Fiber of spherical tangent bundle
$begingroup$
I've been reading a book of tessellations and three-manifolds and it's been said that "the fiber of ST(X) is the 1-sphere and thus ST(X) is a closed 3-manifold", where ST(X) is the spherical tangent bundle of a compact, connected 2-manifold X, that is, the subbundle of TX consisting of vectors of norm 1. Honestly, I don't understand why the assertion between quotations marks is true (why it is a 3-manifold). Could anyone help me?
Many thanks.
geometry algebraic-topology
edited Dec 25 '18 at 23:16
KReiser
10k21435
asked Dec 25 '18 at 22:56
Rubén Fernández FuertesRubén Fernández Fuertes
1087
$endgroup$
add a comment |
$begingroup$
I've been reading a book of tessellations and three-manifolds and it's been said that "the fiber of ST(X) is the 1-sphere and thus ST(X) is a closed 3-manifold", where ST(X) is the spherical tangent bundle of a compact, connected 2-manifold X, that is, the subbundle of TX consisting of vectors of norm 1. Honestly, I don't understand why the assertion between quotations marks is true (why it is a 3-manifold). Could anyone help me?
Many thanks.
geometry algebraic-topology
edited Dec 25 '18 at 23:16
KReiser
10k21435
asked Dec 25 '18 at 22:56
Rubén Fernández FuertesRubén Fernández Fuertes
1087
$endgroup$
1
$begingroup$
Just out of interest, what book is this?
$endgroup$
– Lukas Kofler
Dec 25 '18 at 23:07
1
$begingroup$
Hint: a fiber bundle is locally trivial, and the product of an $n$-dimensional manifold with an $m$-dimensional manifold is an $n+m$-dimensional manifold.
$endgroup$
– KReiser
Dec 25 '18 at 23:16
$begingroup$
The book is Classical Tessellations and Three-Manifolds by José M. Montesinos.
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:24
$begingroup$
Thank you, KReiser!
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:25
add a comment |
$begingroup$
I've been reading a book of tessellations and three-manifolds and it's been said that "the fiber of ST(X) is the 1-sphere and thus ST(X) is a closed 3-manifold", where ST(X) is the spherical tangent bundle of a compact, connected 2-manifold X, that is, the subbundle of TX consisting of vectors of norm 1. Honestly, I don't understand why the assertion between quotations marks is true (why it is a 3-manifold). Could anyone help me?
Many thanks.
geometry algebraic-topology
edited Dec 25 '18 at 23:16
KReiser
10k21435
asked Dec 25 '18 at 22:56
Rubén Fernández FuertesRubén Fernández Fuertes
1087
$endgroup$
I've been reading a book of tessellations and three-manifolds and it's been said that "the fiber of ST(X) is the 1-sphere and thus ST(X) is a closed 3-manifold", where ST(X) is the spherical tangent bundle of a compact, connected 2-manifold X, that is, the subbundle of TX consisting of vectors of norm 1. Honestly, I don't understand why the assertion between quotations marks is true (why it is a 3-manifold). Could anyone help me?
Many thanks.
geometry algebraic-topology
geometry algebraic-topology
edited Dec 25 '18 at 23:16
KReiser
10k21435
asked Dec 25 '18 at 22:56
Rubén Fernández FuertesRubén Fernández Fuertes
1087
edited Dec 25 '18 at 23:16
KReiser
10k21435
asked Dec 25 '18 at 22:56
Rubén Fernández FuertesRubén Fernández Fuertes
1087
edited Dec 25 '18 at 23:16
KReiser
10k21435
edited Dec 25 '18 at 23:16
KReiser
10k21435
edited Dec 25 '18 at 23:16
KReiser
10k21435
10k21435
asked Dec 25 '18 at 22:56
Rubén Fernández FuertesRubén Fernández Fuertes
1087
asked Dec 25 '18 at 22:56
Rubén Fernández FuertesRubén Fernández Fuertes
1087
asked Dec 25 '18 at 22:56
Rubén Fernández FuertesRubén Fernández Fuertes
1087
1087
1
$begingroup$
Just out of interest, what book is this?
$endgroup$
– Lukas Kofler
Dec 25 '18 at 23:07
1
$begingroup$
Hint: a fiber bundle is locally trivial, and the product of an $n$-dimensional manifold with an $m$-dimensional manifold is an $n+m$-dimensional manifold.
$endgroup$
– KReiser
Dec 25 '18 at 23:16
$begingroup$
The book is Classical Tessellations and Three-Manifolds by José M. Montesinos.
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:24
$begingroup$
Thank you, KReiser!
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:25
add a comment |
1
$begingroup$
Just out of interest, what book is this?
