Principal bundles with quotient map
up vote
1
down vote
favorite
I'm trying to prove that if $G$ is a Lie group and $H < G$ a closed subgroup, and we have the quotient map defined as $pi: G rightarrow G/H$, then $(G, pi, G/H, H)$ is a $H$-principal bundle over $G/H$ with total space $G$.
I'm new in this business of fiber bundles and after several hours searching on Internet I didn't find anything that I could use to prove this statement. Any idea or reference?
group-theory differential-geometry lie-groups fiber-bundles principal-bundles
add a comment |
up vote
1
down vote
favorite
I'm trying to prove that if $G$ is a Lie group and $H < G$ a closed subgroup, and we have the quotient map defined as $pi: G rightarrow G/H$, then $(G, pi, G/H, H)$ is a $H$-principal bundle over $G/H$ with total space $G$.
I'm new in this business of fiber bundles and after several hours searching on Internet I didn't find anything that I could use to prove this statement. Any idea or reference?
group-theory differential-geometry lie-groups fiber-bundles principal-bundles
2
A reference that discusses the construction is "The Topology of Fibre Bundles" by N. Steenrod, on page 33 (Section 7.5). He further refers to "Theory of Lie Groups" by C. Chevellay, Proposition 1, page 110. I can try and summarise their content if you would like?
– BenCWBrown
Nov 16 at 22:51
1
Yes, please. Your summary together with that lectures would give me a light in this issue because fiber bundles are so abstract that I'm pretty confused
– Vicky
Nov 16 at 22:54
not exactly what you are looking for however I think that theses notes on bundles and connections are really well written and pedagogical.
– Picaud Vincent
Nov 16 at 23:02
@PicaudVincent I'll take a look at it. Thanks!
– Vicky
Nov 16 at 23:03
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm trying to prove that if $G$ is a Lie group and $H < G$ a closed subgroup, and we have the quotient map defined as $pi: G rightarrow G/H$, then $(G, pi, G/H, H)$ is a $H$-principal bundle over $G/H$ with total space $G$.
I'm new in this business of fiber bundles and after several hours searching on Internet I didn't find anything that I could use to prove this statement. Any idea or reference?
group-theory differential-geometry lie-groups fiber-bundles principal-bundles
I'm trying to prove that if $G$ is a Lie group and $H < G$ a closed subgroup, and we have the quotient map defined as $pi: G rightarrow G/H$, then $(G, pi, G/H, H)$ is a $H$-principal bundle over $G/H$ with total space $G$.
I'm new in this business of fiber bundles and after several hours searching on Internet I didn't find anything that I could use to prove this statement. Any idea or reference?
group-theory differential-geometry lie-groups fiber-bundles principal-bundles
group-theory differential-geometry lie-groups fiber-bundles principal-bundles
edited Nov 16 at 22:45
asked Nov 16 at 22:38
Vicky
1387
1387
2
A reference that discusses the construction is "The Topology of Fibre Bundles" by N. Steenrod, on page 33 (Section 7.5). He further refers to "Theory of Lie Groups" by C. Chevellay, Proposition 1, page 110. I can try and summarise their content if you would like?
– BenCWBrown
Nov 16 at 22:51
1
Yes, please. Your summary together with that lectures would give me a light in this issue because fiber bundles are so abstract that I'm pretty confused
– Vicky
Nov 16 at 22:54
not exactly what you are looking for however I think that theses notes on bundles and connections are really well written and pedagogical.
– Picaud Vincent
Nov 16 at 23:02
@PicaudVincent I'll take a look at it. Thanks!
– Vicky
Nov 16 at 23:03
add a comment |
2
A reference that discusses the construction is "The Topology of Fibre Bundles" by N. Steenrod, on page 33 (Section 7.5). He further refers to "Theory of Lie Groups" by C. Chevellay, Proposition 1, page 110. I can try and summarise their content if you would like?
– BenCWBrown
Nov 16 at 22:51
1
Yes, please. Your summary together with that lectures would give me a light in this issue because fiber bundles are so abstract that I'm pretty confused
– Vicky
Nov 16 at 22:54
not exactly what you are looking for however I think that theses notes on bundles and connections are really well written and pedagogical.
– Picaud Vincent
Nov 16 at 23:02
@PicaudVincent I'll take a look at it. Thanks!
– Vicky
Nov 16 at 23:03
2
2
A reference that discusses the construction is "The Topology of Fibre Bundles" by N. Steenrod, on page 33 (Section 7.5). He further refers to "Theory of Lie Groups" by C. Chevellay, Proposition 1, page 110. I can try and summarise their content if you would like?
– BenCWBrown
Nov 16 at 22:51
A reference that discusses the construction is "The Topology of Fibre Bundles" by N. Steenrod, on page 33 (Section 7.5). He further refers to "Theory of Lie Groups" by C. Chevellay, Proposition 1, page 110. I can try and summarise their content if you would like?
