97 the $k$-handles at $B^{k-1}\times\{0\}$ and $B^{k-1}\times\{1\}$. |
97 the $k$-handles at $B^{k-1}\times\{0\}$ and $B^{k-1}\times\{1\}$. |
98 Let $\eta : E \to [0,1]$, $\eta(x, s) = s$ be the normal coordinate |
98 Let $\eta : E \to [0,1]$, $\eta(x, s) = s$ be the normal coordinate |
99 of $E$. |
99 of $E$. |
100 Let $D_0$ and $D_1$ be the two $k$-handles of $\jj$ adjacent to $E$. |
100 Let $D_0$ and $D_1$ be the two $k$-handles of $\jj$ adjacent to $E$. |
101 There is at most one index $\beta$ such that $C(D_0, \beta) \ne C(D_1, \beta)$. |
101 There is at most one index $\beta$ such that $C(D_0, \beta) \ne C(D_1, \beta)$. |
102 (If there is no such index $\beta$, choose $\beta$ |
102 (If there is no such index, choose $\beta$ |
103 arbitrarily.) |
103 arbitrarily.) |
104 For $p \in E$, define |
104 For $p \in E$, define |
105 \eq{ |
105 \eq{ |
106 u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p(D_0, \alpha) |
106 u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p(D_0, \alpha) |
107 + r_\beta(x) (\eta(p) p(D_0, p) + (1-\eta(p)) p(D_1, p)) \right) . |
107 + r_\beta(x) (\eta(p) p(D_0, p) + (1-\eta(p)) p(D_1, p)) \right) . |
108 } |
108 } |
109 |
109 |
110 \nn{*** resume revising here ***} |
110 |
111 |
111 Now for the general case. |
112 In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
112 Let $E$ be a $k{-}j$-handle. |
113 $\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
113 Let $D_0,\ldots,D_a$ be the $k$-handles adjacent to $E$. |
114 The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
114 There is a subset of cover indices $\cN$, of cardinality $j$, |
115 If we triangulate $R$ (without introducing new vertices), we can linearly extend |
115 such that if $\alpha\notin\cN$ then |
116 a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
116 $p(D_u, \alpha) = p(D_v, \alpha)$ for all $0\le u,v \le a$. |
117 Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
117 For fixed $\beta\in\cN$ let $\{q_{\beta i}\}$ be the set of values of |
118 the $k{-}j$-cell corresponding to $E$. |
118 $p(D_u, \beta)$ for $0\le u \le a$. |
119 For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
119 Recall the product structure $E = B^{k-j}\times B^j$. |
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120 Inductively, we have defined functions $\eta_{\beta i}:\bd B^j \to [0,1]$ such that |
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121 $\sum_i \eta_{\beta i} = 1$ for all $\beta\in \cN$. |
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122 Choose extensions of $\eta_{\beta i}$ to all of $B^j$. |
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123 Via the projection $E\to B^j$, regard $\eta_{\beta i}$ as a function on $E$. |
120 Now define, for $p \in E$, |
124 Now define, for $p \in E$, |
121 \begin{equation} |
125 \begin{equation} |
122 \label{eq:u} |
126 \label{eq:u} |
123 u(t, p, x) = (1-t)p + t \left( |
127 u(t, p, x) = (1-t)p + t \left( |
124 \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
128 \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
125 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
129 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
126 \right) . |
130 \right) . |
127 \end{equation} |
131 \end{equation} |
128 Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
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129 mentioned above. |
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130 |
132 |
131 This completes the definition of $u: I \times P \times X \to P$. |
133 This completes the definition of $u: I \times P \times X \to P$. |
132 |
134 |
133 \medskip |
135 \medskip |
134 |
136 |
135 Next we verify that $u$ has the desired properties. |
137 Next we verify that $u$ affords $F$ the properties claimed in the statement of the lemma. |
136 |
138 |
137 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 $x$. |
139 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 $x$. |
138 Therefore $F$ is a homotopy from $f$ to something. |
140 Therefore $F$ is a homotopy from $f$ to something. |
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141 |
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142 \nn{*** resume revising here ***} |
139 |
143 |
140 Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
144 Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
141 then $F$ is a homotopy through diffeomorphisms. |
145 then $F$ is a homotopy through diffeomorphisms. |
142 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
146 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
143 We have |
147 We have |