42 \item If $f$ restricted to $Q\sub P$ has support $S\sub X$, then the restriction

    42 \item If $f$ has support $S\sub X$, then

    43 $F: (I\times Q)\times X\to T$ also has support $S$.

    43 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$.

    45 [submersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism]

    45 [immersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism]

   107             + r_\beta(x) (\eta(p) p(D_0, p) + (1-\eta(p)) p(D_1, p)) \right) .

   105             + r_\beta(x) (\eta(p) p(D_0, \beta) + (1-\eta(p)) p(D_1, \beta)) \right) .

   128             \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha}

   126             \sum_{\alpha \notin \cN} r_\alpha(x) p(D_0, \alpha)

   142 \nn{*** resume revising here ***}

   140 

   143 

   141 \medskip

   144 Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions,

   142 

   145 then $F$ is a homotopy through diffeomorphisms.

   143 Next we show that for each handle $D$ of $J$, $F(1, \cdot, \cdot) : D\times X \to X$

   146 We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.

   144 is a singular cell adapted to $\cU$.

   145 Let $k-j$ be the index of $D$.

   146 Referring to Equation \ref{eq:u}, we see that $F(1, p, x)$ depends on $p$ only if 

   147 $r_\beta(x) \ne 0$ for some $\beta\in\cN$, i.e.\ only if

   148 $x\in \bigcup_{\beta\in\cN} U_\beta$.

   149 Since the cardinality of $\cN$ is $j$ which is less than or equal to $k$,

   150 this shows that $F(1, \cdot, \cdot) : D\times X \to X$ is adapted to $\cU$.

   151 

   152 \medskip

   153 

   154 Next we show that $F$ does not increase supports.

   155 If $f(p,x) = f(p',x)$ for all $p,p'\in P$,

   156 then 

   157 \[

   158 	F(t, p, x) = f(u(t,p,x),x) = f(u(t',p',x),x) = F(t',p',x)

   159 \]

   160 for all $(t,p)$ and $(t',p')$ in $I\times P$.

   161 

   162 \medskip

   163 

   164 Now for claim 4 of the lemma.

   165 Assume that $X$ and $T$ are smooth manifolds and that $f$ is a family of diffeomorphisms.

   166 We must show that we can choose the $K_\alpha$'s and $u$ so that $F(t, p, \cdot)$ is a 

   167 diffeomorphism for all $t$ and $p$.

   168 It suffices to 

   169 show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.

   152 Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and

   175 Since $f$ is a family of diffeomorphisms and $X$ and $P$ are compact, 

   153 \nn{bounded away from zero, or something like that}.

   176 $\pd{f}{x}$ is non-singular and bounded away from zero.

   154 (Recall that $X$ and $P$ are compact.)

   159 and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s.

   181 and the differences amongst the various $p(D_0,\alpha)$'s and $q_{\beta i}$'s.

   163 \medskip

   185 If we replace ``diffeomorphism" with ``immersion" in the above paragraph, the argument goes

   164 

   186 through essentially unchanged.

   165 Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$

   187 

   166 is a singular cell adapted to $\cU$.

   188 Next we consider the case where $f$ is a family of bi-Lipschitz homeomorphisms.

   167 This will complete the proof of the lemma.

   189 We assume that $f$ is Lipschitz in $P$ direction as well.

   168 \nn{except for boundary issues and the `$P$ is a cell' assumption}

   190 The argument in this case is similar to the one above for diffeomorphisms, with

   169 

   191 bounded partial derivatives replaced by Lipschitz constants.

   170 Let $j$ be the codimension of $D$.

   192 Since $X$ and $P$ are compact, there is a universal bi-Lipschitz constant that works for 

   171 (Or rather, the codimension of its corresponding cell.  From now on we will not make a distinction

   193 $f(p, \cdot)$ for all $p$.

   172 between handle and corresponding cell.)

   194 By choosing the cell decompositions $K_\alpha$ sufficiently fine,

   173 Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$,

   195 we can insure that $u$ has a small Lipschitz constant in the $X$ direction.

   174 where the $j_i$'s are the codimensions of the $K_\alpha$

   196 This allows us to show that $F(t, p, \cdot)$ has a bi-Lipschitz constant

   175 cells of codimension greater than 0 which intersect to form $D$.

   197 close to the universal bi-Lipschitz constant for $f$.

   176 We will show that

   198 

   177 if the relevant $U_\alpha$'s are disjoint, then

   199 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well.

   178 $F(1, \cdot, \cdot) : D\times X \to X$

   179 is a product of singular cells of dimensions $j_1, \ldots, j_m$.

   180 If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular

   181 cells whose dimensions correspond to a partition of the $j_i$'s.

   182 We will consider some simple special cases first, then do the general case.

   183 

   184 First consider the case $j=0$ (and $m=0$).

   185 A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$,

   186 is independent of $p \in P$.

   187 So the corresponding map $D \to \Diff(X)$ is constant.

   188 

   189 Next consider the case $j = 1$ (and $m=1$, $j_1=1$).

   190 Now Equation yyyy applies.

   191 We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$.

   192 It follows that the singular cell $D \to \Diff(X)$ can be written as a product

   193 of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$.

   194 The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set.

   195 

   196 Next case: $j=2$, $m=1$, $j_1 = 2$.

   197 This is similar to the previous case, except that the normal bundle is 2-dimensional instead of

   198 1-dimensional.

   199 We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell

   200 and a 2-cell with support $U_\beta$.

   201 

   202 Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$.

   203 In this case the codimension 2 cell $D$ is the intersection of two

   204 codimension 1 cells, from $K_\beta$ and $K_\gamma$.

   205 We can write $D = D' \times I \times I$, where the normal coordinates are constant

   206 on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$.

   207 If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and

   208 two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively.

   209 If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and

   210 a 2-cell supported on $U_\beta \cup U_\gamma$.

   211 \nn{need to check that this is true}

   212 

   213 \nn{finally, general case...}

   214 

   215 \nn{this completes proof}

   216