Fdtxjr

Know when to turn notes upside-down(eighth notes, sixteen notes, etc.)

Good allowance savings plan?

Current sense amp + op-amp buffer + ADC: Measuring down to 0 with single supply

Running a subshell from the middle of the current command

Does the statement `int val = (++i > ++j) ? ++i : ++j;` invoke undefined behavior?

Why are there 40 737 Max planes in flight when they have been grounded as not airworthy?

Life insurance that covers only simultaneous/dual deaths

How to simplify this time periods definition interface?

Calculus II Professor will not accept my correct integral evaluation that uses a different method, should I bring this up further?

Is it possible to upcast ritual spells?

How to answer questions about my characters?

Identifying the interval from A♭ to D♯

What is this large pipe coming out of my roof?

What is the greatest age difference between a married couple in Tanach?

Science-fiction short story where space navy wanted hospital ships and settlers had guns mounted everywhere

Is a lawful good "antagonist" effective?

Is it true that real estate prices mainly go up?

Importance of differentiation

Can elves maintain concentration in a trance?

Make a transparent 448*448 image

It's a yearly task, alright

Should we release the security issues we found in our product as CVE or we can just update those on weekly release notes?

Why do Australian milk farmers need to protest supermarkets' milk price?

Why doesn't the EU now just force the UK to choose between referendum and no-deal?

Does injectivity of dual imply injectivity of pullback by a surjective submersion?

When is the pullback of a linear injection a surjection on dual space?Does monotonicity imply surjectivity?Projection of fiber bundle is a submersionDetailed proof (submersion) : show that the differential is surjectiveWhy do we need to suppose finite dimension to prove dual of linear map $L$ is injective $Rightarrow$ $L$ is surjective?Pullback of diffrential forms in surjective submersionsDescribe $f$ in terms of the tensor products of $alpha^i$ and $alpha^j$. Is this inner product? What are its coefficients?Derivations of bases of $A_k(V)$ and $L_k(V)$: what's the difference?Matrix-based proof of transformation rule for a wedge product of covectors by wedge product of covectorsWhy is $e^it$ a submersion, and what is the relationship between the derivative $dot h(t)$ and the differential $h_*,t$?

2

1

$begingroup$

My book is An Introduction to Manifolds by Loring W. Tu.

Does Problem 18.8 below follow from an earlier exercise?

Problem 18.8 (Pullback by a surjective submersion)

Problem 10.5a (Injectivity of the dual map)

My thought is that just as we have a dual for k-covectors:

So do we have a dual for k-forms, although that might be what this exercise is trying to prove.

One reason I think Problem 10.5a doesn't apply is that that $pi$ is a submersion is used to prove $pi^*$ is injective (see below), while Problem 10.5a does not seem to say anything explicitly about submersions. Either that or the proof below is just a proof that is alternative, direct and doesn't rely on category theory.

linear-algebra abstract-algebra differential-geometry category-theory differential-forms

share|cite|improve this question

edited Mar 11 at 11:32

Selene Auckland

asked Mar 4 at 6:08

Selene AucklandSelene Auckland

6911

$endgroup$

add a comment | 

2

1

$begingroup$

My book is An Introduction to Manifolds by Loring W. Tu.

Does Problem 18.8 below follow from an earlier exercise?

Problem 18.8 (Pullback by a surjective submersion)

Problem 10.5a (Injectivity of the dual map)

My thought is that just as we have a dual for k-covectors:

So do we have a dual for k-forms, although that might be what this exercise is trying to prove.

One reason I think Problem 10.5a doesn't apply is that that $pi$ is a submersion is used to prove $pi^*$ is injective (see below), while Problem 10.5a does not seem to say anything explicitly about submersions. Either that or the proof below is just a proof that is alternative, direct and doesn't rely on category theory.

linear-algebra abstract-algebra differential-geometry category-theory differential-forms

share|cite|improve this question

edited Mar 11 at 11:32

Selene Auckland

asked Mar 4 at 6:08

Selene AucklandSelene Auckland

6911

$endgroup$

add a comment | 

$begingroup$

My book is An Introduction to Manifolds by Loring W. Tu.

Does Problem 18.8 below follow from an earlier exercise?

