Exercise on uniqueness of smooth substructures from Warner 1.33(a)
If I understand correctly, the following is Theorem 1.32 of Warner's Foundations of Differentiable Manifolds book.
Proposition. Let $Xto Y$ be smooth and $iota :Ato Y$ be an immersed submanifold. Suppose moreover $f(X)subsetiota (A)$. By injectivity of $iota$ there's a unique set function $f_0$ such that $iotacirc f_0=f$.
$iota$ is a topological embedding $implies f_0$ continuous.
$f_0$ continuous $implies f_0$ smooth.
In 1.33(a) we have the following assertion.
Let $M$ be a differentiable manifold and $Asubset M$. Fix a topology on $A$. Then there is at most one differentiable structure on $A$ such that $iota:Ahookrightarrow M$ is an injective immersion.
The hint is to apply Theorem 1.32, but I just don't see it. How to prove this result?
manifolds smooth-manifolds
asked Nov 27 '18 at 0:50
Arrow
5,07621445
|
show 3 more comments
If I understand correctly, the following is Theorem 1.32 of Warner's Foundations of Differentiable Manifolds book.
Proposition. Let $Xto Y$ be smooth and $iota :Ato Y$ be an immersed submanifold. Suppose moreover $f(X)subsetiota (A)$. By injectivity of $iota$ there's a unique set function $f_0$ such that $iotacirc f_0=f$.
$iota$ is a topological embedding $implies f_0$ continuous.
$f_0$ continuous $implies f_0$ smooth.
In 1.33(a) we have the following assertion.
Let $M$ be a differentiable manifold and $Asubset M$. Fix a topology on $A$. Then there is at most one differentiable structure on $A$ such that $iota:Ahookrightarrow M$ is an injective immersion.
The hint is to apply Theorem 1.32, but I just don't see it. How to prove this result?
manifolds smooth-manifolds
asked Nov 27 '18 at 0:50
Arrow
5,07621445
Hint on how to use the hint: Take $Y = M$, $X = A$, and let $f$ and $i$ both be injective immersions. Applying the hint gives a smooth $f_0:Xrightarrow A$. Applying it again with the roles of $X$ and $A$ swapped gives a smooth $g_0:Arightarrow X$...
– Jason DeVito
Nov 27 '18 at 0:59
@JasonDeVito where is the topological embedding ensuring the continuity of $g_0$?
– Arrow
Nov 27 '18 at 1:13
I think I misread Theorem 1.32. Sorry!
– Jason DeVito
Nov 27 '18 at 1:25
@JasonDeVito I think I'm the one missing something, because surely the solution you sketch is along the lines of what needs to happen. Would appreciate some insight!
– Arrow
Nov 27 '18 at 1:27
I get it. The map $f_0$ ends up being the identity map. Since $X=A$ with fixed topology, it is continuous. So you can use part b)
– Jason DeVito
Nov 27 '18 at 1:39
|
show 3 more comments
If I understand correctly, the following is Theorem 1.32 of Warner's Foundations of Differentiable Manifolds book.
Proposition. Let $Xto Y$ be smooth and $iota :Ato Y$ be an immersed submanifold. Suppose moreover $f(X)subsetiota (A)$. By injectivity of $iota$ there's a unique set function $f_0$ such that $iotacirc f_0=f$.
$iota$ is a topological embedding $implies f_0$ continuous.
$f_0$ continuous $implies f_0$ smooth.
In 1.33(a) we have the following assertion.
Let $M$ be a differentiable manifold and $Asubset M$. Fix a topology on $A$. Then there is at most one differentiable structure on $A$ such that $iota:Ahookrightarrow M$ is an injective immersion.
The hint is to apply Theorem 1.32, but I just don't see it. How to prove this result?
manifolds smooth-manifolds
asked Nov 27 '18 at 0:50
Arrow
5,07621445
If I understand correctly, the following is Theorem 1.32 of Warner's Foundations of Differentiable Manifolds book.
Proposition. Let $Xto Y$ be smooth and $iota :Ato Y$ be an immersed submanifold. Suppose moreover $f(X)subsetiota (A)$. By injectivity of $iota$ there's a unique set function $f_0$ such that $iotacirc f_0=f$.
$iota$ is a topological embedding $implies f_0$ continuous.
$f_0$ continuous $implies f_0$ smooth.
In 1.33(a) we have the following assertion.
Let $M$ be a differentiable manifold and $Asubset M$. Fix a topology on $A$. Then there is at most one differentiable structure on $A$ such that $iota:Ahookrightarrow M$ is an injective immersion.
The hint is to apply Theorem 1.32, but I just don't see it. How to prove this result?
manifolds smooth-manifolds
manifolds smooth-manifolds
asked Nov 27 '18 at 0:50
Arrow
5,07621445
asked Nov 27 '18 at 0:50
Arrow
5,07621445
asked Nov 27 '18 at 0:50
Arrow
5,07621445
asked Nov 27 '18 at 0:50
Arrow
5,07621445
asked Nov 27 '18 at 0:50
Arrow
5,07621445
5,07621445
Hint on how to use the hint: Take $Y = M$, $X = A$, and let $f$ and $i$ both be injective immersions. Applying the hint gives a smooth $f_0:Xrightarrow A$. Applying it again with the roles of $X$ and $A$ swapped gives a smooth $g_0:Arightarrow X$...
