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Let $ G \to E \to B $ be a $ G $ principal bundle, where $ G $ is a compact Lie group. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of the bundle, also admits a transitive action by some Lie group?

Note that this conjecture does not hold for general bundles with homogeneous fiber over homogeneous base. For example there are two non principal circle bundles over $ K^2 $ whose total spaces are not homogeneous. Indeed these are the other two of the four non orientable compact flat three manifolds, they can also be viewed as the other two of the four mapping tori of $ K^2 $. They are non principal bundles $ S^1 \to S^1 \rtimes_b K^2 \to K^2 $ both with non orientable total space. For $ b=0 $ this is the mapping torus of the Y homoeomorphism of $ K^2 $, it has first homology $ \mathbb{Z}^2 \times C_2 \times C_2 $. For $ b=1 $ this is the mapping torus of $ K^2 $ for the mapping class corresponding to the combination of a Dehn twist and a Y homoemorphism. It has first homology $ \mathbb{Z}^2 \times C_4 $. $ E^3 $ geometry. Both have total space which are not homogeneous see the argument given here Is a mapping torus of a solvmanifold always a solvmanifold? and the arguments given here https://math.stackexchange.com/a/4374850/758507

Ian Gershon Teixeira
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  • One special case would be when $E$ admits a locally flat principal connection, since this allows you to horizontally lift the Lie algebra action from $B$ to $E$, giving rise to a transitive $\widetilde{H}\times G$-action on $E$ where $\widetilde{H}$ is the universal cover of the group $H$ acting on $B$. – Kajelad Dec 21 '23 at 20:00
  • @MarianoSuárez-Álvarez good point I've removed that claim – Ian Gershon Teixeira Jan 02 '24 at 17:06
  • Check if the Hopf fibration is a counterexample that disproves the conjecture. It looks like a principal $U(1)$ bundle with fiber $S^1$ over a homogenous manifold $S^2$, the manifold (not the total space) admitting transitive action by $SO(3)$, but at best it's locally homogenous as a total space. I'm not an expert though so maybe verify this – Barri Jan 17 '24 at 23:07
  • @Barri this is not a counterexample the total space of the hopf fibration is the 3 sphere, many groups act transitively on the 3 sphere – Ian Gershon Teixeira Jan 18 '24 at 02:01
  • @IanGershonTeixeira that's true, the question in the title is if the total space is 'homogenous', which I took to mean that the action is transitive and preserves the fiber structure. There are transitive maps that do (or don't) preserve structure in various ways. I found that the Hopf fibration is 'pointwise' transitive and 'fiberwise' transitive. It's not clear to me if these conditions are weaker than 'homogenous' (i.e. maps that exchange pairs of points may not preserve fibers, and maps that preserve fibers may not be transitive on all pairs of points) – Barri Jan 18 '24 at 15:17

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