Orientation and line bundle
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I'm stuck on the following problem:
Given $X$ a manifold and $L$ a line bundle over $X$, considerer $Z$
the null-section and define $Or(L) := (L setminus Z) ,
/,mathbb{R}_+^*$. Show that the trivialisation $L_{|U} simeq U times mathbb{R}$ induces a homeomorphism $pi^{-1}(U) simeq U
times Or(mathbb{R})$ (where $Or(mathbb{R})$ is nothing more that
${pm 1 }$), where $pi : L to X$ is the canonical
projection. Conclude that $Or(L)$ is given a unique smooth manifold
structure such that $eta : Or(L) to X$ is a local diffeomorphism.
Here, $eta$ is the continuous surjection induced by $pi$. Indeed, $pi$ is trivially constant over the fibers of the quotient map, so that it goes down continuously to the said quotient $Or(L)$.
Thanks for the help!
differential-geometry
asked 17 hours ago
Hermès
1,799612
add a comment |
up vote
0
down vote
favorite
I'm stuck on the following problem:
Given $X$ a manifold and $L$ a line bundle over $X$, considerer $Z$
the null-section and define $Or(L) := (L setminus Z) ,
/,mathbb{R}_+^*$. Show that the trivialisation $L_{|U} simeq U times mathbb{R}$ induces a homeomorphism $pi^{-1}(U) simeq U
times Or(mathbb{R})$ (where $Or(mathbb{R})$ is nothing more that
${pm 1 }$), where $pi : L to X$ is the canonical
projection. Conclude that $Or(L)$ is given a unique smooth manifold
structure such that $eta : Or(L) to X$ is a local diffeomorphism.
Here, $eta$ is the continuous surjection induced by $pi$. Indeed, $pi$ is trivially constant over the fibers of the quotient map, so that it goes down continuously to the said quotient $Or(L)$.
Thanks for the help!
differential-geometry
asked 17 hours ago
Hermès
1,799612
Something's wrong in the statement here. You mean that $eta^{-1}(U)cong Utimes text{Or}(Bbb R)$?
– Ted Shifrin
6 hours ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm stuck on the following problem:
Given $X$ a manifold and $L$ a line bundle over $X$, considerer $Z$
the null-section and define $Or(L) := (L setminus Z) ,
/,mathbb{R}_+^*$. Show that the trivialisation $L_{|U} simeq U times mathbb{R}$ induces a homeomorphism $pi^{-1}(U) simeq U
times Or(mathbb{R})$ (where $Or(mathbb{R})$ is nothing more that
${pm 1 }$), where $pi : L to X$ is the canonical
projection. Conclude that $Or(L)$ is given a unique smooth manifold
structure such that $eta : Or(L) to X$ is a local diffeomorphism.
Here, $eta$ is the continuous surjection induced by $pi$. Indeed, $pi$ is trivially constant over the fibers of the quotient map, so that it goes down continuously to the said quotient $Or(L)$.
Thanks for the help!
differential-geometry
asked 17 hours ago
Hermès
1,799612
I'm stuck on the following problem:
Given $X$ a manifold and $L$ a line bundle over $X$, considerer $Z$
the null-section and define $Or(L) := (L setminus Z) ,
/,mathbb{R}_+^*$. Show that the trivialisation $L_{|U} simeq U times mathbb{R}$ induces a homeomorphism $pi^{-1}(U) simeq U
times Or(mathbb{R})$ (where $Or(mathbb{R})$ is nothing more that
${pm 1 }$), where $pi : L to X$ is the canonical
projection. Conclude that $Or(L)$ is given a unique smooth manifold
structure such that $eta : Or(L) to X$ is a local diffeomorphism.
Here, $eta$ is the continuous surjection induced by $pi$. Indeed, $pi$ is trivially constant over the fibers of the quotient map, so that it goes down continuously to the said quotient $Or(L)$.
Thanks for the help!
differential-geometry
differential-geometry
asked 17 hours ago
Hermès
1,799612
asked 17 hours ago
Hermès
1,799612
edited 15 hours ago
asked 17 hours ago
Hermès
1,799612
asked 17 hours ago
Hermès
1,799612
asked 17 hours ago
Hermès
1,799612
1,799612
Something's wrong in the statement here. You mean that $eta^{-1}(U)cong Utimes text{Or}(Bbb R)$?
– Ted Shifrin
6 hours ago
add a comment |
Something's wrong in the statement here. You mean that $eta^{-1}(U)cong Utimes text{Or}(Bbb R)$?
– Ted Shifrin
6 hours ago
Something's wrong in the statement here. You mean that $eta^{-1}(U)cong Utimes text{Or}(Bbb R)$?
– Ted Shifrin
6 hours ago
Something's wrong in the statement here. You mean that $eta^{-1}(U)cong Utimes text{Or}(Bbb R)$?
– Ted Shifrin
6 hours ago
add a comment |
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Something's wrong in the statement here. You mean that $eta^{-1}(U)cong Utimes text{Or}(Bbb R)$?
– Ted Shifrin
6 hours ago