1 \nn{Here's the ``explicit'' version.} |
1 \nn{Here's the ``explicit'' version.} |
2 |
2 |
3 Fix a finite open cover of $X$, say $(U_l)_{l=1}^L$, along with an |
3 Fix a finite open cover of $X$, say $(U_l)_{l=1}^L$, along with an |
4 associated partition of unity $(r_l)$. |
4 associated partition of unity $(r_l)$. |
5 |
5 |
6 We'll define the homotopy $H:I \times P \times X \To X$ via a function |
6 We'll define the homotopy $H:I \times P \times X \to X$ via a function |
7 $u:I \times P \times X \To P$, with |
7 $u:I \times P \times X \to P$, with |
8 \begin{equation*} |
8 \begin{equation*} |
9 H(t,p,x) = F(u(t,p,x),x). |
9 H(t,p,x) = F(u(t,p,x),x). |
10 \end{equation*} |
10 \end{equation*} |
11 |
11 |
12 To begin, we'll define a function $u'' : I \times P \times X \To P$, and |
12 To begin, we'll define a function $u'' : I \times P \times X \to P$, and |
13 a corresponding homotopy $H''$. This homotopy will just be a homotopy of |
13 a corresponding homotopy $H''$. This homotopy will just be a homotopy of |
14 $F$ through families of maps, not through families of diffeomorphisms. On |
14 $F$ through families of maps, not through families of diffeomorphisms. On |
15 the other hand, it will be quite simple to describe, and we'll later |
15 the other hand, it will be quite simple to describe, and we'll later |
16 explain how to build the desired function $u$ out of it. |
16 explain how to build the desired function $u$ out of it. |
17 |
17 |
18 For each $l = 1, \ldots, L$, pick some $C^\infty$ function $f_l : I \To |
18 For each $l = 1, \ldots, L$, pick some $C^\infty$ function $f_l : I \to |
19 I$ which is identically $0$ on a neighborhood of the closed interval $[0,\frac{l-1}{L}]$ |
19 I$ which is identically $0$ on a neighborhood of the closed interval $[0,\frac{l-1}{L}]$ |
20 and identically $1$ on a neighborhood of the closed interval $[\frac{l}{L},1]$. (Monotonic? |
20 and identically $1$ on a neighborhood of the closed interval $[\frac{l}{L},1]$. (Monotonic? |
21 Fix a bound for the derivative?) We'll extend it to a function on |
21 Fix a bound for the derivative?) We'll extend it to a function on |
22 $k$-tuples $f_l : I^k \To I^k$ pointwise. |
22 $k$-tuples $f_l : I^k \to I^k$ pointwise. |
23 |
23 |
24 Define $$u''(t,p,x) = \sum_{l=1}^L r_l(x) u_l(t,p),$$ with |
24 Define $$u''(t,p,x) = \sum_{l=1}^L r_l(x) u_l(t,p),$$ with |
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 $p$. |
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 $p$. |
26 |
26 |
27 Let's now establish some properties of $u''$ and $H''$. First, |
27 Let's now establish some properties of $u''$ and $H''$. First, |