |
|
2169 arc (45:65:3) node[above] {$\bar{B}$} |
|
2170 arc (65:90:3) node[below] {$C$} |
|
2171 arc (90:135:3) node[circle,fill=black,inner sep=2pt] {} |
|
2172 arc (-135:-90:3) node[below] {$A$} |
|
2173 arc (-90:-45:3); |
|
2174 \draw[fill] (150:1.5) circle (2pt) node[below=4pt] {$D'$}; |
|
2175 \node[green!50!brown] at (-2,0) {\scalebox{2.0}{$f'\uparrow $}}; |
|
2176 \node[green!50!brown] at (0.2,0.8) {\scalebox{2.0}{$\psi^+\uparrow $}}; |
|
2177 \end{tikzpicture} |
2140 \end{equation*} |
2178 \end{equation*} |
2141 \caption{Moving $B$ from bottom to top} |
2179 \caption{Moving $B$ from bottom to top} |
2142 \label{jun23c} |
2180 \label{jun23c} |
2143 \end{figure} |
2181 \end{figure} |
2144 Let $D' = B\cap C$. |
2182 Let $D' = B\cap C$. |