Proving surjectivity of Hopf map and existence of Hopf Circles on $mathbb{S}^3$
$begingroup$
I am using the Hopf Map $f: mathbb{S}^3 rightarrow mathbb{S}^2$ where $f(a,b,c,d) = (2(ab+cd),2(ad-cb),(a^2+c^2)-(b^2+d^2))$
My question is
1.) How can I prove surjectivity of the map? I am aware of the primary definition of surjectivity but I am unsure how to apply in this case.
I tried using the trivial method of finding an inverse but the method is very complicated.
2.) How can I show that a point that the preimage of $q in mathbb{S}^2$ is a circle in $mathbb{S}^3$?
I realize that the solution is dependent on finding the inverse function as hinted in the first part, but since the function itself was complicated to find, hence I had no progress in this part.
differential-geometry
asked Nov 25 '18 at 22:04
mathnoob123mathnoob123
693417
$endgroup$
add a comment |
$begingroup$
I am using the Hopf Map $f: mathbb{S}^3 rightarrow mathbb{S}^2$ where $f(a,b,c,d) = (2(ab+cd),2(ad-cb),(a^2+c^2)-(b^2+d^2))$
My question is
1.) How can I prove surjectivity of the map? I am aware of the primary definition of surjectivity but I am unsure how to apply in this case.
I tried using the trivial method of finding an inverse but the method is very complicated.
2.) How can I show that a point that the preimage of $q in mathbb{S}^2$ is a circle in $mathbb{S}^3$?
I realize that the solution is dependent on finding the inverse function as hinted in the first part, but since the function itself was complicated to find, hence I had no progress in this part.
differential-geometry
asked Nov 25 '18 at 22:04
mathnoob123mathnoob123
693417
$endgroup$
$begingroup$
It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
$endgroup$
– Brian Shin
Nov 26 '18 at 0:57
$begingroup$
@BrianShin How can I demonstrate the equivalence of the two maps?
$endgroup$
– mathnoob123
Nov 26 '18 at 8:09
add a comment |
$begingroup$
I am using the Hopf Map $f: mathbb{S}^3 rightarrow mathbb{S}^2$ where $f(a,b,c,d) = (2(ab+cd),2(ad-cb),(a^2+c^2)-(b^2+d^2))$
My question is
1.) How can I prove surjectivity of the map? I am aware of the primary definition of surjectivity but I am unsure how to apply in this case.
I tried using the trivial method of finding an inverse but the method is very complicated.
2.) How can I show that a point that the preimage of $q in mathbb{S}^2$ is a circle in $mathbb{S}^3$?
I realize that the solution is dependent on finding the inverse function as hinted in the first part, but since the function itself was complicated to find, hence I had no progress in this part.
differential-geometry
asked Nov 25 '18 at 22:04
mathnoob123mathnoob123
693417
$endgroup$
I am using the Hopf Map $f: mathbb{S}^3 rightarrow mathbb{S}^2$ where $f(a,b,c,d) = (2(ab+cd),2(ad-cb),(a^2+c^2)-(b^2+d^2))$
My question is
1.) How can I prove surjectivity of the map? I am aware of the primary definition of surjectivity but I am unsure how to apply in this case.
I tried using the trivial method of finding an inverse but the method is very complicated.
2.) How can I show that a point that the preimage of $q in mathbb{S}^2$ is a circle in $mathbb{S}^3$?
I realize that the solution is dependent on finding the inverse function as hinted in the first part, but since the function itself was complicated to find, hence I had no progress in this part.
differential-geometry
differential-geometry
asked Nov 25 '18 at 22:04
mathnoob123mathnoob123
693417
asked Nov 25 '18 at 22:04
mathnoob123mathnoob123
693417
edited Nov 25 '18 at 22:58
mathnoob123
asked Nov 25 '18 at 22:04
mathnoob123mathnoob123
693417
asked Nov 25 '18 at 22:04
mathnoob123mathnoob123
693417
asked Nov 25 '18 at 22:04
mathnoob123mathnoob123
693417
693417
$begingroup$
It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
$endgroup$
– Brian Shin
Nov 26 '18 at 0:57
$begingroup$
@BrianShin How can I demonstrate the equivalence of the two maps?
$endgroup$
– mathnoob123
Nov 26 '18 at 8:09
add a comment |
$begingroup$
It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
$endgroup$
– Brian Shin
Nov 26 '18 at 0:57
$begingroup$
@BrianShin How can I demonstrate the equivalence of the two maps?
$endgroup$
– mathnoob123
Nov 26 '18 at 8:09
$begingroup$
It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
$endgroup$
– Brian Shin
Nov 26 '18 at 0:57
$begingroup$
It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
$endgroup$
– Brian Shin
Nov 26 '18 at 0:57
$begingroup$
@BrianShin How can I demonstrate the equivalence of the two maps?
$endgroup$
– mathnoob123
Nov 26 '18 at 8:09
$begingroup$
@BrianShin How can I demonstrate the equivalence of the two maps?
$endgroup$
– mathnoob123
Nov 26 '18 at 8:09
add a comment |
0
active
oldest
votes
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
});
}
});
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%2f3013479%2fproving-surjectivity-of-hopf-map-and-existence-of-hopf-circles-on-mathbbs3%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
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%2f3013479%2fproving-surjectivity-of-hopf-map-and-existence-of-hopf-circles-on-mathbbs3%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
$begingroup$
It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
$endgroup$
– Brian Shin
Nov 26 '18 at 0:57
$begingroup$
@BrianShin How can I demonstrate the equivalence of the two maps?
$endgroup$
– mathnoob123
Nov 26 '18 at 8:09