Problem 18.8 (Pullback by a surjective submersion)

Problem 10.5a (Injectivity of the dual map)

My thought is that just as we have a dual for k-covectors:

So do we have a dual for k-forms, although that might be what this exercise is trying to prove.

One reason I think Problem 10.5a doesn't apply is that that $pi$ is a submersion is used to prove $pi^*$ is injective (see below), while Problem 10.5a does not seem to say anything explicitly about submersions. Either that or the proof below is just a proof that is alternative, direct and doesn't rely on category theory.

linear-algebra abstract-algebra differential-geometry category-theory differential-forms

share|cite|improve this question

edited Mar 11 at 11:32

Selene Auckland

asked Mar 4 at 6:08

Selene AucklandSelene Auckland

6911

$endgroup$

share|cite|improve this question

edited Mar 11 at 11:32

Selene Auckland

asked Mar 4 at 6:08

Selene AucklandSelene Auckland

6911

share|cite|improve this question

edited Mar 11 at 11:32

Selene Auckland

asked Mar 4 at 6:08

Selene AucklandSelene Auckland

6911

asked Mar 4 at 6:08

Selene AucklandSelene Auckland

6911

asked Mar 4 at 6:08

Selene AucklandSelene Auckland

6911

1 Answer
1

active

oldest

votes

2

$begingroup$

Problem 18.8 can be proven in the following way (where I hope to not miss anything important hiding under the surface; let me know if you find issues in the proof!).

You should first note that the pullback map $pi^*$ is the dual map to the map $pi$. Now, a general fact from linear algebra is that for any linear map $f$, $$ftext injective Longleftrightarrow f^*text surjective,\ ftext surjective Longleftrightarrow f^*text injective. $$ So in your case, if $pi$ is surjective, $pi^*$ is an injective homomorphism of vector spaces and you are left showing that it respects the algebra structure on $Omega^*(M)$, i.e. that everything fits well with the differential. I think that you will need the property of being a submersion in that. Try to proceed with what I wrote, if you can't, I will work out a solution and put it here.

share|cite|improve this answer

answered Mar 4 at 9:03

JamesJames

1

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I didn't realise that you put problem 10.5a, sorry. I do indeed agree that this suffices provided that you know that $pi^*$ is a morphism of algebras not only of vector spaces. If you know that already, you should be fine proving problem 10.5a.
  $endgroup$
  – James
  Mar 4 at 10:49
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  That’s what I meant, yes
  $endgroup$
  – James
  Mar 5 at 5:26
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @SeleneAuckland If this answered your question, please consider accepting the answer.
  $endgroup$
  – James
  Mar 6 at 7:44
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks! Wait, I just realized a potential flaw. $pi$ is not necessarily a linear map right?
  $endgroup$
  – Selene Auckland
  Mar 7 at 9:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @Selene: You are correct that $pi$ is not necessarily linear. In fact,$tildeM$ and $M$ need not be vector spaces, so "linearity" is meaningless! On the other hand, saying $pi$ is a submersion means that $pi_ast: T_p tildeMrightarrow T_pi(p) M$ is a surjective linear map. So, apply what James said, except using $pi_ast$ instead of $pi$.
  $endgroup$
  – Jason DeVito
  Mar 11 at 15:58
  

  
  
  
  

 | 
show 3 more comments

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3134509%2fdoes-injectivity-of-dual-imply-injectivity-of-pullback-by-a-surjective-submersio%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

2

$begingroup$

Problem 18.8 can be proven in the following way (where I hope to not miss anything important hiding under the surface; let me know if you find issues in the proof!).

You should first note that the pullback map $pi^*$ is the dual map to the map $pi$. Now, a general fact from linear algebra is that for any linear map $f$, $$ftext injective Longleftrightarrow f^*text surjective,\ ftext surjective Longleftrightarrow f^*text injective. $$ So in your case, if $pi$ is surjective, $pi^*$ is an injective homomorphism of vector spaces and you are left showing that it respects the algebra structure on $Omega^*(M)$, i.e. that everything fits well with the differential. I think that you will need the property of being a submersion in that. Try to proceed with what I wrote, if you can't, I will work out a solution and put it here.

