298 \item A field $r \in \cC(X \setmin B; c)$ |
298 \item A field $r \in \cC(X \setmin B; c)$ |
299 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
299 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
300 \item A local relation field $u \in U(B; c)$ |
300 \item A local relation field $u \in U(B; c)$ |
301 (same $c$ as previous bullet). |
301 (same $c$ as previous bullet). |
302 \end{itemize} |
302 \end{itemize} |
303 %(Note that the the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$, |
303 %(Note that the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$, |
304 %so we will omit $c$ from the notation.) |
304 %so we will omit $c$ from the notation.) |
305 Define $\bc_1(X)$ to be the space of all finite linear combinations of |
305 Define $\bc_1(X)$ to be the space of all finite linear combinations of |
306 1-blob diagrams, modulo the simple relations relating labels of 0-cells and |
306 1-blob diagrams, modulo the simple relations relating labels of 0-cells and |
307 also the label ($u$ above) of the blob. |
307 also the label ($u$ above) of the blob. |
308 \nn{maybe spell this out in more detail} |
308 \nn{maybe spell this out in more detail} |
964 |
964 |
965 In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
965 In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
966 $\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
966 $\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
967 The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
967 The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
968 If we triangulate $R$ (without introducing new vertices), we can linearly extend |
968 If we triangulate $R$ (without introducing new vertices), we can linearly extend |
969 a map from the the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
969 a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
970 Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
970 Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
971 the $k{-}j$-cell corresponding to $E$. |
971 the $k{-}j$-cell corresponding to $E$. |
972 For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
972 For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
973 Now define, for $p \in E$, |
973 Now define, for $p \in E$, |
974 \eq{ |
974 \eq{ |
987 Next we verify that $u$ has the desired properties. |
987 Next we verify that $u$ has the desired properties. |
988 |
988 |
989 Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$. |
989 Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$. |
990 Therefore $F$ is a homotopy from $f$ to something. |
990 Therefore $F$ is a homotopy from $f$ to something. |
991 |
991 |
992 Next we show that the the $K_\alpha$'s are sufficiently fine cell decompositions, |
992 Next we show that the $K_\alpha$'s are sufficiently fine cell decompositions, |
993 then $F$ is a homotopy through diffeomorphisms. |
993 then $F$ is a homotopy through diffeomorphisms. |
994 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
994 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
995 We have |
995 We have |
996 \eq{ |
996 \eq{ |
997 % \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) . |
997 % \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) . |