equal
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replaced
1965 Let $X_1$ and $X_2$ be the two components of $Y\times I$ (pinched) cut along |
1965 Let $X_1$ and $X_2$ be the two components of $Y\times I$ (pinched) cut along |
1966 $D\times I$. |
1966 $D\times I$. |
1967 (Here we are overloading notation and letting $D$ denote both a decorated and an undecorated |
1967 (Here we are overloading notation and letting $D$ denote both a decorated and an undecorated |
1968 manifold.) |
1968 manifold.) |
1969 We have $\bd X_i = Y_i \cup \ol{Y}_i \cup (D\times I)$ |
1969 We have $\bd X_i = Y_i \cup \ol{Y}_i \cup (D\times I)$ |
1970 (see Figure xxxx). |
1970 (see Figure \ref{jun23a}). |
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1971 \begin{figure}[t] |
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1972 \begin{equation*} |
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1973 \mathfig{.6}{tempkw/jun23a} |
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1974 \end{equation*} |
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1975 \caption{$Y\times I$ sliced open} |
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1976 \label{jun23a} |
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1977 \end{figure} |
1971 Given $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ and $v\in\cS(D\times I)$ |
1978 Given $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ and $v\in\cS(D\times I)$ |
1972 which agree on their boundaries, we can evaluate |
1979 which agree on their boundaries, we can evaluate |
1973 \[ |
1980 \[ |
1974 z_{Y_i}(a_i\bullet b_i\bullet v) \in \c . |
1981 z_{Y_i}(a_i\bullet b_i\bullet v) \in \c . |
1975 \] |
1982 \] |
2006 \[ |
2013 \[ |
2007 \cS(C\cup \ol{B}) \stackrel{f\ot\id}{\longrightarrow} |
2014 \cS(C\cup \ol{B}) \stackrel{f\ot\id}{\longrightarrow} |
2008 \cS(A\cup B\cup \ol{B}) \stackrel{\id\ot\psi}{\longrightarrow} |
2015 \cS(A\cup B\cup \ol{B}) \stackrel{\id\ot\psi}{\longrightarrow} |
2009 \cS(A\cup(D\times I)) \stackrel{\cong}{\longrightarrow} \cS(A) . |
2016 \cS(A\cup(D\times I)) \stackrel{\cong}{\longrightarrow} \cS(A) . |
2010 \] |
2017 \] |
2011 (See Figure xxxx.) |
2018 (See Figure \ref{jun23b}.) |
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2019 \begin{figure}[t] |
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2020 \begin{equation*} |
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2021 \mathfig{.5}{tempkw/jun23b} |
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2022 \end{equation*} |
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2023 \caption{Moving $B$ from top to bottom} |
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2024 \label{jun23b} |
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2025 \end{figure} |
2012 Let $D' = B\cap C$. |
2026 Let $D' = B\cap C$. |
2013 Using the inner products there is an adjoint map |
2027 Using the inner products there is an adjoint map |
2014 \[ |
2028 \[ |
2015 \psi^\dagger: \cS(D'\times I) \to \cS(\ol{B})\ot\cS(B) . |
2029 \psi^\dagger: \cS(D'\times I) \to \cS(\ol{B})\ot\cS(B) . |
2016 \] |
2030 \] |
2020 \cS(C) \stackrel{\cong}{\longrightarrow} |
2034 \cS(C) \stackrel{\cong}{\longrightarrow} |
2021 \cS(C\cup(D'\times I)) \stackrel{\id\ot\psi^\dagger}{\longrightarrow} |
2035 \cS(C\cup(D'\times I)) \stackrel{\id\ot\psi^\dagger}{\longrightarrow} |
2022 \cS(C\cup \ol{B}\cup B) \stackrel{f'\ot\id}{\longrightarrow} |
2036 \cS(C\cup \ol{B}\cup B) \stackrel{f'\ot\id}{\longrightarrow} |
2023 \cS(A\cup B) . |
2037 \cS(A\cup B) . |
2024 \] |
2038 \] |
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2039 (See Figure \ref{jun23c}.) |
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2040 \begin{figure}[t] |
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2041 \begin{equation*} |
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2042 \mathfig{.5}{tempkw/jun23c} |
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2043 \end{equation*} |
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2044 \caption{Moving $B$ from bottom to top} |
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2045 \label{jun23c} |
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2046 \end{figure} |
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2047 Let $D' = B\cap C$. |
2025 It is not hard too show that the above two maps are mutually inverse. |
2048 It is not hard too show that the above two maps are mutually inverse. |
2026 |
2049 |
2027 \begin{lem} |
2050 \begin{lem} |
2028 Any two choices of $E$ and $E'$ are related by a series of modifications as above. |
2051 Any two choices of $E$ and $E'$ are related by a series of modifications as above. |
2029 \end{lem} |
2052 \end{lem} |
2047 As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
2070 As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
2048 inverse, so we have invariance under this movie move. |
2071 inverse, so we have invariance under this movie move. |
2049 |
2072 |
2050 The second movie move replaces to successive pushes in the same direction, |
2073 The second movie move replaces to successive pushes in the same direction, |
2051 across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$. |
2074 across $B_1$ and $B_2$, say, with a single push across $B_1\cup B_2$. |
2052 (See Figure xxxx.) |
2075 (See Figure \ref{jun23d}.) |
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2076 \begin{figure}[t] |
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2077 \begin{equation*} |
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2078 \mathfig{.9}{tempkw/jun23d} |
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2079 \end{equation*} |
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2080 \caption{A movie move} |
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2081 \label{jun23d} |
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2082 \end{figure} |
2053 Invariance under this movie move follows from the compatibility of the inner |
2083 Invariance under this movie move follows from the compatibility of the inner |
2054 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
2084 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
2055 |
2085 |
2056 If $n\ge 2$, these two movie move suffice: |
2086 If $n\ge 2$, these two movie move suffice: |
2057 |
2087 |