Fibre Bundles – Which Principal Bundles Are Locally Trivial?

fibre-bundlestopological-groups

If $H$ is a closed subgroup of a topological group $G$, then the orbit map $G\to G/H$ is a principal bundle, yet somewhat surprisingly, it need not be locally trivial.
In the wikipedia article on fiber bundles it is claimed that if $H$ is a Lie group, then $G\to G/H$ is locally trivial. Is the claim true, and if so, what is the reference?

Remarks:

  1. That $G\to G/H$ is a principal bundle is explained e.g. in Husemoller's "Fiber bundles", example 2.4 in the 3rd edition. In the same section one can also find a definition of a principal bundle (which does not require local triviality).

  2. A simple example when $G\to G/H$ is not locally trivial can be found in
    the paper
    of Karube [On the local cross-sections in locally compact groups,
    J. Math. Soc. Japan 10 1958 343–347]. In the example $G$ is the product of infinitly many circles, and $H$ is the product of their order $2$ subgroups; there can be no
    cross-section because $G$ is locally-connected and $H$ is not, so $G$ is not even locally homeomorphic to $H\times G/H$.

  3. In the same paper Karube proves that $G\to G/H$ is locally trivial
    in a number of cases, including when $G$ is locally compact, and $H$ is a Lie group.

UPDATE: If $H$ is a Lie group, Palais's paper mentioned in his answer actually
characterises the principal $H$-bundles that are locally trivial;
details are below.

For a topological group $H$ acting freely and by homeomorphisms on a space $X$, we
let $X^\ast$ be the subsets of $X\times X$ consisting of pairs
$(x,hx)$ where $x\in X$ and $h\in H$.
Since $H$ acts freely, there is a map
$t: X^\ast\to H$ given by $t(x,hx)=h$.

Theorem 4.1 of Palais's paper says that if the space $X$ is
completely regular, and
if $H$ is a Lie group, then the free $H$-space $X$ is locally trivial if
and only if the map $t$ is continuous.

Note that in the terminology of Husemoller's "Fiber bundles" book
continuity of $t$ is assumed in the definition of a $H$-principal bundle,
thus Husemoller's $H$-principal bundles are all locally trivial
(provided $H$ is a Lie group and $X$ is completely regular).

If $X$ is a topological group and $H$ is a subgroup, then
continuity of $t$ follows from continuity of multiplication and inverse in $X$.
It is fun to see why Palais's result doesn't show that
the $\mathbb Z$-action on $S^1$ by irrational rotation is a principal bundle:
here $X=S^1$, and $H$ is the subgroup $\{e^{in}: n\in \mathbb Z\}$ with the subspace topology. The map $t$ is continuous, but $H$ is not a Lie group.

Best Answer

...if $H$ is a Lie group, then $G \to G/H$ is locally trivial. Is the claim true, and if so, what is the reference?

Yes, it is true. See the Corollary in section 4.1 of: "On the Existence of Slices for Actions of Non-compact Lie Groups", which you can download here: http://vmm.math.uci.edu/ExistenceOfSlices.pdf

This is a paper originally published in the March 1961 Annals of Math.

The Corollary says that "If X is a topological group and G is a closed Lie subgroup of X then the fibering of X by left G cosets is locally trivial."

Related Question