The fundamental group of a topological space is isomorphic with its connected component fundamental group
$begingroup$
can you help me with this problem of fundamental groups?
suppose that $X$ is a topological space, let's fix a point on $X$ like $pin{X}$. Thus I want to prove that the fundamental group of $X$ at $p$ is isomorphic to the fundamental group of connected component of $X$ at $p$.
Should I build a bijection between them and prove its existence? or should I use theorems of the fundamental theorems and covering spaces... to prove this proposition?
I really need help to solve this problem, thank you all!
algebraic-topology covering-spaces fundamental-groups
edited Dec 7 '18 at 20:42
Bernard
122k741116
asked Dec 7 '18 at 20:38
pershina oladpershina olad
9410
$endgroup$
add a comment |
$begingroup$
can you help me with this problem of fundamental groups?
suppose that $X$ is a topological space, let's fix a point on $X$ like $pin{X}$. Thus I want to prove that the fundamental group of $X$ at $p$ is isomorphic to the fundamental group of connected component of $X$ at $p$.
Should I build a bijection between them and prove its existence? or should I use theorems of the fundamental theorems and covering spaces... to prove this proposition?
I really need help to solve this problem, thank you all!
algebraic-topology covering-spaces fundamental-groups
edited Dec 7 '18 at 20:42
Bernard
122k741116
asked Dec 7 '18 at 20:38
pershina oladpershina olad
9410
$endgroup$
1
$begingroup$
You don't need covering spaces. Where must the image of a path $alpha: [0,1] to X$ live?
$endgroup$
– Randall
Dec 7 '18 at 20:40
$begingroup$
@Randall .thank you for your guidance .the image is in the fundamental group of connected component of $X$
$endgroup$
– pershina olad
Dec 7 '18 at 20:50
$begingroup$
Yep. That correspondence gives an iso.
$endgroup$
– Randall
Dec 7 '18 at 20:53
2
$begingroup$
@pershinaolad I would suggest that you write up an answer to your own question, now that you've figured it out.
$endgroup$
– Kyle Miller
Dec 8 '18 at 3:39
$begingroup$
Seconded........
$endgroup$
– Randall
Dec 8 '18 at 3:53
add a comment |
$begingroup$
can you help me with this problem of fundamental groups?
suppose that $X$ is a topological space, let's fix a point on $X$ like $pin{X}$. Thus I want to prove that the fundamental group of $X$ at $p$ is isomorphic to the fundamental group of connected component of $X$ at $p$.
Should I build a bijection between them and prove its existence? or should I use theorems of the fundamental theorems and covering spaces... to prove this proposition?
I really need help to solve this problem, thank you all!
algebraic-topology covering-spaces fundamental-groups
edited Dec 7 '18 at 20:42
Bernard
122k741116
asked Dec 7 '18 at 20:38
pershina oladpershina olad
9410
$endgroup$
can you help me with this problem of fundamental groups?
suppose that $X$ is a topological space, let's fix a point on $X$ like $pin{X}$. Thus I want to prove that the fundamental group of $X$ at $p$ is isomorphic to the fundamental group of connected component of $X$ at $p$.
Should I build a bijection between them and prove its existence? or should I use theorems of the fundamental theorems and covering spaces... to prove this proposition?
I really need help to solve this problem, thank you all!
algebraic-topology covering-spaces fundamental-groups
algebraic-topology covering-spaces fundamental-groups
edited Dec 7 '18 at 20:42
Bernard
122k741116
asked Dec 7 '18 at 20:38
pershina oladpershina olad
9410
edited Dec 7 '18 at 20:42
Bernard
122k741116
asked Dec 7 '18 at 20:38
pershina oladpershina olad
9410
edited Dec 7 '18 at 20:42
Bernard
122k741116
edited Dec 7 '18 at 20:42
Bernard
122k741116
edited Dec 7 '18 at 20:42
Bernard
122k741116
122k741116
asked Dec 7 '18 at 20:38
pershina oladpershina olad
9410
asked Dec 7 '18 at 20:38
pershina oladpershina olad
9410
asked Dec 7 '18 at 20:38
pershina oladpershina olad
9410
9410
1
$begingroup$
You don't need covering spaces. Where must the image of a path $alpha: [0,1] to X$ live?
