Understanding sections of fiber bundles
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I need some explanation of the meaning of sections of the following situation of fiber bundles;
Let $G$ be a topological group such that $G=Ntimes Z$ where $N$ is normal and $Z$ is central subgroups. Let $H<G$ be a closed subgroup and let $J:=N_G(H^0)$ be the normalizer of the identity component of $H$ then $Z<J$. Now consider the following fibration:
$$G/Hto G/J$$
$$gHmapsto gJ$$
The fiber here is $J/H$.
I was reading somewhere the following "since $N$-orbits intersect the fiber $J/H$ in only one point i.e, $eH$, then $N$-orbits are sections of the fibration" Here $N$ is a normal subgroup of $G$.
I don't really understand this. It will be very helpful if you explain it to me. Thanks
group-theory differential-geometry algebraic-topology lie-groups
asked Nov 22 at 2:03
Amrat A
31818
|
show 4 more comments
up vote
0
down vote
favorite
I need some explanation of the meaning of sections of the following situation of fiber bundles;
Let $G$ be a topological group such that $G=Ntimes Z$ where $N$ is normal and $Z$ is central subgroups. Let $H<G$ be a closed subgroup and let $J:=N_G(H^0)$ be the normalizer of the identity component of $H$ then $Z<J$. Now consider the following fibration:
$$G/Hto G/J$$
$$gHmapsto gJ$$
The fiber here is $J/H$.
I was reading somewhere the following "since $N$-orbits intersect the fiber $J/H$ in only one point i.e, $eH$, then $N$-orbits are sections of the fibration" Here $N$ is a normal subgroup of $G$.
I don't really understand this. It will be very helpful if you explain it to me. Thanks
group-theory differential-geometry algebraic-topology lie-groups
asked Nov 22 at 2:03
Amrat A
31818
1
what is $N$ here ?
– Tsemo Aristide
Nov 22 at 2:11
A normal subgroup of $G$. I will add this to the question.
– Amrat A
Nov 22 at 2:12
$N$ is really any normal subgroup ?
– Tsemo Aristide
Nov 22 at 2:15
In fact, $G=Ntimes Z$ where $Z$ is central. Sorry maybe this is important.
– Amrat A
Nov 22 at 2:16
1
what is the relation between $J$ and $Z$ is $Jsubset Z$?
– Tsemo Aristide
Nov 22 at 2:18
|
show 4 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I need some explanation of the meaning of sections of the following situation of fiber bundles;
Let $G$ be a topological group such that $G=Ntimes Z$ where $N$ is normal and $Z$ is central subgroups. Let $H<G$ be a closed subgroup and let $J:=N_G(H^0)$ be the normalizer of the identity component of $H$ then $Z<J$. Now consider the following fibration:
$$G/Hto G/J$$
$$gHmapsto gJ$$
The fiber here is $J/H$.
I was reading somewhere the following "since $N$-orbits intersect the fiber $J/H$ in only one point i.e, $eH$, then $N$-orbits are sections of the fibration" Here $N$ is a normal subgroup of $G$.
I don't really understand this. It will be very helpful if you explain it to me. Thanks
group-theory differential-geometry algebraic-topology lie-groups
asked Nov 22 at 2:03
Amrat A
31818
I need some explanation of the meaning of sections of the following situation of fiber bundles;
Let $G$ be a topological group such that $G=Ntimes Z$ where $N$ is normal and $Z$ is central subgroups. Let $H<G$ be a closed subgroup and let $J:=N_G(H^0)$ be the normalizer of the identity component of $H$ then $Z<J$. Now consider the following fibration:
$$G/Hto G/J$$
$$gHmapsto gJ$$
The fiber here is $J/H$.
I was reading somewhere the following "since $N$-orbits intersect the fiber $J/H$ in only one point i.e, $eH$, then $N$-orbits are sections of the fibration" Here $N$ is a normal subgroup of $G$.
I don't really understand this. It will be very helpful if you explain it to me. Thanks
group-theory differential-geometry algebraic-topology lie-groups
group-theory differential-geometry algebraic-topology lie-groups
asked Nov 22 at 2:03
Amrat A
31818
asked Nov 22 at 2:03
Amrat A
31818
edited Nov 22 at 2:40
asked Nov 22 at 2:03
Amrat A
31818
asked Nov 22 at 2:03
Amrat A
31818
asked Nov 22 at 2:03
Amrat A
31818
31818
1
what is $N$ here ?
– Tsemo Aristide
Nov 22 at 2:11
A normal subgroup of $G$. I will add this to the question.
– Amrat A
Nov 22 at 2:12
$N$ is really any normal subgroup ?
– Tsemo Aristide
Nov 22 at 2:15
In fact, $G=Ntimes Z$ where $Z$ is central. Sorry maybe this is important.
– Amrat A
Nov 22 at 2:16
1
what is the relation between $J$ and $Z$ is $Jsubset Z$?
– Tsemo Aristide
Nov 22 at 2:18
|
show 4 more comments
1
what is $N$ here ?
– Tsemo Aristide
Nov 22 at 2:11
A normal subgroup of $G$. I will add this to the question.
– Amrat A
Nov 22 at 2:12
$N$ is really any normal subgroup ?
– Tsemo Aristide
Nov 22 at 2:15
In fact, $G=Ntimes Z$ where $Z$ is central. Sorry maybe this is important.
– Amrat A
Nov 22 at 2:16
1
what is the relation between $J$ and $Z$ is $Jsubset Z$?
– Tsemo Aristide
Nov 22 at 2:18
1
1
what is $N$ here ?
– Tsemo Aristide
Nov 22 at 2:11
what is $N$ here ?
– Tsemo Aristide
Nov 22 at 2:11
A normal subgroup of $G$. I will add this to the question.
– Amrat A
Nov 22 at 2:12
A normal subgroup of $G$. I will add this to the question.
– Amrat A
Nov 22 at 2:12
$N$ is really any normal subgroup ?
– Tsemo Aristide
Nov 22 at 2:15
$N$ is really any normal subgroup ?
– Tsemo Aristide
Nov 22 at 2:15
In fact, $G=Ntimes Z$ where $Z$ is central. Sorry maybe this is important.
– Amrat A
Nov 22 at 2:16
In fact, $G=Ntimes Z$ where $Z$ is central. Sorry maybe this is important.
– Amrat A
Nov 22 at 2:16
1
1
what is the relation between $J$ and $Z$ is $Jsubset Z$?
– Tsemo Aristide
Nov 22 at 2:18
what is the relation between $J$ and $Z$ is $Jsubset Z$?
– Tsemo Aristide
Nov 22 at 2:18
|
show 4 more comments
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1
what is $N$ here ?
– Tsemo Aristide
Nov 22 at 2:11
A normal subgroup of $G$. I will add this to the question.
– Amrat A
Nov 22 at 2:12
$N$ is really any normal subgroup ?
– Tsemo Aristide
Nov 22 at 2:15
In fact, $G=Ntimes Z$ where $Z$ is central. Sorry maybe this is important.
– Amrat A
Nov 22 at 2:16
1
what is the relation between $J$ and $Z$ is $Jsubset Z$?
– Tsemo Aristide
Nov 22 at 2:18