deducing standard Brouwer fixed point theorem
$begingroup$
This is a corollary in Dold, Algebraic Topology.(Cor 2.4,2.5,2.6) The goal is to see whether standard brouwer fixed point theorem can be deduced from 2.5 or 2.6.
Cor 2.4 $S^{n-1}$ is not a retraction of $D^n$ where $D^n$ is the closed $n-$disk.
Cor 2.5 If $f:D^nto R^n$ is continuous, then either $f(y)=0$ for some $y$ or $f(z)=lambda z$ for some $zin S^{n-1}$ and some $lambda>0$.
Cor 2.6("Brouwer fixed point Thm": Consider $g(x)-x$ as a function in Cor 2.5) If $g:D^nto R^n$ is continuous, then either $g(y)=y$ for some $yin D^n$ or $g(z)=(1+lambda)z$ for some $zin S^{n-1}$.
Standard Brouwer fixed point Thm: Continuous function $f:D^nto D^n$ must have at least 1 fixed point.
$textbf{Q:}$ I do not see how to deduce Standard Brouwer fixed point Thm from Cor 2.6 or 2.5 without going through contradiction via constructing retraction.(In other words, I do not know how to eliminate $g(z)=(1+lambda) z$ situation in Cor 2.6 in general without going through standard argument.) I suspect that I can suppose no such fixed point. Then I can pick a sequence of points $z_i$ along with a sequence of functions defined as the following.
Start with $z_1$. Suppose $f_1=f$ does not have fixed point. From Cor 2.6, I can pick out $z_1$ s.t $f(z_1)=lambda_1 z_1$ with $lambda_1>1$. Then consider $f_2=frac{f}{lambda_1}$. If $f_2$ has no fixed point, then pick out $z_2$. Iterate this procedure. I can hope the sequence converging to $0$ as each time $f_i$ shrinks. ($textbf{How do I see this sequence of function do converge to $0$?}$ Note that $lambda_i$ are varying and I do not have growth estimation of $lambda_i$.) Suppose this holds. I have $0$ as limiting point. Then I want to say $0$ is my fixed point to get desired contradiction.
general-topology algebraic-topology
asked Dec 21 '18 at 23:15
user45765user45765
2,6322724
$endgroup$
add a comment |
$begingroup$
This is a corollary in Dold, Algebraic Topology.(Cor 2.4,2.5,2.6) The goal is to see whether standard brouwer fixed point theorem can be deduced from 2.5 or 2.6.
Cor 2.4 $S^{n-1}$ is not a retraction of $D^n$ where $D^n$ is the closed $n-$disk.
Cor 2.5 If $f:D^nto R^n$ is continuous, then either $f(y)=0$ for some $y$ or $f(z)=lambda z$ for some $zin S^{n-1}$ and some $lambda>0$.
Cor 2.6("Brouwer fixed point Thm": Consider $g(x)-x$ as a function in Cor 2.5) If $g:D^nto R^n$ is continuous, then either $g(y)=y$ for some $yin D^n$ or $g(z)=(1+lambda)z$ for some $zin S^{n-1}$.
Standard Brouwer fixed point Thm: Continuous function $f:D^nto D^n$ must have at least 1 fixed point.
$textbf{Q:}$ I do not see how to deduce Standard Brouwer fixed point Thm from Cor 2.6 or 2.5 without going through contradiction via constructing retraction.(In other words, I do not know how to eliminate $g(z)=(1+lambda) z$ situation in Cor 2.6 in general without going through standard argument.) I suspect that I can suppose no such fixed point. Then I can pick a sequence of points $z_i$ along with a sequence of functions defined as the following.