$endgroup$
– Lukas Kofler
Dec 25 '18 at 23:07
1
$begingroup$
Hint: a fiber bundle is locally trivial, and the product of an $n$-dimensional manifold with an $m$-dimensional manifold is an $n+m$-dimensional manifold.
$endgroup$
– KReiser
Dec 25 '18 at 23:16
$begingroup$
The book is Classical Tessellations and Three-Manifolds by José M. Montesinos.
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:24
$begingroup$
Thank you, KReiser!
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:25
1
1
$begingroup$
Just out of interest, what book is this?
$endgroup$
– Lukas Kofler
Dec 25 '18 at 23:07
$begingroup$
Just out of interest, what book is this?
$endgroup$
– Lukas Kofler
Dec 25 '18 at 23:07
1
1
$begingroup$
Hint: a fiber bundle is locally trivial, and the product of an $n$-dimensional manifold with an $m$-dimensional manifold is an $n+m$-dimensional manifold.
$endgroup$
– KReiser
Dec 25 '18 at 23:16
$begingroup$
Hint: a fiber bundle is locally trivial, and the product of an $n$-dimensional manifold with an $m$-dimensional manifold is an $n+m$-dimensional manifold.
$endgroup$
– KReiser
Dec 25 '18 at 23:16
$begingroup$
The book is Classical Tessellations and Three-Manifolds by José M. Montesinos.
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:24
$begingroup$
The book is Classical Tessellations and Three-Manifolds by José M. Montesinos.
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:24
$begingroup$
Thank you, KReiser!
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:25
$begingroup$
Thank you, KReiser!
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:25
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Fiber bundles $F to E to X$ are locally trivializable, meaning there is an open neighborhood $U$ of any $xin X$ Such that the inverse image of $U$ under the projection is diffeomorphic to $U times F$, where $F$ is the fiber. In this case, we may choose $U$ to be diffeomorphic to an open neighborhood of $mathbb{R}^2$. Since the tangent space $V$ to a point $x in X$ is isomorphic to $mathbb{R}^2$, we have $F cong S^1$. This tells us that $ST(X)$ is a $3$-manifold.
As for why it's closed, since $X$ is compact and $ST(X)to X$ has compact fibers, $ST(X)$ will be compact as well.
answered Dec 25 '18 at 23:17
leibnewtzleibnewtz
2,6411717
$endgroup$
$begingroup$
In the first line, what is E?
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:34
$begingroup$
$E$ is the total space of the bundle and $F$ is the fiber. The notation $F to E to X$ is standard notation for a fiber bundle
$endgroup$
– leibnewtz
Dec 26 '18 at 7:13
$begingroup$
Perfect. Many thanks.
$endgroup$
– Rubén Fernández Fuertes
Dec 26 '18 at 17:11
add a comment |
Your Answer
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Fiber bundles $F to E to X$ are locally trivializable, meaning there is an open neighborhood $U$ of any $xin X$ Such that the inverse image of $U$ under the projection is diffeomorphic to $U times F$, where $F$ is the fiber. In this case, we may choose $U$ to be diffeomorphic to an open neighborhood of $mathbb{R}^2$. Since the tangent space $V$ to a point $x in X$ is isomorphic to $mathbb{R}^2$, we have $F cong S^1$. This tells us that $ST(X)$ is a $3$-manifold.
As for why it's closed, since $X$ is compact and $ST(X)to X$ has compact fibers, $ST(X)$ will be compact as well.
answered Dec 25 '18 at 23:17
leibnewtzleibnewtz
2,6411717
$endgroup$
$begingroup$
In the first line, what is E?
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:34
$begingroup$
$E$ is the total space of the bundle and $F$ is the fiber. The notation $F to E to X$ is standard notation for a fiber bundle
$endgroup$
– leibnewtz
Dec 26 '18 at 7:13
$begingroup$
Perfect. Many thanks.
$endgroup$
– Rubén Fernández Fuertes
Dec 26 '18 at 17:11
add a comment |
$begingroup$
Fiber bundles $F to E to X$ are locally trivializable, meaning there is an open neighborhood $U$ of any $xin X$ Such that the inverse image of $U$ under the projection is diffeomorphic to $U times F$, where $F$ is the fiber. In this case, we may choose $U$ to be diffeomorphic to an open neighborhood of $mathbb{R}^2$. Since the tangent space $V$ to a point $x in X$ is isomorphic to $mathbb{R}^2$, we have $F cong S^1$. This tells us that $ST(X)$ is a $3$-manifold.
