author | Kevin Walker <kevin@canyon23.net> |
Mon, 10 May 2010 14:14:19 -0700 | |
changeset 260 | 971234b03c4a |
parent 245 | 7537032ad5a0 |
child 271 | cb40431c8a65 |
permissions | -rw-r--r-- |
169 | 1 |
%!TEX root = ../../blob1.tex |
98 | 2 |
|
3 |
\section{Families of Diffeomorphisms} \label{sec:localising} |
|
4 |
||
245
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
5 |
|
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
6 |
\medskip |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
7 |
\hrule |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
8 |
\medskip |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
9 |
\nn{the following was removed from earlier section; it should be reincorporated somehwere |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
10 |
in this section} |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
11 |
|
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
12 |
Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
13 |
A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
14 |
{\it adapted to $\cU$} if there is a factorization |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
15 |
\eq{ |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
16 |
P = P_1 \times \cdots \times P_m |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
17 |
} |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
18 |
(for some $m \le k$) |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
19 |
and families of homeomorphisms |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
20 |
\eq{ |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
21 |
f_i : P_i \times X \to X |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
22 |
} |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
23 |
such that |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
24 |
\begin{itemize} |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
25 |
\item each $f_i$ is supported on some connected $V_i \sub X$; |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
26 |
\item the sets $V_i$ are mutually disjoint; |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
27 |
\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
28 |
where $k_i = \dim(P_i)$; and |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
29 |
\item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
30 |
for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$. |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
31 |
\end{itemize} |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
32 |
A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
33 |
of singular cells, each of which is adapted to $\cU$. |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
34 |
\medskip |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
35 |
\hrule |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
36 |
\medskip |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
37 |
\nn{another refugee:} |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
38 |
|
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
39 |
We will actually prove the following more general result. |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
40 |
Let $S$ and $T$ be an arbitrary topological spaces. |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
41 |
%\nn{might need to restrict $S$; the proof uses partition of unity on $S$; |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
42 |
%check this; or maybe just restrict the cover} |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
43 |
Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
44 |
from $S$ to $T$. |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
45 |
Let $\cU$ be an open cover of $S$ which affords a partition of unity. |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
46 |
\nn{for some $S$ and $\cU$ there is no partition of unity? like if $S$ is not paracompact? |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
47 |
in any case, in our applications $S$ will always be a manifold} |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
48 |
|
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
49 |
\begin{lemma} \label{extension_lemma_b} |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
50 |
Let $x \in CM_k(S, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
51 |
Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
52 |
Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
53 |
If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
54 |
chains of smooth maps or immersions. |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
55 |
\end{lemma} |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
56 |
|
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
57 |
\medskip |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
58 |
\hrule |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
59 |
\medskip |
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
60 |
|
7537032ad5a0
more evmap.tex; also testing using hg from office computer; also
Kevin Walker <kevin@canyon23.net>
parents:
210
diff
changeset
|
61 |
|
194 | 62 |
In this appendix we provide the proof of |
210 | 63 |
\nn{should change this to the more general \ref{extension_lemma_b}} |
98 | 64 |
|
194 | 65 |
\begin{lem*}[Restatement of Lemma \ref{extension_lemma}] |
66 |
Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
|
67 |
Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
68 |
Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
|
69 |
\end{lem*} |
|
98 | 70 |
|
71 |
\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
|
72 |
later draft} |
|
73 |
||
74 |
\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
|
75 |
about boundary later} |
|
76 |
||
194 | 77 |
\begin{proof} |
78 |
||
98 | 79 |
Recall that we are given |
80 |
an open cover $\cU = \{U_\alpha\}$ and an |
|
81 |
$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
|
82 |
We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
83 |
||
84 |
Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
|
85 |
||
86 |
As a first approximation to the argument we will eventually make, let's replace $x$ |
|
87 |
with a single singular cell |
|
88 |
\eq{ |
|
89 |
f: P \times X \to X . |
|
90 |
} |
|
91 |
Also, we'll ignore for now issues around $\bd P$. |
|
92 |
||
93 |
Our homotopy will have the form |
|
94 |
\eqar{ |
|
95 |
F: I \times P \times X &\to& X \\ |
|
96 |
(t, p, x) &\mapsto& f(u(t, p, x), x) |
|
97 |
} |
|
98 |
for some function |
|
99 |
\eq{ |
|
100 |
u : I \times P \times X \to P . |
|
101 |
} |
|
102 |
First we describe $u$, then we argue that it does what we want it to do. |
|
103 |
||
104 |
For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
|
105 |
The various $K_\alpha$ should be in general position with respect to each other. |
|
106 |
We will see below that the $K_\alpha$'s need to be sufficiently fine in order |
|
107 |
to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not |
|
108 |
merely a homotopy through maps $X\to X$. |
