Suppose I have a surjective homomorphism of topological groups $f:E \to G$. Let K be the kernel of f. The topological group K acts on E in an obvious way. When is this a fiber bundle over G? (It will then be a K-principal bundle over G).
I'm writing a paper where I make a claim about when this holds. I thought I had a reference but when I went looking for it, my claim was not in the reference.
I don't want to consider examples that are too pathological so lets assume everything is Hausdorff and paracompact. (However if people are familiar with the more general setting, I'd be curious about that too!).
Clearly a necessary condition is that $G \cong E/K$, so let's assume this is the case. By homogeneity it is enough to show that f admits a local section in a neighborhood of the identity element of G. So my question is equivalent to asking if there are conditions I can impose on E and G which will ensure that f admit local sections near the identity of G.
I know by work of G. Segal ("Cohomology of Topological Groups" Symposia Math. Vol IV 1970 pg 377, in the appendix) that if G is abelian and locally contractible then the sequence $$G \to EG \to BG$$ is of this kind.
I want to know:
- Does this hold when K is locally contractible?
- What if K is globally contractible?
- Are there any simple (but not tautological) conditions I can put on K, E, or G to make this hold?
I'd also like to know some reasonable examples where this fails to be a principal bundle (if there are any).