Which principlal bundles are locally trivial?

2011-03-01T17:01:31Z

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:

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).
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 $H\times G/H$.
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.

Answer by Dick Palais for Which principlal bundles are locally trivial?

2011-03-01T18:41:00Z

...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."

Which principlal bundles are locally trivial? - MathOverflow

Which principlal bundles are locally trivial?

Answer by Dick Palais for Which principlal bundles are locally trivial?