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Nontrivial U(1) bundles of a 3-manifold
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If G is a Lie group which is (1) a connected,
(2) simply connected
(3) compact,
then a G bundle on a 3-manifold is necessarily trivial.
However, U(1) bundle does not satisfy this (2) simply connected criterion.
Can we construct explicit nontrivial U(1) bundles of a 3-manifold? For the following examples:
$S^3$
$mathbbT^3$
$S^2 times S^1$
$D^2 times S^1$
$D^3$
($D^d$ is a $d$-disk.)
It looks that it is easier to do on $S^2 times S^1$ if we consider a nontrivial Chern number $c_1$ over the $S^2$ (?). How about other cases?
general-topology differential-geometry differential-topology geometric-topology fiber-bundles
asked Jul 22 at 0:54
wonderich
1,65321226
add a comment |Â
up vote
2
down vote
favorite
If G is a Lie group which is (1) a connected,
(2) simply connected
(3) compact,
then a G bundle on a 3-manifold is necessarily trivial.
However, U(1) bundle does not satisfy this (2) simply connected criterion.
Can we construct explicit nontrivial U(1) bundles of a 3-manifold? For the following examples:
$S^3$
$mathbbT^3$
$S^2 times S^1$
$D^2 times S^1$
$D^3$
($D^d$ is a $d$-disk.)
It looks that it is easier to do on $S^2 times S^1$ if we consider a nontrivial Chern number $c_1$ over the $S^2$ (?). How about other cases?
general-topology differential-geometry differential-topology geometric-topology fiber-bundles
asked Jul 22 at 0:54
wonderich
1,65321226
5
$U(1)$-bundles are classified by $H^1(-, U(1)) cong H^2(-, mathbbZ)$, so you just have to compute $H^2$ and this is straightforward in all of your examples using the Kunneth formula.
– Qiaochu Yuan
Jul 22 at 2:17
thanks for the nice comment +1
– wonderich
Jul 22 at 2:24
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
If G is a Lie group which is (1) a connected,
(2) simply connected
(3) compact,
then a G bundle on a 3-manifold is necessarily trivial.
However, U(1) bundle does not satisfy this (2) simply connected criterion.
Can we construct explicit nontrivial U(1) bundles of a 3-manifold? For the following examples:
$S^3$
$mathbbT^3$
$S^2 times S^1$
$D^2 times S^1$
$D^3$
($D^d$ is a $d$-disk.)
It looks that it is easier to do on $S^2 times S^1$ if we consider a nontrivial Chern number $c_1$ over the $S^2$ (?). How about other cases?
general-topology differential-geometry differential-topology geometric-topology fiber-bundles
asked Jul 22 at 0:54
wonderich
1,65321226
If G is a Lie group which is (1) a connected,
(2) simply connected
(3) compact,
then a G bundle on a 3-manifold is necessarily trivial.
However, U(1) bundle does not satisfy this (2) simply connected criterion.
Can we construct explicit nontrivial U(1) bundles of a 3-manifold? For the following examples:
$S^3$
$mathbbT^3$
$S^2 times S^1$
$D^2 times S^1$
$D^3$
($D^d$ is a $d$-disk.)
It looks that it is easier to do on $S^2 times S^1$ if we consider a nontrivial Chern number $c_1$ over the $S^2$ (?). How about other cases?
general-topology differential-geometry differential-topology geometric-topology fiber-bundles
asked Jul 22 at 0:54
wonderich
1,65321226
asked Jul 22 at 0:54
wonderich
1,65321226
asked Jul 22 at 0:54
wonderich
1,65321226
asked Jul 22 at 0:54
wonderich
1,65321226
1,65321226
5
$U(1)$-bundles are classified by $H^1(-, U(1)) cong H^2(-, mathbbZ)$, so you just have to compute $H^2$ and this is straightforward in all of your examples using the Kunneth formula.
– Qiaochu Yuan
Jul 22 at 2:17
thanks for the nice comment +1
– wonderich
Jul 22 at 2:24
add a comment |Â
5
$U(1)$-bundles are classified by $H^1(-, U(1)) cong H^2(-, mathbbZ)$, so you just have to compute $H^2$ and this is straightforward in all of your examples using the Kunneth formula.
– Qiaochu Yuan
Jul 22 at 2:17
thanks for the nice comment +1
– wonderich
Jul 22 at 2:24
5
5
$U(1)$-bundles are classified by $H^1(-, U(1)) cong H^2(-, mathbbZ)$, so you just have to compute $H^2$ and this is straightforward in all of your examples using the Kunneth formula.
– Qiaochu Yuan
Jul 22 at 2:17
$U(1)$-bundles are classified by $H^1(-, U(1)) cong H^2(-, mathbbZ)$, so you just have to compute $H^2$ and this is straightforward in all of your examples using the Kunneth formula.
– Qiaochu Yuan
Jul 22 at 2:17
thanks for the nice comment +1
– wonderich
Jul 22 at 2:24
thanks for the nice comment +1
– wonderich
Jul 22 at 2:24
add a comment |Â
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5
$U(1)$-bundles are classified by $H^1(-, U(1)) cong H^2(-, mathbbZ)$, so you just have to compute $H^2$ and this is straightforward in all of your examples using the Kunneth formula.
– Qiaochu Yuan
Jul 22 at 2:17
thanks for the nice comment +1
– wonderich
Jul 22 at 2:24