7 Let $\cU = \{U_\alpha\}$ be an open cover of $X$ which affords a partition of |
7 Let $\cU = \{U_\alpha\}$ be an open cover of $X$ which affords a partition of |
8 unity $\{r_\alpha\}$. |
8 unity $\{r_\alpha\}$. |
9 (That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$; |
9 (That is, $r_\alpha : X \to [0,1]$; $r_\alpha(x) = 0$ if $x\notin U_\alpha$; |
10 for fixed $x$, $r_\alpha(x) \ne 0$ for only finitely many $\alpha$; and $\sum_\alpha r_\alpha = 1$.) |
10 for fixed $x$, $r_\alpha(x) \ne 0$ for only finitely many $\alpha$; and $\sum_\alpha r_\alpha = 1$.) |
11 Since $X$ is compact, we will further assume that $r_\alpha = 0$ (globally) |
11 Since $X$ is compact, we will further assume that $r_\alpha = 0$ (globally) |
12 for all but finitely many $\alpha$. |
12 for all but finitely many $\alpha$. \nn{Can't we just assume $\cU$ is finite? Is something subtler happening? -S} |
13 |
13 |
14 Let |
14 Consider $C_*(\Maps(X\to T))$, the singular chains on the space of continuous maps from $X$ to $T$. |
15 \[ |
15 $C_k(\Maps(X \to T))$ is generated by continuous maps |
16 CM_*(X, T) \deq C_*(\Maps(X\to T)) , |
|
17 \] |
|
18 the singular chains on the space of continuous maps from $X$ to $T$. |
|
19 $CM_k(X, T)$ is generated by continuous maps |
|
20 \[ |
16 \[ |
21 f: P\times X \to T , |
17 f: P\times X \to T , |
22 \] |
18 \] |
23 where $P$ is some convex linear polyhedron in $\r^k$. |
19 where $P$ is some convex linear polyhedron in $\r^k$. |
24 Recall that $f$ is {\it supported} on $S\sub X$ if $f(p, x)$ does not depend on $p$ when |
20 Recall that $f$ is {\it supported} on $S\sub X$ if $f(p, x)$ does not depend on $p$ when |
25 $x \notin S$, and that $f$ is {\it adapted} to $\cU$ if |
21 $x \notin S$, and that $f$ is {\it adapted} to $\cU$ if |
26 $f$ is supported on the union of at most $k$ of the $U_\alpha$'s. |
22 $f$ is supported on the union of at most $k$ of the $U_\alpha$'s. |
27 A chain $c \in CM_*(X, T)$ is adapted to $\cU$ if it is a linear combination of |
23 A chain $c \in C_*(\Maps(X \to T))$ is adapted to $\cU$ if it is a linear combination of |
28 generators which are adapted. |
24 generators which are adapted. |
29 |
25 |
30 \begin{lemma} \label{basic_adaptation_lemma} |
26 \begin{lemma} \label{basic_adaptation_lemma} |
31 Let $f: P\times X \to T$, as above. |
27 Let $f: P\times X \to T$, as above. |
32 Then there exists |
28 Then there exists |
38 \item $F(0, \cdot, \cdot) = f$ . |
34 \item $F(0, \cdot, \cdot) = f$ . |
39 \item We can decompose $P = \cup_i D_i$ so that |
35 \item We can decompose $P = \cup_i D_i$ so that |
40 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. |
36 the restrictions $F(1, \cdot, \cdot) : D_i\times X\to T$ are all adapted to $\cU$. |
41 \item If $f$ has support $S\sub X$, then |
37 \item If $f$ has support $S\sub X$, then |
42 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$. |
38 $F: (I\times P)\times X\to T$ (a $k{+}1$-parameter family of maps) also has support $S$. |
43 Furthermore, if $Q\sub\bd P$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$ |
39 Furthermore, if $Q$ is a convex linear subpolyhedron of $\bd P$ and $f$ restricted to $Q$ |
44 has support $S'$, then |
40 has support $S' \subset X$, then |
45 $F: (I\times Q)\times X\to T$ also has support $S'$. |
41 $F: (I\times Q)\times X\to T$ also has support $S'$. |
46 \item If for all $p\in P$ we have $f(p, \cdot):X\to T$ is a |
42 \item Suppose both $X$ and $T$ are smooth manifolds, metric spaces, or PL manifolds, and let $\cX$ denote the subspace of $\Maps(X \to T)$ consisting of immersions or of diffeomorphisms (in the smooth case), bi-Lipschitz homeomorphisms (in the metric case), or PL homeomorphisms (in the PL case). |
47 [immersion, diffeomorphism, PL homeomorphism, bi-Lipschitz homeomorphism] |
43 If $f$ is smooth, Lipschitz or PL, as appropriate, and $f(p, \cdot):X\to T$ is in $\cX$ for all $p \in P$ |
48 then the same is true of $F(t, p, \cdot)$ for all $t\in I$ and $p\in P$. |
44 then $F(t, p, \cdot)$ is also in $\cX$ for all $t\in I$ and $p\in P$. |
49 (Of course we must assume that $X$ and $T$ are the appropriate |
|
50 sort of manifolds for this to make sense.) |
|
51 \end{enumerate} |
45 \end{enumerate} |
52 \end{lemma} |
46 \end{lemma} |
53 |
47 |
54 \begin{proof} |
48 \begin{proof} |
55 Our homotopy will have the form |
49 Our homotopy will have the form |
78 corresponding $i$-handles of $\jj$. |
72 corresponding $i$-handles of $\jj$. |