share|cite|improve this answer

answered Mar 4 at 9:03

JamesJames

1

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I didn't realise that you put problem 10.5a, sorry. I do indeed agree that this suffices provided that you know that $pi^*$ is a morphism of algebras not only of vector spaces. If you know that already, you should be fine proving problem 10.5a.
  $endgroup$
  – James
  Mar 4 at 10:49
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  That’s what I meant, yes
  $endgroup$
  – James
  Mar 5 at 5:26
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @SeleneAuckland If this answered your question, please consider accepting the answer.
  $endgroup$
  – James
  Mar 6 at 7:44
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks! Wait, I just realized a potential flaw. $pi$ is not necessarily a linear map right?
  $endgroup$
  – Selene Auckland
  Mar 7 at 9:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @Selene: You are correct that $pi$ is not necessarily linear. In fact,$tildeM$ and $M$ need not be vector spaces, so "linearity" is meaningless! On the other hand, saying $pi$ is a submersion means that $pi_ast: T_p tildeMrightarrow T_pi(p) M$ is a surjective linear map. So, apply what James said, except using $pi_ast$ instead of $pi$.
  $endgroup$
  – Jason DeVito
  Mar 11 at 15:58
  

  
  
  
  

 | 
show 3 more comments

2

$begingroup$

Problem 18.8 can be proven in the following way (where I hope to not miss anything important hiding under the surface; let me know if you find issues in the proof!).

You should first note that the pullback map $pi^*$ is the dual map to the map $pi$. Now, a general fact from linear algebra is that for any linear map $f$, $$ftext injective Longleftrightarrow f^*text surjective,\ ftext surjective Longleftrightarrow f^*text injective. $$ So in your case, if $pi$ is surjective, $pi^*$ is an injective homomorphism of vector spaces and you are left showing that it respects the algebra structure on $Omega^*(M)$, i.e. that everything fits well with the differential. I think that you will need the property of being a submersion in that. Try to proceed with what I wrote, if you can't, I will work out a solution and put it here.

share|cite|improve this answer

answered Mar 4 at 9:03

JamesJames

1

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I didn't realise that you put problem 10.5a, sorry. I do indeed agree that this suffices provided that you know that $pi^*$ is a morphism of algebras not only of vector spaces. If you know that already, you should be fine proving problem 10.5a.
  $endgroup$
  – James
  Mar 4 at 10:49
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  That’s what I meant, yes
  $endgroup$
  – James
  Mar 5 at 5:26
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @SeleneAuckland If this answered your question, please consider accepting the answer.
  $endgroup$
  – James
  Mar 6 at 7:44
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks! Wait, I just realized a potential flaw. $pi$ is not necessarily a linear map right?
  $endgroup$
  – Selene Auckland
  Mar 7 at 9:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @Selene: You are correct that $pi$ is not necessarily linear. In fact,$tildeM$ and $M$ need not be vector spaces, so "linearity" is meaningless! On the other hand, saying $pi$ is a submersion means that $pi_ast: T_p tildeMrightarrow T_pi(p) M$ is a surjective linear map. So, apply what James said, except using $pi_ast$ instead of $pi$.
  $endgroup$
  – Jason DeVito
  Mar 11 at 15:58
  

  
  
  
  

 | 
show 3 more comments

$begingroup$

Problem 18.8 can be proven in the following way (where I hope to not miss anything important hiding under the surface; let me know if you find issues in the proof!).

You should first note that the pullback map $pi^*$ is the dual map to the map $pi$. Now, a general fact from linear algebra is that for any linear map $f$, $$ftext injective Longleftrightarrow f^*text surjective,\ ftext surjective Longleftrightarrow f^*text injective. $$ So in your case, if $pi$ is surjective, $pi^*$ is an injective homomorphism of vector spaces and you are left showing that it respects the algebra structure on $Omega^*(M)$, i.e. that everything fits well with the differential. I think that you will need the property of being a submersion in that. Try to proceed with what I wrote, if you can't, I will work out a solution and put it here.

share|cite|improve this answer

answered Mar 4 at 9:03

JamesJames

1

$endgroup$

share|cite|improve this answer

answered Mar 4 at 9:03

JamesJames

1

answered Mar 4 at 9:03

JamesJames

1

answered Mar 4 at 9:03

JamesJames

1