Simply connected and direct productscontractible and simply connectedIs a path connected subspace of a simply connected space simply connected?Is a simply connected set connected?How to compute the fundamental group from first homology group?Is there a non-simply-connected space with trivial first homology group?Question about simply connected spaces.Does trivial fundamental group imply contractible?Use Seifert–van Kampen to show $X ast Y$ is simply connected if $X$ is path connected.locally simply-connected vs. semilocally simply-connectedProve “Contractible implies simply connected” using tools in Munkres Topology. Context is theta-space.
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Simply connected and direct products
contractible and simply connectedIs a path connected subspace of a simply connected space simply connected?Is a simply connected set connected?How to compute the fundamental group from first homology group?Is there a non-simply-connected space with trivial first homology group?Question about simply connected spaces.Does trivial fundamental group imply contractible?Use Seifert–van Kampen to show $X ast Y$ is simply connected if $X$ is path connected.locally simply-connected vs. semilocally simply-connectedProve “Contractible implies simply connected” using tools in Munkres Topology. Context is theta-space.
$begingroup$
A topological space $X$ is simply connected if and only if $X$ is path-connected and the fundamental group of $X$ at each point is trivial, that is, $pi(X,x_0) = 0$ for any $x_0 in X$.
Now, we know that if $X,Y$ are simply connected, then $X times Y$ is simply connected. My question: Is it also true the converse?
Since $X times Y$ is path-connected we get $X,Y$ are path-connected. There exist a way to assert $pi(X) = pi(Y) = 0$?
Thanks!
algebraic-topology fundamental-groups
asked Mar 21 at 19:45
user183002user183002
312
$endgroup$
add a comment |
$begingroup$
A topological space $X$ is simply connected if and only if $X$ is path-connected and the fundamental group of $X$ at each point is trivial, that is, $pi(X,x_0) = 0$ for any $x_0 in X$.
Now, we know that if $X,Y$ are simply connected, then $X times Y$ is simply connected. My question: Is it also true the converse?
Since $X times Y$ is path-connected we get $X,Y$ are path-connected. There exist a way to assert $pi(X) = pi(Y) = 0$?
Thanks!
algebraic-topology fundamental-groups
asked Mar 21 at 19:45
user183002user183002
312
$endgroup$
add a comment |
$begingroup$
A topological space $X$ is simply connected if and only if $X$ is path-connected and the fundamental group of $X$ at each point is trivial, that is, $pi(X,x_0) = 0$ for any $x_0 in X$.
Now, we know that if $X,Y$ are simply connected, then $X times Y$ is simply connected. My question: Is it also true the converse?
Since $X times Y$ is path-connected we get $X,Y$ are path-connected. There exist a way to assert $pi(X) = pi(Y) = 0$?
Thanks!
algebraic-topology fundamental-groups
asked Mar 21 at 19:45
user183002user183002
312
$endgroup$
A topological space $X$ is simply connected if and only if $X$ is path-connected and the fundamental group of $X$ at each point is trivial, that is, $pi(X,x_0) = 0$ for any $x_0 in X$.
Now, we know that if $X,Y$ are simply connected, then $X times Y$ is simply connected. My question: Is it also true the converse?
Since $X times Y$ is path-connected we get $X,Y$ are path-connected. There exist a way to assert $pi(X) = pi(Y) = 0$?
Thanks!
algebraic-topology fundamental-groups
algebraic-topology fundamental-groups
asked Mar 21 at 19:45
user183002user183002
312
asked Mar 21 at 19:45
user183002user183002
312
asked Mar 21 at 19:45
user183002user183002
312
asked Mar 21 at 19:45
user183002user183002
312
asked Mar 21 at 19:45
user183002user183002
312
312
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: $pi_1(X times Y, (x_0, y_0)) cong pi_1(X, x_0) times pi_1(Y, y_0)$.
answered Mar 21 at 20:09
JHFJHF
4,9611026
$endgroup$
$begingroup$
I know that result... But I´m not sure that $pi(X times Y) = 0$ implies $pi(X) = 0$.
$endgroup$
– user183002
Mar 21 at 20:17
1
$begingroup$
Try counting the size of the groups on either side of the isomorphism. We know that $pi_1(X times Y)$ has one element, so...
$endgroup$
– JHF
Mar 21 at 20:24
$begingroup$
So....... It thas implies $pi(X)$ only has one element?
$endgroup$
– user183002
Mar 21 at 20:26
1
$begingroup$
Yes. The product of two groups is trivial iff the two factors are trivial.
$endgroup$
– JHF
Mar 21 at 20:28
add a comment |
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1 Answer
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1 Answer
1
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oldest
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votes
$begingroup$
Hint: $pi_1(X times Y, (x_0, y_0)) cong pi_1(X, x_0) times pi_1(Y, y_0)$.
answered Mar 21 at 20:09
JHFJHF
4,9611026
$endgroup$
$begingroup$
I know that result... But I´m not sure that $pi(X times Y) = 0$ implies $pi(X) = 0$.