– Jason DeVito
Nov 27 '18 at 0:59
@JasonDeVito where is the topological embedding ensuring the continuity of $g_0$?
– Arrow
Nov 27 '18 at 1:13
I think I misread Theorem 1.32. Sorry!
– Jason DeVito
Nov 27 '18 at 1:25
@JasonDeVito I think I'm the one missing something, because surely the solution you sketch is along the lines of what needs to happen. Would appreciate some insight!
– Arrow
Nov 27 '18 at 1:27
I get it. The map $f_0$ ends up being the identity map. Since $X=A$ with fixed topology, it is continuous. So you can use part b)
– Jason DeVito
Nov 27 '18 at 1:39
|
show 3 more comments
Hint on how to use the hint: Take $Y = M$, $X = A$, and let $f$ and $i$ both be injective immersions. Applying the hint gives a smooth $f_0:Xrightarrow A$. Applying it again with the roles of $X$ and $A$ swapped gives a smooth $g_0:Arightarrow X$...
– Jason DeVito
Nov 27 '18 at 0:59
@JasonDeVito where is the topological embedding ensuring the continuity of $g_0$?
– Arrow
Nov 27 '18 at 1:13
I think I misread Theorem 1.32. Sorry!
– Jason DeVito
Nov 27 '18 at 1:25
@JasonDeVito I think I'm the one missing something, because surely the solution you sketch is along the lines of what needs to happen. Would appreciate some insight!
– Arrow
Nov 27 '18 at 1:27
I get it. The map $f_0$ ends up being the identity map. Since $X=A$ with fixed topology, it is continuous. So you can use part b)
– Jason DeVito
Nov 27 '18 at 1:39
Hint on how to use the hint: Take $Y = M$, $X = A$, and let $f$ and $i$ both be injective immersions. Applying the hint gives a smooth $f_0:Xrightarrow A$. Applying it again with the roles of $X$ and $A$ swapped gives a smooth $g_0:Arightarrow X$...
– Jason DeVito
Nov 27 '18 at 0:59
Hint on how to use the hint: Take $Y = M$, $X = A$, and let $f$ and $i$ both be injective immersions. Applying the hint gives a smooth $f_0:Xrightarrow A$. Applying it again with the roles of $X$ and $A$ swapped gives a smooth $g_0:Arightarrow X$...
– Jason DeVito
Nov 27 '18 at 0:59
@JasonDeVito where is the topological embedding ensuring the continuity of $g_0$?
– Arrow
Nov 27 '18 at 1:13
@JasonDeVito where is the topological embedding ensuring the continuity of $g_0$?
– Arrow
Nov 27 '18 at 1:13
I think I misread Theorem 1.32. Sorry!
– Jason DeVito
Nov 27 '18 at 1:25
I think I misread Theorem 1.32. Sorry!
– Jason DeVito
Nov 27 '18 at 1:25
@JasonDeVito I think I'm the one missing something, because surely the solution you sketch is along the lines of what needs to happen. Would appreciate some insight!
– Arrow
Nov 27 '18 at 1:27
@JasonDeVito I think I'm the one missing something, because surely the solution you sketch is along the lines of what needs to happen. Would appreciate some insight!
– Arrow
Nov 27 '18 at 1:27
I get it. The map $f_0$ ends up being the identity map. Since $X=A$ with fixed topology, it is continuous. So you can use part b)
– Jason DeVito
Nov 27 '18 at 1:39
I get it. The map $f_0$ ends up being the identity map. Since $X=A$ with fixed topology, it is continuous. So you can use part b)
– Jason DeVito
Nov 27 '18 at 1:39
|
show 3 more comments
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Hint on how to use the hint: Take $Y = M$, $X = A$, and let $f$ and $i$ both be injective immersions. Applying the hint gives a smooth $f_0:Xrightarrow A$. Applying it again with the roles of $X$ and $A$ swapped gives a smooth $g_0:Arightarrow X$...
– Jason DeVito
Nov 27 '18 at 0:59
@JasonDeVito where is the topological embedding ensuring the continuity of $g_0$?
– Arrow
Nov 27 '18 at 1:13
I think I misread Theorem 1.32. Sorry!
– Jason DeVito
Nov 27 '18 at 1:25
@JasonDeVito I think I'm the one missing something, because surely the solution you sketch is along the lines of what needs to happen. Would appreciate some insight!
– Arrow
Nov 27 '18 at 1:27
I get it. The map $f_0$ ends up being the identity map. Since $X=A$ with fixed topology, it is continuous. So you can use part b)
– Jason DeVito
Nov 27 '18 at 1:39