As for why it's closed, since $X$ is compact and $ST(X)to X$ has compact fibers, $ST(X)$ will be compact as well.
answered Dec 25 '18 at 23:17
leibnewtzleibnewtz
2,6411717
$endgroup$
$begingroup$
In the first line, what is E?
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:34
$begingroup$
$E$ is the total space of the bundle and $F$ is the fiber. The notation $F to E to X$ is standard notation for a fiber bundle
$endgroup$
– leibnewtz
Dec 26 '18 at 7:13
$begingroup$
Perfect. Many thanks.
$endgroup$
– Rubén Fernández Fuertes
Dec 26 '18 at 17:11
add a comment |
$begingroup$
Fiber bundles $F to E to X$ are locally trivializable, meaning there is an open neighborhood $U$ of any $xin X$ Such that the inverse image of $U$ under the projection is diffeomorphic to $U times F$, where $F$ is the fiber. In this case, we may choose $U$ to be diffeomorphic to an open neighborhood of $mathbb{R}^2$. Since the tangent space $V$ to a point $x in X$ is isomorphic to $mathbb{R}^2$, we have $F cong S^1$. This tells us that $ST(X)$ is a $3$-manifold.
As for why it's closed, since $X$ is compact and $ST(X)to X$ has compact fibers, $ST(X)$ will be compact as well.
answered Dec 25 '18 at 23:17
leibnewtzleibnewtz
2,6411717
$endgroup$
Fiber bundles $F to E to X$ are locally trivializable, meaning there is an open neighborhood $U$ of any $xin X$ Such that the inverse image of $U$ under the projection is diffeomorphic to $U times F$, where $F$ is the fiber. In this case, we may choose $U$ to be diffeomorphic to an open neighborhood of $mathbb{R}^2$. Since the tangent space $V$ to a point $x in X$ is isomorphic to $mathbb{R}^2$, we have $F cong S^1$. This tells us that $ST(X)$ is a $3$-manifold.
As for why it's closed, since $X$ is compact and $ST(X)to X$ has compact fibers, $ST(X)$ will be compact as well.
answered Dec 25 '18 at 23:17
leibnewtzleibnewtz
2,6411717
answered Dec 25 '18 at 23:17
leibnewtzleibnewtz
2,6411717
answered Dec 25 '18 at 23:17
leibnewtzleibnewtz
2,6411717
answered Dec 25 '18 at 23:17
leibnewtzleibnewtz
2,6411717
2,6411717
$begingroup$
In the first line, what is E?
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:34
$begingroup$
$E$ is the total space of the bundle and $F$ is the fiber. The notation $F to E to X$ is standard notation for a fiber bundle
$endgroup$
– leibnewtz
Dec 26 '18 at 7:13
$begingroup$
Perfect. Many thanks.
$endgroup$
– Rubén Fernández Fuertes
Dec 26 '18 at 17:11
add a comment |
$begingroup$
In the first line, what is E?
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:34
$begingroup$
$E$ is the total space of the bundle and $F$ is the fiber. The notation $F to E to X$ is standard notation for a fiber bundle
$endgroup$
– leibnewtz
Dec 26 '18 at 7:13
$begingroup$
Perfect. Many thanks.
$endgroup$
– Rubén Fernández Fuertes
Dec 26 '18 at 17:11
$begingroup$
In the first line, what is E?
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:34
$begingroup$
In the first line, what is E?
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:34
$begingroup$
$E$ is the total space of the bundle and $F$ is the fiber. The notation $F to E to X$ is standard notation for a fiber bundle
$endgroup$
– leibnewtz
Dec 26 '18 at 7:13
$begingroup$
$E$ is the total space of the bundle and $F$ is the fiber. The notation $F to E to X$ is standard notation for a fiber bundle
$endgroup$
– leibnewtz
Dec 26 '18 at 7:13
$begingroup$
Perfect. Many thanks.
$endgroup$
– Rubén Fernández Fuertes
Dec 26 '18 at 17:11
$begingroup$
Perfect. Many thanks.
$endgroup$
– Rubén Fernández Fuertes
Dec 26 '18 at 17:11
add a comment |
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1
$begingroup$
Just out of interest, what book is this?
$endgroup$
– Lukas Kofler
Dec 25 '18 at 23:07
1
$begingroup$
Hint: a fiber bundle is locally trivial, and the product of an $n$-dimensional manifold with an $m$-dimensional manifold is an $n+m$-dimensional manifold.
$endgroup$
– KReiser
Dec 25 '18 at 23:16
$begingroup$
The book is Classical Tessellations and Three-Manifolds by José M. Montesinos.
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:24
$begingroup$
Thank you, KReiser!
$endgroup$
– Rubén Fernández Fuertes
Dec 25 '18 at 23:25