|
109 |
||
110 |
Let $L$ be the union of all the $K_\alpha$'s. |
|
111 |
$L$ is itself a cell decomposition of $P$. |
|
112 |
\nn{next two sentences not needed?} |
|
113 |
To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$, |
|
114 |
where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$. |
|
115 |
Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$. |
|
116 |
||
117 |
Let $J$ denote the handle decomposition of $P$ corresponding to $L$. |
|
118 |
Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and, |
|
119 |
more importantly, a $k{-}i$-dimensional normal coordinate. |
|
120 |
||
121 |
For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$. |
|
122 |
Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding |
|
123 |
$k$-cell of $L$. |
|
124 |
To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
|
125 |
which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
|
126 |
||
127 |
For $p \in D$ we define |
|
128 |
\eq{ |
|
129 |
u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
|
130 |
} |
|
131 |
(Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
|
132 |
makes sense.) |
|
133 |
||
134 |
So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
|
135 |
For handles of $J$ of index less than $k$, we will define $u$ to |
|
136 |
interpolate between the values on $k$-handles defined above. |
|
137 |
||
138 |
If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
|
139 |
of $E$. |
|
140 |
In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
|
141 |
with a $k$-handle. |
|
142 |
Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell |
|
143 |
corresponding to $E$. |
|
144 |
Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ |
|
145 |
adjacent to the $k{-}1$-cell corresponding to $E$. |
|
146 |
For $p \in E$, define |
|
147 |
\eq{ |
|
148 |
u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} |
|
149 |
+ r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . |
|
150 |
} |
|
151 |
||
152 |
In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
|
153 |
$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
|
154 |
The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
|
155 |
If we triangulate $R$ (without introducing new vertices), we can linearly extend |
|
156 |
a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
|
157 |
Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
|
158 |
the $k{-}j$-cell corresponding to $E$. |
|
159 |
For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
|
160 |
Now define, for $p \in E$, |
|
194 | 161 |
\begin{equation} |
162 |
\label{eq:u} |
|
98 | 163 |
u(t, p, x) = (1-t)p + t \left( |
164 |
\sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
|
165 |
+ \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
|
166 |
\right) . |
|
194 | 167 |
\end{equation} |
98 | 168 |
Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
169 |
mentioned above. |
|
170 |
||
171 |
This completes the definition of $u: I \times P \times X \to P$. |
|
172 |
||
173 |
\medskip |
|
174 |
||
175 |
Next we verify that $u$ has the desired properties. |
|
176 |
||
177 |
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 $x$. |
|
178 |
Therefore $F$ is a homotopy from $f$ to something. |
|
179 |
||
180 |
Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
|
181 |
then $F$ is a homotopy through diffeomorphisms. |
|
182 |
We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
|
183 |
We have |
|
184 |
\eq{ |
|
185 |
% \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) . |
|
186 |
\pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
|
187 |
} |
|
188 |
Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
|
189 |
\nn{bounded away from zero, or something like that}. |
|
190 |
(Recall that $X$ and $P$ are compact.) |
|
191 |
Also, $\pd{f}{p}$ is bounded. |
|
192 |
So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
|
194 | 193 |
It follows from Equation \eqref{eq:u} above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
98 | 194 |
(which is bounded) |
195 |
and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. |
|
196 |
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. |
|
198 |
||
199 |
\medskip |
|
200 |
||
201 |
Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
|
202 |
is a singular cell adapted to $\cU$. |
|
203 |
This will complete the proof of the lemma. |
|
204 |
\nn{except for boundary issues and the `$P$ is a cell' assumption} |
|
205 |
||
206 |
Let $j$ be the codimension of $D$. |
|
207 |
(Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
|
208 |
between handle and corresponding cell.) |
|
209 |
Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
|
210 |
where the $j_i$'s are the codimensions of the $K_\alpha$ |
|
211 |
cells of codimension greater than 0 which intersect to form $D$. |
|
212 |
We will show that |
|
213 |
if the relevant $U_\alpha$'s are disjoint, then |
|
214 |
$F(1, \cdot, \cdot) : D\times X \to X$ |
|
215 |
is a product of singular cells of dimensions $j_1, \ldots, j_m$. |
|
216 |
If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular |
|
217 |
cells whose dimensions correspond to a partition of the $j_i$'s. |
|
218 |
We will consider some simple special cases first, then do the general case. |
|
219 |
||
220 |
First consider the case $j=0$ (and $m=0$). |
|
221 |
A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$, |
|
222 |
is independent of $p \in P$. |
|
223 |
So the corresponding map $D \to \Diff(X)$ is constant. |
|
224 |
||
225 |
Next consider the case $j = 1$ (and $m=1$, $j_1=1$). |
|
226 |
Now Equation yyyy applies. |
|
227 |
We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$. |
|
228 |
It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
|
229 |
of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
|
230 |
The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
|
231 |
||
232 |
Next case: $j=2$, $m=1$, $j_1 = 2$. |
|
233 |
This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
|
234 |
1-dimensional. |
|
235 |
We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
|
236 |
and a 2-cell with support $U_\beta$. |
|
237 |
||
238 |
Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. |
|
239 |
In this case the codimension 2 cell $D$ is the intersection of two |
|
240 |
codimension 1 cells, from $K_\beta$ and $K_\gamma$. |
|
241 |
We can write $D = D' \times I \times I$, where the normal coordinates are constant |
|
242 |
on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$. |
|
243 |
If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and |
|
244 |
two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively. |
|
245 |
If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and |
|
246 |
a 2-cell supported on $U_\beta \cup U_\gamma$. |
|
247 |
\nn{need to check that this is true} |
|
248 |
||
249 |
\nn{finally, general case...} |
|
250 |
||
251 |
\nn{this completes proof} |
|
252 |
||
194 | 253 |
\end{proof} |
98 | 254 |
|
194 | 255 |
\input{text/appendixes/explicit.tex} |
256 |