The fundamental group of a topological space is isomorphic with its connected component fundamental group
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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
119k740113
asked Dec 7 '18 at 20:38
pershina oladpershina olad
8410
$endgroup$
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
119k740113
asked Dec 7 '18 at 20:38
pershina oladpershina olad
8410
$endgroup$
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
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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
119k740113
asked Dec 7 '18 at 20:38
pershina oladpershina olad
8410
$endgroup$
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
119k740113
asked Dec 7 '18 at 20:38
pershina oladpershina olad
8410
edited Dec 7 '18 at 20:42
Bernard
119k740113
asked Dec 7 '18 at 20:38
pershina oladpershina olad
8410
edited Dec 7 '18 at 20:42
Bernard
119k740113
edited Dec 7 '18 at 20:42
Bernard
119k740113
edited Dec 7 '18 at 20:42
Bernard
119k740113
119k740113
asked Dec 7 '18 at 20:38
pershina oladpershina olad
8410
asked Dec 7 '18 at 20:38
pershina oladpershina olad
8410
asked Dec 7 '18 at 20:38
pershina oladpershina olad
8410
8410
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 |
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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