79 |
73 |
80 For each (top-dimensional) $k$-cell $C$ of each $K_\alpha$, choose a point $p(C) \in C \sub P$. |
74 For each (top-dimensional) $k$-cell $C$ of each $K_\alpha$, choose a point $p(C) \in C \sub P$. |
81 If $C$ meets a subpolyhedron $Q$ of $\bd P$, we require that $p(C)\in Q$. |
75 If $C$ meets a subpolyhedron $Q$ of $\bd P$, we require that $p(C)\in Q$. |
82 (It follows that if $C$ meets both $Q$ and $Q'$, then $p(C)\in Q\cap Q'$. |
76 (It follows that if $C$ meets both $Q$ and $Q'$, then $p(C)\in Q\cap Q'$. |
83 This puts some mild constraints on the choice of $K_\alpha$.) |
77 Ensuring this is possible corresponds to some mild constraints on the choice of the $K_\alpha$.) |
84 |
78 |
85 Let $D$ be a $k$-handle of $\jj$. |
79 Let $D$ be a $k$-handle of $\jj$. |
86 For each $\alpha$ let $C(D, \alpha)$ be the $k$-cell of $K_\alpha$ which contains $D$ |
80 For each $\alpha$ let $C(D, \alpha)$ be the $k$-cell of $K_\alpha$ which contains $D$ |
87 and let $p(D, \alpha) = p(C(D, \alpha))$. |
81 and let $p(D, \alpha) = p(C(D, \alpha))$. |
88 |
82 |
132 \sum_{\alpha \notin \cN} r_\alpha(x) p(D_0, \alpha) |
126 \sum_{\alpha \notin \cN} r_\alpha(x) p(D_0, \alpha) |
133 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
127 + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
134 \right) . |
128 \right) . |
135 \end{equation} |
129 \end{equation} |
136 |
130 |
137 This completes the definition of $u: I \times P \times X \to P$. |
131 This completes the definition of $u: I \times P \times X \to P$. The formulas above are consistent: for $p$ at the boundary between a $k-j$-handle and a $k-(j+1)$-handle the corresponding expressions in Equation \eqref{eq:u} agree, since one of the normal coordinates becomes $0$ or $1$. |
138 Note that if $Q\sub \bd P$ is a convex linear subpolyhedron, then $u(I\times Q\times X) \sub Q$. |
132 Note that if $Q\sub \bd P$ is a convex linear subpolyhedron, then $u(I\times Q\times X) \sub Q$. |
139 |
133 |
140 \medskip |
134 \medskip |
141 |
135 |
142 Next we verify that $u$ affords $F$ the properties claimed in the statement of the lemma. |
136 Next we verify that $u$ affords $F$ the properties claimed in the statement of the lemma. |
148 \medskip |
142 \medskip |
149 |
143 |
150 Next we show that for each handle $D$ of $J$, $F(1, \cdot, \cdot) : D\times X \to X$ |
144 Next we show that for each handle $D$ of $J$, $F(1, \cdot, \cdot) : D\times X \to X$ |
151 is a singular cell adapted to $\cU$. |
145 is a singular cell adapted to $\cU$. |
152 Let $k-j$ be the index of $D$. |
146 Let $k-j$ be the index of $D$. |
153 Referring to Equation \ref{eq:u}, we see that $F(1, p, x)$ depends on $p$ only if |
147 Referring to Equation \eqref{eq:u}, we see that $F(1, p, x)$ depends on $p$ only if |
154 $r_\beta(x) \ne 0$ for some $\beta\in\cN$, i.e.\ only if |
148 $r_\beta(x) \ne 0$ for some $\beta\in\cN$, i.e.\ only if |
155 $x\in \bigcup_{\beta\in\cN} U_\beta$. |
149 $x\in \bigcup_{\beta\in\cN} U_\beta$. |
156 Since the cardinality of $\cN$ is $j$ which is less than or equal to $k$, |
150 Since the cardinality of $\cN$ is $j$ which is less than or equal to $k$, |
157 this shows that $F(1, \cdot, \cdot) : D\times X \to X$ is adapted to $\cU$. |
151 this shows that $F(1, \cdot, \cdot) : D\times X \to X$ is adapted to $\cU$. |
158 |
152 |
174 (Recall that we arranged above that $u(I\times Q\times X) \sub Q$.) |
168 (Recall that we arranged above that $u(I\times Q\times X) \sub Q$.) |
175 |
169 |
176 \medskip |
170 \medskip |
177 |
171 |
178 Now for claim 4 of the lemma. |
172 Now for claim 4 of the lemma. |
179 Assume that $X$ and $T$ are smooth manifolds and that $f$ is a family of diffeomorphisms. |
173 Assume that $X$ and $T$ are smooth manifolds and that $f$ is a smooth family of diffeomorphisms. |
180 We must show that we can choose the $K_\alpha$'s and $u$ so that $F(t, p, \cdot)$ is a |
174 We must show that we can choose the $K_\alpha$'s and $u$ so that $F(t, p, \cdot)$ is a |
181 diffeomorphism for all $t$ and $p$. |
175 diffeomorphism for all $t$ and $p$. |
182 It suffices to |
176 It suffices to |
183 show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
177 show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
184 We have |
178 We have |
186 % \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) . |
180 % \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) . |
187 \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