– BenCWBrown
Nov 16 at 22:51
1
1
Yes, please. Your summary together with that lectures would give me a light in this issue because fiber bundles are so abstract that I'm pretty confused
– Vicky
Nov 16 at 22:54
Yes, please. Your summary together with that lectures would give me a light in this issue because fiber bundles are so abstract that I'm pretty confused
– Vicky
Nov 16 at 22:54
not exactly what you are looking for however I think that theses notes on bundles and connections are really well written and pedagogical.
– Picaud Vincent
Nov 16 at 23:02
not exactly what you are looking for however I think that theses notes on bundles and connections are really well written and pedagogical.
– Picaud Vincent
Nov 16 at 23:02
@PicaudVincent I'll take a look at it. Thanks!
– Vicky
Nov 16 at 23:03
@PicaudVincent I'll take a look at it. Thanks!
– Vicky
Nov 16 at 23:03
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
This is summarising pages 30 - 33 in The Topology of Fibre Bundles by Steenrod. I will omit proofs as they are quite long. There is a Corollary on p. 31 (to a Theorem on p. 30), which is relevant to your question:
Let $H$ be a closed subgroup of $B$ (not assuming $B$ is a Lie group yet, just a topological group), and consider the projection $p:B rightarrow B/G$. If there exists a section $s:B/G rightarrow B$, then $B$ is a fibre bundle over $B/G$ which assigns to each $b in B$ the coset $bG$. The fibre of the bundle is $G$ and the group is $G$ acting on the fibre by left translations. (Proof is on p. 31 - 32).
Now introducing Lie groups, a requirement following this construction that a section $s:G/H rightarrow G$ exists is treated in p. 110, Prop. 1, of Theory of Lie Groups by C. Chevalley. We also require that the maps are now smooth and that the projection map $p:G rightarrow G/H$ is of maximal rank everywhere in $G$ (again discussed in Steenrod). Then the above theorem applies to Lie groups too.
First, I want to say thanks for your effort; and second, I'm having problems understanding the references you give. Could you give me something in more detail, not so abstract? I know that maybe it is too much, but I'm stuck. Apologies
– Vicky
Nov 17 at 0:47
Okay I'll give it a shot!
– BenCWBrown
Nov 17 at 0:48
I appreciate it. Really thanks!
– Vicky
Nov 17 at 0:49
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
This is summarising pages 30 - 33 in The Topology of Fibre Bundles by Steenrod. I will omit proofs as they are quite long. There is a Corollary on p. 31 (to a Theorem on p. 30), which is relevant to your question:
Let $H$ be a closed subgroup of $B$ (not assuming $B$ is a Lie group yet, just a topological group), and consider the projection $p:B rightarrow B/G$. If there exists a section $s:B/G rightarrow B$, then $B$ is a fibre bundle over $B/G$ which assigns to each $b in B$ the coset $bG$. The fibre of the bundle is $G$ and the group is $G$ acting on the fibre by left translations. (Proof is on p. 31 - 32).
Now introducing Lie groups, a requirement following this construction that a section $s:G/H rightarrow G$ exists is treated in p. 110, Prop. 1, of Theory of Lie Groups by C. Chevalley. We also require that the maps are now smooth and that the projection map $p:G rightarrow G/H$ is of maximal rank everywhere in $G$ (again discussed in Steenrod). Then the above theorem applies to Lie groups too.
First, I want to say thanks for your effort; and second, I'm having problems understanding the references you give. Could you give me something in more detail, not so abstract? I know that maybe it is too much, but I'm stuck. Apologies
– Vicky
Nov 17 at 0:47
Okay I'll give it a shot!
– BenCWBrown
Nov 17 at 0:48
I appreciate it. Really thanks!
– Vicky
Nov 17 at 0:49
add a comment |
up vote
1
down vote
This is summarising pages 30 - 33 in The Topology of Fibre Bundles by Steenrod. I will omit proofs as they are quite long. There is a Corollary on p. 31 (to a Theorem on p. 30), which is relevant to your question:
Let $H$ be a closed subgroup of $B$ (not assuming $B$ is a Lie group yet, just a topological group), and consider the projection $p:B rightarrow B/G$. If there exists a section $s:B/G rightarrow B$, then $B$ is a fibre bundle over $B/G$ which assigns to each $b in B$ the coset $bG$. The fibre of the bundle is $G$ and the group is $G$ acting on the fibre by left translations. (Proof is on p. 31 - 32).
Now introducing Lie groups, a requirement following this construction that a section $s:G/H rightarrow G$ exists is treated in p. 110, Prop. 1, of Theory of Lie Groups by C. Chevalley. We also require that the maps are now smooth and that the projection map $p:G rightarrow G/H$ is of maximal rank everywhere in $G$ (again discussed in Steenrod). Then the above theorem applies to Lie groups too.