$endgroup$
– user183002
Mar 21 at 20:17
1
$begingroup$
Try counting the size of the groups on either side of the isomorphism. We know that $pi_1(X times Y)$ has one element, so...
$endgroup$
– JHF
Mar 21 at 20:24
$begingroup$
So....... It thas implies $pi(X)$ only has one element?
$endgroup$
– user183002
Mar 21 at 20:26
1
$begingroup$
Yes. The product of two groups is trivial iff the two factors are trivial.
$endgroup$
– JHF
Mar 21 at 20:28
add a comment |
$begingroup$
Hint: $pi_1(X times Y, (x_0, y_0)) cong pi_1(X, x_0) times pi_1(Y, y_0)$.
answered Mar 21 at 20:09
JHFJHF
4,9611026
$endgroup$
$begingroup$
I know that result... But I´m not sure that $pi(X times Y) = 0$ implies $pi(X) = 0$.
$endgroup$
– user183002
Mar 21 at 20:17
1
$begingroup$
Try counting the size of the groups on either side of the isomorphism. We know that $pi_1(X times Y)$ has one element, so...
$endgroup$
– JHF
Mar 21 at 20:24
$begingroup$
So....... It thas implies $pi(X)$ only has one element?
$endgroup$
– user183002
Mar 21 at 20:26
1
$begingroup$
Yes. The product of two groups is trivial iff the two factors are trivial.
$endgroup$
– JHF
Mar 21 at 20:28
add a comment |
$begingroup$
Hint: $pi_1(X times Y, (x_0, y_0)) cong pi_1(X, x_0) times pi_1(Y, y_0)$.
answered Mar 21 at 20:09
JHFJHF
4,9611026
$endgroup$
Hint: $pi_1(X times Y, (x_0, y_0)) cong pi_1(X, x_0) times pi_1(Y, y_0)$.
answered Mar 21 at 20:09
JHFJHF
4,9611026
answered Mar 21 at 20:09
JHFJHF
4,9611026
answered Mar 21 at 20:09
JHFJHF
4,9611026
answered Mar 21 at 20:09
JHFJHF
4,9611026
4,9611026
$begingroup$
I know that result... But I´m not sure that $pi(X times Y) = 0$ implies $pi(X) = 0$.
$endgroup$
– user183002
Mar 21 at 20:17
1
$begingroup$
Try counting the size of the groups on either side of the isomorphism. We know that $pi_1(X times Y)$ has one element, so...
$endgroup$
– JHF
Mar 21 at 20:24
$begingroup$
So....... It thas implies $pi(X)$ only has one element?
$endgroup$
– user183002
Mar 21 at 20:26
1
$begingroup$
Yes. The product of two groups is trivial iff the two factors are trivial.
$endgroup$
– JHF
Mar 21 at 20:28
add a comment |
$begingroup$
I know that result... But I´m not sure that $pi(X times Y) = 0$ implies $pi(X) = 0$.
$endgroup$
– user183002
Mar 21 at 20:17
1
$begingroup$
Try counting the size of the groups on either side of the isomorphism. We know that $pi_1(X times Y)$ has one element, so...
$endgroup$
– JHF
Mar 21 at 20:24
$begingroup$
So....... It thas implies $pi(X)$ only has one element?
$endgroup$
– user183002
Mar 21 at 20:26
1
$begingroup$
Yes. The product of two groups is trivial iff the two factors are trivial.
$endgroup$
– JHF
Mar 21 at 20:28
$begingroup$
I know that result... But I´m not sure that $pi(X times Y) = 0$ implies $pi(X) = 0$.
$endgroup$
– user183002
Mar 21 at 20:17
$begingroup$
I know that result... But I´m not sure that $pi(X times Y) = 0$ implies $pi(X) = 0$.
$endgroup$
– user183002
Mar 21 at 20:17
1
1
$begingroup$
Try counting the size of the groups on either side of the isomorphism. We know that $pi_1(X times Y)$ has one element, so...
$endgroup$
– JHF
Mar 21 at 20:24
$begingroup$
Try counting the size of the groups on either side of the isomorphism. We know that $pi_1(X times Y)$ has one element, so...
$endgroup$
– JHF
Mar 21 at 20:24
$begingroup$
So....... It thas implies $pi(X)$ only has one element?
$endgroup$
– user183002
Mar 21 at 20:26
$begingroup$
So....... It thas implies $pi(X)$ only has one element?
$endgroup$
– user183002
Mar 21 at 20:26
1
1
$begingroup$
Yes. The product of two groups is trivial iff the two factors are trivial.
$endgroup$
– JHF
Mar 21 at 20:28
$begingroup$
Yes. The product of two groups is trivial iff the two factors are trivial.
$endgroup$
– JHF
Mar 21 at 20:28
add a comment |
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