181 \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
188 } |
182 } |
189 Since $f$ is a family of diffeomorphisms and $X$ and $P$ are compact, |
183 Since $f$ is a family of diffeomorphisms and $X$ and $P$ are compact, |
190 $\pd{f}{x}$ is non-singular and bounded away from zero. |
184 $\pd{f}{x}$ is non-singular and bounded away from zero. |
191 Also, $\pd{f}{p}$ is bounded. |
185 Also, since $f$ is smooth $\pd{f}{p}$ is bounded. |
192 So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
186 Thus if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
193 It follows from Equation \eqref{eq:u} above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
187 It follows from Equation \eqref{eq:u} above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
194 (which is bounded) |
188 (which is bounded) |
195 and the differences amongst the various $p(D_0,\alpha)$'s and $q_{\beta i}$'s. |
189 and the differences amongst the various $p(D_0,\alpha)$'s and $q_{\beta i}$'s. |
196 These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
190 These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
197 This completes the proof that $F$ is a homotopy through diffeomorphisms. |
191 This completes the proof that $F$ is a homotopy through diffeomorphisms. |
198 |
192 |
199 If we replace ``diffeomorphism" with ``immersion" in the above paragraph, the argument goes |
193 If we replace ``diffeomorphism" with ``immersion" in the above paragraph, the argument goes |
200 through essentially unchanged. |
194 through essentially unchanged. |
201 |
195 |
202 Next we consider the case where $f$ is a family of bi-Lipschitz homeomorphisms. |
196 Next we consider the case where $f$ is a family of bi-Lipschitz homeomorphisms. |
203 We assume that $f$ is Lipschitz in $P$ direction as well. |
197 Recall that we assume that $f$ is Lipschitz in the $P$ direction as well. |
204 The argument in this case is similar to the one above for diffeomorphisms, with |
198 The argument in this case is similar to the one above for diffeomorphisms, with |
205 bounded partial derivatives replaced by Lipschitz constants. |
199 bounded partial derivatives replaced by Lipschitz constants. |
206 Since $X$ and $P$ are compact, there is a universal bi-Lipschitz constant that works for |
200 Since $X$ and $P$ are compact, there is a universal bi-Lipschitz constant that works for |
207 $f(p, \cdot)$ for all $p$. |
201 $f(p, \cdot)$ for all $p$. |
208 By choosing the cell decompositions $K_\alpha$ sufficiently fine, |
202 By choosing the cell decompositions $K_\alpha$ sufficiently fine, |
212 |
206 |
213 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
207 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
214 \end{proof} |
208 \end{proof} |
215 |
209 |
216 \begin{lemma} |
210 \begin{lemma} |
217 Let $CM_*(X\to T)$ be the singular chains on the space of continuous maps |
211 Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, bi-Lipschitz homeomorphisms or PL homeomorphisms. |
218 [resp. immersions, diffeomorphisms, PL homeomorphisms, bi-Lipschitz homeomorphisms] |
212 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$ |
219 from $X$ to $T$, and let $G_*\sub CM_*(X\to T)$ denote the chains adapted to an open cover $\cU$ |
|
220 of $X$. |
213 of $X$. |
221 Then $G_*$ is a strong deformation retract of $CM_*(X\to T)$. |
214 Then $G_*$ is a strong deformation retract of $\cX_*$. |
222 \end{lemma} |
215 \end{lemma} |
223 \begin{proof} |
216 \begin{proof} |
224 If suffices to show that given a generator $f:P\times X\to T$ of $CM_k(X\to T)$ with |
217 If suffices to show that given a generator $f:P\times X\to T$ of $\cX_k$ with |
225 $\bd f \in G_{k-1}$ there exists $h\in CM_{k+1}(X\to T)$ with $\bd h = f + g$ and $g \in G_k$. |
218 $\bd f \in G_{k-1}$ there exists $h\in \cX_{k+1}$ with $\bd h = f + g$ and $g \in G_k$. |
226 This is exactly what Lemma \ref{basic_adaptation_lemma} |
219 This is exactly what Lemma \ref{basic_adaptation_lemma} |
227 gives us. |
220 gives us. |
228 More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$. |
221 More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$. |
229 Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}. |
222 Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}. |
230 Then $\bd F$ is equal to $f$ plus $F(1, \cdot, \cdot)$ plus the restrictions of $F$ to $I\times Q_i$. |
223 Then $\bd F$ is equal to $f$ plus $F(1, \cdot, \cdot)$ plus the restrictions of $F$ to $I\times Q_i$. |