Start with $z_1$. Suppose $f_1=f$ does not have fixed point. From Cor 2.6, I can pick out $z_1$ s.t $f(z_1)=lambda_1 z_1$ with $lambda_1>1$. Then consider $f_2=frac{f}{lambda_1}$. If $f_2$ has no fixed point, then pick out $z_2$. Iterate this procedure. I can hope the sequence converging to $0$ as each time $f_i$ shrinks. ($textbf{How do I see this sequence of function do converge to $0$?}$ Note that $lambda_i$ are varying and I do not have growth estimation of $lambda_i$.) Suppose this holds. I have $0$ as limiting point. Then I want to say $0$ is my fixed point to get desired contradiction.
general-topology algebraic-topology
asked Dec 21 '18 at 23:15
user45765user45765
2,6322724
$endgroup$
add a comment |
$begingroup$
This is a corollary in Dold, Algebraic Topology.(Cor 2.4,2.5,2.6) The goal is to see whether standard brouwer fixed point theorem can be deduced from 2.5 or 2.6.
Cor 2.4 $S^{n-1}$ is not a retraction of $D^n$ where $D^n$ is the closed $n-$disk.
Cor 2.5 If $f:D^nto R^n$ is continuous, then either $f(y)=0$ for some $y$ or $f(z)=lambda z$ for some $zin S^{n-1}$ and some $lambda>0$.
Cor 2.6("Brouwer fixed point Thm": Consider $g(x)-x$ as a function in Cor 2.5) If $g:D^nto R^n$ is continuous, then either $g(y)=y$ for some $yin D^n$ or $g(z)=(1+lambda)z$ for some $zin S^{n-1}$.
Standard Brouwer fixed point Thm: Continuous function $f:D^nto D^n$ must have at least 1 fixed point.
$textbf{Q:}$ I do not see how to deduce Standard Brouwer fixed point Thm from Cor 2.6 or 2.5 without going through contradiction via constructing retraction.(In other words, I do not know how to eliminate $g(z)=(1+lambda) z$ situation in Cor 2.6 in general without going through standard argument.) I suspect that I can suppose no such fixed point. Then I can pick a sequence of points $z_i$ along with a sequence of functions defined as the following.
Start with $z_1$. Suppose $f_1=f$ does not have fixed point. From Cor 2.6, I can pick out $z_1$ s.t $f(z_1)=lambda_1 z_1$ with $lambda_1>1$. Then consider $f_2=frac{f}{lambda_1}$. If $f_2$ has no fixed point, then pick out $z_2$. Iterate this procedure. I can hope the sequence converging to $0$ as each time $f_i$ shrinks. ($textbf{How do I see this sequence of function do converge to $0$?}$ Note that $lambda_i$ are varying and I do not have growth estimation of $lambda_i$.) Suppose this holds. I have $0$ as limiting point. Then I want to say $0$ is my fixed point to get desired contradiction.
general-topology algebraic-topology
asked Dec 21 '18 at 23:15
user45765user45765
2,6322724
$endgroup$
This is a corollary in Dold, Algebraic Topology.(Cor 2.4,2.5,2.6) The goal is to see whether standard brouwer fixed point theorem can be deduced from 2.5 or 2.6.
Cor 2.4 $S^{n-1}$ is not a retraction of $D^n$ where $D^n$ is the closed $n-$disk.
Cor 2.5 If $f:D^nto R^n$ is continuous, then either $f(y)=0$ for some $y$ or $f(z)=lambda z$ for some $zin S^{n-1}$ and some $lambda>0$.
Cor 2.6("Brouwer fixed point Thm": Consider $g(x)-x$ as a function in Cor 2.5) If $g:D^nto R^n$ is continuous, then either $g(y)=y$ for some $yin D^n$ or $g(z)=(1+lambda)z$ for some $zin S^{n-1}$.
Standard Brouwer fixed point Thm: Continuous function $f:D^nto D^n$ must have at least 1 fixed point.
$textbf{Q:}$ I do not see how to deduce Standard Brouwer fixed point Thm from Cor 2.6 or 2.5 without going through contradiction via constructing retraction.(In other words, I do not know how to eliminate $g(z)=(1+lambda) z$ situation in Cor 2.6 in general without going through standard argument.) I suspect that I can suppose no such fixed point. Then I can pick a sequence of points $z_i$ along with a sequence of functions defined as the following.
