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1 \nn{Here's the ``explicit'' version.} |
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2 |
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3 Fix a finite open cover of $X$, say $(U_l)_{l=1}^L$, along with an |
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4 associated partition of unity $(r_l)$. |
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5 |
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6 We'll define the homotopy $H:I \times P \times X \To X$ via a function |
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7 $u:I \times P \times X \To P$, with |
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8 \begin{equation*} |
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9 H(t,p,x) = F(u(t,p,x),x). |
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10 \end{equation*} |
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11 |
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12 To begin, we'll define a function $u'' : I \times P \times X \To P$, and |
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13 a corresponding homotopy $H''$. This homotopy will just be a homotopy of |
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14 $F$ through families of maps, not through families of diffeomorphisms. On |
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15 the other hand, it will be quite simple to describe, and we'll later |
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16 explain how to build the desired function $u$ out of it. |
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17 |
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18 For each $l = 1, \ldots, L$, pick some $C^\infty$ function $f_l : I \To |
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19 I$ which is identically $0$ on a neighborhood of the closed interval $[0,\frac{l-1}{L}]$ |
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20 and identically $1$ on a neighborhood of the closed interval $[\frac{l}{L},1]$. (Monotonic? |
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21 Fix a bound for the derivative?) We'll extend it to a function on |
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22 $k$-tuples $f_l : I^k \To I^k$ pointwise. |
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23 |
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24 Define $$u''(t,p,x) = \sum_{l=1}^L r_l(x) u_l(t,p),$$ with |
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25 $$u_l(t,p) = t f_l(p) + (1-t)p.$$ Notice that the $i$-th component of $u''(t,p,x)$ depends only on the $i$-th component of $p$. |
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26 |
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27 Let's now establish some properties of $u''$ and $H''$. First, |
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28 \begin{align*} |
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29 H''(0,p,x) & = F(u''(0,p,x),x) \\ |
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30 & = F(\sum_{l=1}^L r_l(x) p, x) \\ |
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31 & = F(p,x). |
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32 \end{align*} |
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33 Next, calculate the derivatives |
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34 \begin{align*} |
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35 \partial_{p_i} H''(1,p,x) & = \partial_{p_i}u''(1,p,x) \partial_1 F(u(1,p,x),x) \\ |
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36 \intertext{and} |
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37 \partial_{p_i}u''(1,p,x) & = \sum_{l=1}^L r_l(x) \partial_{p_i} f_l(p). |
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38 \end{align*} |
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39 Now $\partial_{p_i} f_l(p) = 0$ unless $\frac{l-1}{L} < p_i < \frac{l}{L}$, and $r_l(x) = 0$ unless $x \in U_l$, |
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40 so we conclude that for a fixed $p$, $\partial_p H''(1,p,x) = 0$ for all $x$ outside the union of $k$ open sets from the open cover, namely |
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41 $\bigcup_{i=1}^k U_{l_i}$ where for each $i$, we choose $l_i$ so $\frac{l_i -1}{L} \leq p_i \leq \frac{l_i}{L}$. It may be helpful to refer to Figure \ref{fig:supports}. |
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42 |
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43 \begin{figure}[!ht] |
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44 \begin{equation*} |
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45 \mathfig{0.5}{explicit/supports} |
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46 \end{equation*} |
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47 \caption{The supports of the derivatives {\color{green}$\partial_p f_1$}, {\color{blue}$\partial_p f_2$} and {\color{red}$\partial_p f_3$}, illustrating the case $k=2$, $L=3$. Notice that any |
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48 point $p$ lies in the intersection of at most $k$ supports. The support of $\partial_p u''(1,p,x)$ is contained in the union of these supports.} |
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49 \label{fig:supports} |
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50 \end{figure} |
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51 |
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52 Unfortunately, $H''$ does not have the desired property that it's a homotopy through diffeomorphisms. To achieve this, we'll paste together several copies |
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53 of the map $u''$. First, glue together $2^k$ copies, defining $u':I \times P \times X$ by |
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54 \begin{align*} |
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55 u'(t,p,x)_i & = |
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56 \begin{cases} |
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57 \frac{1}{2} u''(t, 2p_i, x)_i & \text{if $0 \leq p_i \leq \frac{1}{2}$} \\ |
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58 1-\frac{1}{2} u''(t, 2-2p_i, x)_i & \text{if $\frac{1}{2} \leq p_i \leq 1$}. |
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59 \end{cases} |
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60 \end{align*} |
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61 (Note that we're abusing notation somewhat, using the fact that $u''(t,p,x)_i$ depends on $p$ only through $p_i$.) |
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62 To see what's going on here, it may be helpful to look at Figure \ref{fig:supports_4}, which shows the support of $\partial_p u'(1,p,x)$. |
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63 \begin{figure}[!ht] |
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64 \begin{equation*} |
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65 \mathfig{0.4}{explicit/supports_4} \qquad \qquad \mathfig{0.4}{explicit/supports_36} |
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66 \end{equation*} |
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67 \caption{The supports of $\partial_p u'(1,p,x)$ and of $\partial_p u(1,p,x)$ (with $K=3$) are subsets of the indicated region.} |
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68 \label{fig:supports_4} |
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69 \end{figure} |
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70 |
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71 Second, pick some $K$, and define |
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72 \begin{align*} |
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73 u(t,p,x) & = \frac{\floor{K p}}{K} + \frac{1}{K} u'\left(t, K \left(p - \frac{\floor{K p}}{K}\right), x\right). |
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74 \end{align*} |
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75 |
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76 \todo{Explain that the localisation property survives for $u'$ and $u$.} |
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77 |
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78 We now check that by making $K$ large enough, $H$ becomes a homotopy through diffeomorphisms. We start with |
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79 $$\partial_x H(t,p,x) = \partial_x u(t,p,x) \partial_1 F(u(t,p,x), x) + \partial_2 F(u(t,p,x), x)$$ |
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80 and observe that since $F(p, -)$ is a diffeomorphism, the second term $\partial_2 F(u(t,p,x), x)$ is bounded away from $0$. Thus if we can control the |
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81 size of the first term $\partial_x u(t,p,x) \partial_1 F(u(t,p,x), x)$ we're done. The factor $\partial_1 F(u(t,p,x), x)$ is bounded, and we |
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82 calculate \todo{err... this is a mess, and probably wrong.} |
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83 \begin{align*} |
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84 \partial_x u(t,p,x)_i & = \partial_x \frac{1}{K} u'\left(t, K\left(p - \frac{\floor{K p}}{K}\right), x\right)_i \\ |
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85 & = \pm \frac{1}{2 K} \partial_x u''\left(t, (1\mp1)\pm 2K\left(p_i-\frac{\floor{K p}}{K}\right), x\right)_i \\ |
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86 & = \pm \frac{1}{2 K} \sum_{l=1}^L (\partial_x r_l(x)) u_l\left(t, (1\mp1)\pm 2K\left(p_i-\frac{\floor{K p}}{K}\right)\right)_i. \\ |
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87 \intertext{Since the target of $u_l$ is just the unit cube $I^k$, we can make the estimate} |
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88 \norm{\partial_x u(t,p,x)_i} & \leq \frac{1}{2 K} \sum_{l=1}^L \norm{\partial_x r_l(x)}. |
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89 \end{align*} |
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90 The sum here is bounded, so for large enough $K$ this is small enough that $\partial_x H(t,p,x)$ is never zero. |