1926 We will show that if the sphere modules are equipped with a compatible family of |
1926 We will show that if the sphere modules are equipped with a compatible family of |
1927 non-degenerate inner products, then there is a coherent family of isomorphisms |
1927 non-degenerate inner products, then there is a coherent family of isomorphisms |
1928 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
1928 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
1929 This will allow us to define $\cS(X; e)$ independently of the choice of $E$. |
1929 This will allow us to define $\cS(X; e)$ independently of the choice of $E$. |
1930 |
1930 |
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1931 First we must define ``inner product", ``non-degenerate" and ``compatible". |
1931 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
1932 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
1932 (We assume we are working in the unoriented category.) |
1933 (We assume we are working in the unoriented category.) |
1933 Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
1934 Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
1934 along their common boundary. |
1935 along their common boundary. |
1935 An {\it inner product} on $\cS(Y)$ is a dual vector |
1936 An {\it inner product} on $\cS(Y)$ is a dual vector |
1938 \] |
1939 \] |
1939 We will also use the notation |
1940 We will also use the notation |
1940 \[ |
1941 \[ |
1941 \langle a, b\rangle \deq z_Y(a\bullet \ol{b}) \in \c . |
1942 \langle a, b\rangle \deq z_Y(a\bullet \ol{b}) \in \c . |
1942 \] |
1943 \] |
1943 An inner product is {\it non-degenerate} if |
1944 An inner product induces a linear map |
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1945 \begin{eqnarray*} |
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1946 \varphi: \cS(Y) &\to& \cS(Y)^* \\ |
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1947 a &\mapsto& \langle a, \cdot \rangle |
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1948 \end{eqnarray*} |
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1949 which satisfies, for all morphisms $e$ of $\cS(\bd Y)$, |
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1950 \[ |
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1951 \varphi(ae)(b) = \langle ae, b \rangle = z_Y(a\bullet e\bullet b) = |
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1952 \langle a, eb \rangle = \varphi(a)(eb) . |
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1953 \] |
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1954 In other words, $\varphi$ is a map of $\cS(\bd Y)$ modules. |
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1955 An inner product is {\it non-degenerate} if $\varphi$ is an isomorphism. |
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1956 This implies that $\cS(Y; c)$ is finite dimensional for all boundary conditions $c$. |
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1957 (One can think of these inner products as giving some duality in dimension $n{+}1$; |
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1958 heretofore we have only assumed duality in dimensions 0 through $n$.) |
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1959 |
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1960 Next we define compatibility. |
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1961 Let $Y = Y_1\cup Y_2$ with $D = Y_1\cap Y_2$. |
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1962 Let $X_1$ and $X_2$ be the two components of $Y\times I$ (pinched) cut along |
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1963 $D\times I$. |
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1964 (Here we are overloading notation and letting $D$ denote both a decorated and an undecorated |
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1965 manifold.) |
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1966 We have $\bd X_i = Y_i \cup \ol{Y}_i \cup (D\times I)$ |
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1967 (see Figure xxxx). |
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1968 Given $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ and $v\in\cS(D\times I)$ |
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1969 which agree on their boundaries, we can evaluate |
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1970 \[ |
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1971 z_{Y_i}(a_i\bullet b_i\bullet v) \in \c . |
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1972 \] |
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1973 (This requires a choice of homeomorphism $Y_i \cup \ol{Y}_i \cup (D\times I) \cong |
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1974 Y_i \cup \ol{Y}_i$, but the value of $z_{Y_i}$ is independent of this choice.) |
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1975 We can think of $z_{Y_i}$ as giving a function |
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1976 \[ |
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1977 \psi_i : \cS(Y_i) \ot \cS(\ol{Y}_i) \to \cS(D\times I)^* |
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1978 \stackrel{\varphi\inv}{\longrightarrow} \cS(D\times I) . |
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1979 \] |
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1980 We can now finally define a family of inner products to be {\it compatible} if |
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1981 for all decompositions $Y = Y_1\cup Y_2$ as above and all $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ |
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1982 we have |
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1983 \[ |
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1984 z_Y(a_1\bullet a_2\bullet b_1\bullet b_2) = |
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1985 z_{D\times I}(\psi_1(a_1\ot b_1)\bullet \psi_2(a_2\ot b_2)) . |
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1986 \] |
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1987 In other words, the inner product on $Y$ is determined by the inner products on |
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1988 $Y_1$, $Y_2$ and $D\times I$. |
1944 |
1989 |
1945 \nn{...} |
1990 \nn{...} |
1946 |
1991 |
1947 \medskip |
1992 \medskip |
1948 \hrule |
1993 \hrule |