Start with $z_1$. Suppose $f_1=f$ does not have fixed point. From Cor 2.6, I can pick out $z_1$ s.t $f(z_1)=lambda_1 z_1$ with $lambda_1>1$. Then consider $f_2=frac{f}{lambda_1}$. If $f_2$ has no fixed point, then pick out $z_2$. Iterate this procedure. I can hope the sequence converging to $0$ as each time $f_i$ shrinks. ($textbf{How do I see this sequence of function do converge to $0$?}$ Note that $lambda_i$ are varying and I do not have growth estimation of $lambda_i$.) Suppose this holds. I have $0$ as limiting point. Then I want to say $0$ is my fixed point to get desired contradiction.
general-topology algebraic-topology
general-topology algebraic-topology
asked Dec 21 '18 at 23:15
user45765user45765
2,6322724
asked Dec 21 '18 at 23:15
user45765user45765
2,6322724
asked Dec 21 '18 at 23:15
user45765user45765
2,6322724
asked Dec 21 '18 at 23:15
user45765user45765
2,6322724
asked Dec 21 '18 at 23:15
user45765user45765
2,6322724
2,6322724
add a comment |
add a comment |
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Note that ${(1+lambda)z mid lambda > 0 text{ and } z in partial D^n} cap D^n = varnothing$?
answered Dec 21 '18 at 23:25
Eric TowersEric Towers
32.8k22370
$endgroup$
$begingroup$
That was dumb of me. The last conclusion was to rule out range. Thanks.
$endgroup$
– user45765
Dec 21 '18 at 23:30
add a comment |
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$begingroup$
Note that ${(1+lambda)z mid lambda > 0 text{ and } z in partial D^n} cap D^n = varnothing$?
answered Dec 21 '18 at 23:25
Eric TowersEric Towers
32.8k22370
$endgroup$
$begingroup$
That was dumb of me. The last conclusion was to rule out range. Thanks.
$endgroup$
– user45765
Dec 21 '18 at 23:30
add a comment |
$begingroup$
Note that ${(1+lambda)z mid lambda > 0 text{ and } z in partial D^n} cap D^n = varnothing$?
answered Dec 21 '18 at 23:25
Eric TowersEric Towers
32.8k22370
$endgroup$
$begingroup$
That was dumb of me. The last conclusion was to rule out range. Thanks.
$endgroup$
– user45765
Dec 21 '18 at 23:30
add a comment |
$begingroup$
Note that ${(1+lambda)z mid lambda > 0 text{ and } z in partial D^n} cap D^n = varnothing$?
answered Dec 21 '18 at 23:25
Eric TowersEric Towers
32.8k22370
$endgroup$
Note that ${(1+lambda)z mid lambda > 0 text{ and } z in partial D^n} cap D^n = varnothing$?
answered Dec 21 '18 at 23:25
Eric TowersEric Towers
32.8k22370
answered Dec 21 '18 at 23:25
Eric TowersEric Towers
32.8k22370
answered Dec 21 '18 at 23:25
Eric TowersEric Towers
32.8k22370
answered Dec 21 '18 at 23:25
Eric TowersEric Towers
32.8k22370
32.8k22370
$begingroup$
That was dumb of me. The last conclusion was to rule out range. Thanks.
$endgroup$
– user45765
Dec 21 '18 at 23:30
add a comment |
$begingroup$
That was dumb of me. The last conclusion was to rule out range. Thanks.
$endgroup$
– user45765
Dec 21 '18 at 23:30
$begingroup$
That was dumb of me. The last conclusion was to rule out range. Thanks.
$endgroup$
– user45765
Dec 21 '18 at 23:30
$begingroup$
That was dumb of me. The last conclusion was to rule out range. Thanks.
$endgroup$
– user45765
Dec 21 '18 at 23:30
add a comment |
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