$endgroup$
– Randall
Dec 7 '18 at 20:40
$begingroup$
@Randall .thank you for your guidance .the image is in the fundamental group of connected component of $X$
$endgroup$
– pershina olad
Dec 7 '18 at 20:50
$begingroup$
Yep. That correspondence gives an iso.
$endgroup$
– Randall
Dec 7 '18 at 20:53
2
$begingroup$
@pershinaolad I would suggest that you write up an answer to your own question, now that you've figured it out.
$endgroup$
– Kyle Miller
Dec 8 '18 at 3:39
$begingroup$
Seconded........
$endgroup$
– Randall
Dec 8 '18 at 3:53
add a comment |
1
$begingroup$
You don't need covering spaces. Where must the image of a path $alpha: [0,1] to X$ live?
$endgroup$
– Randall
Dec 7 '18 at 20:40
$begingroup$
@Randall .thank you for your guidance .the image is in the fundamental group of connected component of $X$
$endgroup$
– pershina olad
Dec 7 '18 at 20:50
$begingroup$
Yep. That correspondence gives an iso.
$endgroup$
– Randall
Dec 7 '18 at 20:53
2
$begingroup$
@pershinaolad I would suggest that you write up an answer to your own question, now that you've figured it out.
$endgroup$
– Kyle Miller
Dec 8 '18 at 3:39
$begingroup$
Seconded........
$endgroup$
– Randall
Dec 8 '18 at 3:53
1
1
$begingroup$
You don't need covering spaces. Where must the image of a path $alpha: [0,1] to X$ live?
$endgroup$
– Randall
Dec 7 '18 at 20:40
$begingroup$
You don't need covering spaces. Where must the image of a path $alpha: [0,1] to X$ live?
$endgroup$
– Randall
Dec 7 '18 at 20:40
$begingroup$
@Randall .thank you for your guidance .the image is in the fundamental group of connected component of $X$
$endgroup$
– pershina olad
Dec 7 '18 at 20:50
$begingroup$
@Randall .thank you for your guidance .the image is in the fundamental group of connected component of $X$
$endgroup$
– pershina olad
Dec 7 '18 at 20:50
$begingroup$
Yep. That correspondence gives an iso.
$endgroup$
– Randall
Dec 7 '18 at 20:53
$begingroup$
Yep. That correspondence gives an iso.
$endgroup$
– Randall
Dec 7 '18 at 20:53
2
2
$begingroup$
@pershinaolad I would suggest that you write up an answer to your own question, now that you've figured it out.
$endgroup$
– Kyle Miller
Dec 8 '18 at 3:39
$begingroup$
@pershinaolad I would suggest that you write up an answer to your own question, now that you've figured it out.
$endgroup$
– Kyle Miller
Dec 8 '18 at 3:39
$begingroup$
Seconded........
$endgroup$
– Randall
Dec 8 '18 at 3:53
$begingroup$
Seconded........
$endgroup$
– Randall
Dec 8 '18 at 3:53
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%2f3030348%2fthe-fundamental-group-of-a-topological-space-is-isomorphic-with-its-connected-co%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%2f3030348%2fthe-fundamental-group-of-a-topological-space-is-isomorphic-with-its-connected-co%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
1
$begingroup$
You don't need covering spaces. Where must the image of a path $alpha: [0,1] to X$ live?
$endgroup$
– Randall
Dec 7 '18 at 20:40
$begingroup$
@Randall .thank you for your guidance .the image is in the fundamental group of connected component of $X$
$endgroup$
– pershina olad
Dec 7 '18 at 20:50
$begingroup$
Yep. That correspondence gives an iso.
$endgroup$
– Randall
Dec 7 '18 at 20:53
2
$begingroup$
@pershinaolad I would suggest that you write up an answer to your own question, now that you've figured it out.
$endgroup$
– Kyle Miller
Dec 8 '18 at 3:39
$begingroup$
Seconded........
$endgroup$
– Randall
Dec 8 '18 at 3:53