First, I want to say thanks for your effort; and second, I'm having problems understanding the references you give. Could you give me something in more detail, not so abstract? I know that maybe it is too much, but I'm stuck. Apologies
– Vicky
Nov 17 at 0:47
Okay I'll give it a shot!
– BenCWBrown
Nov 17 at 0:48
I appreciate it. Really thanks!
– Vicky
Nov 17 at 0:49
add a comment |
up vote
1
down vote
up vote
1
down vote
This is summarising pages 30 - 33 in The Topology of Fibre Bundles by Steenrod. I will omit proofs as they are quite long. There is a Corollary on p. 31 (to a Theorem on p. 30), which is relevant to your question:
Let $H$ be a closed subgroup of $B$ (not assuming $B$ is a Lie group yet, just a topological group), and consider the projection $p:B rightarrow B/G$. If there exists a section $s:B/G rightarrow B$, then $B$ is a fibre bundle over $B/G$ which assigns to each $b in B$ the coset $bG$. The fibre of the bundle is $G$ and the group is $G$ acting on the fibre by left translations. (Proof is on p. 31 - 32).
Now introducing Lie groups, a requirement following this construction that a section $s:G/H rightarrow G$ exists is treated in p. 110, Prop. 1, of Theory of Lie Groups by C. Chevalley. We also require that the maps are now smooth and that the projection map $p:G rightarrow G/H$ is of maximal rank everywhere in $G$ (again discussed in Steenrod). Then the above theorem applies to Lie groups too.
This is summarising pages 30 - 33 in The Topology of Fibre Bundles by Steenrod. I will omit proofs as they are quite long. There is a Corollary on p. 31 (to a Theorem on p. 30), which is relevant to your question:
Let $H$ be a closed subgroup of $B$ (not assuming $B$ is a Lie group yet, just a topological group), and consider the projection $p:B rightarrow B/G$. If there exists a section $s:B/G rightarrow B$, then $B$ is a fibre bundle over $B/G$ which assigns to each $b in B$ the coset $bG$. The fibre of the bundle is $G$ and the group is $G$ acting on the fibre by left translations. (Proof is on p. 31 - 32).
Now introducing Lie groups, a requirement following this construction that a section $s:G/H rightarrow G$ exists is treated in p. 110, Prop. 1, of Theory of Lie Groups by C. Chevalley. We also require that the maps are now smooth and that the projection map $p:G rightarrow G/H$ is of maximal rank everywhere in $G$ (again discussed in Steenrod). Then the above theorem applies to Lie groups too.
answered Nov 16 at 23:28
BenCWBrown
3557
3557
First, I want to say thanks for your effort; and second, I'm having problems understanding the references you give. Could you give me something in more detail, not so abstract? I know that maybe it is too much, but I'm stuck. Apologies
– Vicky
Nov 17 at 0:47
Okay I'll give it a shot!
– BenCWBrown
Nov 17 at 0:48
I appreciate it. Really thanks!
– Vicky
Nov 17 at 0:49
add a comment |
First, I want to say thanks for your effort; and second, I'm having problems understanding the references you give. Could you give me something in more detail, not so abstract? I know that maybe it is too much, but I'm stuck. Apologies
– Vicky
Nov 17 at 0:47
Okay I'll give it a shot!
– BenCWBrown
Nov 17 at 0:48
I appreciate it. Really thanks!
– Vicky
Nov 17 at 0:49
First, I want to say thanks for your effort; and second, I'm having problems understanding the references you give. Could you give me something in more detail, not so abstract? I know that maybe it is too much, but I'm stuck. Apologies
– Vicky
Nov 17 at 0:47
First, I want to say thanks for your effort; and second, I'm having problems understanding the references you give. Could you give me something in more detail, not so abstract? I know that maybe it is too much, but I'm stuck. Apologies
– Vicky
Nov 17 at 0:47
Okay I'll give it a shot!
– BenCWBrown
Nov 17 at 0:48
Okay I'll give it a shot!
– BenCWBrown
Nov 17 at 0:48
I appreciate it. Really thanks!
– Vicky
Nov 17 at 0:49
I appreciate it. Really thanks!
– Vicky
Nov 17 at 0:49
add a comment |
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3001718%2fprincipal-bundles-with-quotient-map%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
2
A reference that discusses the construction is "The Topology of Fibre Bundles" by N. Steenrod, on page 33 (Section 7.5). He further refers to "Theory of Lie Groups" by C. Chevellay, Proposition 1, page 110. I can try and summarise their content if you would like?
– BenCWBrown
Nov 16 at 22:51
1
Yes, please. Your summary together with that lectures would give me a light in this issue because fiber bundles are so abstract that I'm pretty confused
– Vicky
Nov 16 at 22:54
not exactly what you are looking for however I think that theses notes on bundles and connections are really well written and pedagogical.
– Picaud Vincent
Nov 16 at 23:02
@PicaudVincent I'll take a look at it. Thanks!
– Vicky
Nov 16 at 23:03