map to a product is a submersion
up vote
1
down vote
favorite
Let $f:Yrightarrow W,g:Zrightarrow W$ are submersions (so that $Ytimes_W Z$ is a manifold).
Suppose $F:Xrightarrow Ytimes_WZ$ is a smooth map such that $pr_1circ F:Xrightarrow Y$ is submersion and $pr_2circ F:Xrightarrow Z$ is a submersion. Then, I want to prove that $F$ is a submersion.
I think this has to be true. Can some one give a quick proof.
Here $Ytimes_WZ$ is the pullback manifold $Ytimes_WZ={(y,z):f(y)=g(z)}subseteq Ytimes Z$.
differential-geometry smooth-manifolds
asked Nov 22 at 12:06
Praphulla Koushik
26215
add a comment |
up vote
1
down vote
favorite
Let $f:Yrightarrow W,g:Zrightarrow W$ are submersions (so that $Ytimes_W Z$ is a manifold).
Suppose $F:Xrightarrow Ytimes_WZ$ is a smooth map such that $pr_1circ F:Xrightarrow Y$ is submersion and $pr_2circ F:Xrightarrow Z$ is a submersion. Then, I want to prove that $F$ is a submersion.
I think this has to be true. Can some one give a quick proof.
Here $Ytimes_WZ$ is the pullback manifold $Ytimes_WZ={(y,z):f(y)=g(z)}subseteq Ytimes Z$.
differential-geometry smooth-manifolds
asked Nov 22 at 12:06
Praphulla Koushik
26215
What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 at 19:25
@Laz I have added the details.
– Praphulla Koushik
Nov 22 at 19:59
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $f:Yrightarrow W,g:Zrightarrow W$ are submersions (so that $Ytimes_W Z$ is a manifold).
Suppose $F:Xrightarrow Ytimes_WZ$ is a smooth map such that $pr_1circ F:Xrightarrow Y$ is submersion and $pr_2circ F:Xrightarrow Z$ is a submersion. Then, I want to prove that $F$ is a submersion.
I think this has to be true. Can some one give a quick proof.
Here $Ytimes_WZ$ is the pullback manifold $Ytimes_WZ={(y,z):f(y)=g(z)}subseteq Ytimes Z$.
differential-geometry smooth-manifolds
asked Nov 22 at 12:06
Praphulla Koushik
26215
Let $f:Yrightarrow W,g:Zrightarrow W$ are submersions (so that $Ytimes_W Z$ is a manifold).
Suppose $F:Xrightarrow Ytimes_WZ$ is a smooth map such that $pr_1circ F:Xrightarrow Y$ is submersion and $pr_2circ F:Xrightarrow Z$ is a submersion. Then, I want to prove that $F$ is a submersion.
I think this has to be true. Can some one give a quick proof.
Here $Ytimes_WZ$ is the pullback manifold $Ytimes_WZ={(y,z):f(y)=g(z)}subseteq Ytimes Z$.
differential-geometry smooth-manifolds
differential-geometry smooth-manifolds
asked Nov 22 at 12:06
Praphulla Koushik
26215
asked Nov 22 at 12:06
Praphulla Koushik
26215
edited Nov 22 at 19:58
asked Nov 22 at 12:06
Praphulla Koushik
26215
asked Nov 22 at 12:06
Praphulla Koushik
26215
asked Nov 22 at 12:06
Praphulla Koushik
26215
26215
What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 at 19:25
@Laz I have added the details.
– Praphulla Koushik
Nov 22 at 19:59
add a comment |
What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 at 19:25
@Laz I have added the details.
– Praphulla Koushik
Nov 22 at 19:59
What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 at 19:25
What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 at 19:25
@Laz I have added the details.
– Praphulla Koushik
Nov 22 at 19:59
@Laz I have added the details.
– Praphulla Koushik
Nov 22 at 19:59
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
I am sorry but it is not true. Let us consider the following counterexample. Let $W={*}$ be the singleton, let $Y=Z=X$ be a manifold of dimension greater or equal to $1$ and let $f=g$ be the unique possible map. Then the pullback $Ytimes_W Z$ is simply the usual product $Xtimes X$. Finally, let $F=Delta : x in X mapsto (x,x)in Xtimes X$ be the diagonal map, which is not a submersion. However, $pr_1 circ F = pr_2 circ F = id_X$ is a submersion.
answered Nov 30 at 3:04
Dante Grevino
8817
add a comment |
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',
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
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009052%2fmap-to-a-product-is-a-submersion%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
I am sorry but it is not true. Let us consider the following counterexample. Let $W={*}$ be the singleton, let $Y=Z=X$ be a manifold of dimension greater or equal to $1$ and let $f=g$ be the unique possible map. Then the pullback $Ytimes_W Z$ is simply the usual product $Xtimes X$. Finally, let $F=Delta : x in X mapsto (x,x)in Xtimes X$ be the diagonal map, which is not a submersion. However, $pr_1 circ F = pr_2 circ F = id_X$ is a submersion.
answered Nov 30 at 3:04
Dante Grevino
8817
add a comment |
up vote
1
down vote
accepted
I am sorry but it is not true. Let us consider the following counterexample. Let $W={*}$ be the singleton, let $Y=Z=X$ be a manifold of dimension greater or equal to $1$ and let $f=g$ be the unique possible map. Then the pullback $Ytimes_W Z$ is simply the usual product $Xtimes X$. Finally, let $F=Delta : x in X mapsto (x,x)in Xtimes X$ be the diagonal map, which is not a submersion. However, $pr_1 circ F = pr_2 circ F = id_X$ is a submersion.
answered Nov 30 at 3:04
Dante Grevino
8817
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
I am sorry but it is not true. Let us consider the following counterexample. Let $W={*}$ be the singleton, let $Y=Z=X$ be a manifold of dimension greater or equal to $1$ and let $f=g$ be the unique possible map. Then the pullback $Ytimes_W Z$ is simply the usual product $Xtimes X$. Finally, let $F=Delta : x in X mapsto (x,x)in Xtimes X$ be the diagonal map, which is not a submersion. However, $pr_1 circ F = pr_2 circ F = id_X$ is a submersion.
answered Nov 30 at 3:04
Dante Grevino
8817
I am sorry but it is not true. Let us consider the following counterexample. Let $W={*}$ be the singleton, let $Y=Z=X$ be a manifold of dimension greater or equal to $1$ and let $f=g$ be the unique possible map. Then the pullback $Ytimes_W Z$ is simply the usual product $Xtimes X$. Finally, let $F=Delta : x in X mapsto (x,x)in Xtimes X$ be the diagonal map, which is not a submersion. However, $pr_1 circ F = pr_2 circ F = id_X$ is a submersion.
answered Nov 30 at 3:04
Dante Grevino
8817
answered Nov 30 at 3:04
Dante Grevino
8817
answered Nov 30 at 3:04
Dante Grevino
8817
answered Nov 30 at 3:04
Dante Grevino
8817
8817
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009052%2fmap-to-a-product-is-a-submersion%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
What do you mean by $Ytimes_W Z$? The only way I've seen this notation is as an attaching map. Is that what you mean?
– Laz
Nov 22 at 19:25
@Laz I have added the details.
– Praphulla Koushik
Nov 22 at 19:59