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?

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    $\begingroup$ 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? $\endgroup$ – BenCWBrown Nov 16 '18 at 22:51
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    $\begingroup$ 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 $\endgroup$ – Vicky Nov 16 '18 at 22:54
  • $\begingroup$ not exactly what you are looking for however I think that theses notes on bundles and connections are really well written and pedagogical. $\endgroup$ – Picaud Vincent Nov 16 '18 at 23:02
  • $\begingroup$ @PicaudVincent I'll take a look at it. Thanks! $\endgroup$ – Vicky Nov 16 '18 at 23:03

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.

  • $\begingroup$ 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 $\endgroup$ – Vicky Nov 17 '18 at 0:47
  • $\begingroup$ Okay I'll give it a shot! $\endgroup$ – BenCWBrown Nov 17 '18 at 0:48
  • $\begingroup$ I appreciate it. Really thanks! $\endgroup$ – Vicky Nov 17 '18 at 0:49

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