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I am considering the following situation.

Let $M_5$ be a 5-dimensional manifold which is an $S^1$ principal bundle over 4-manifold $M_4$. For instance, $M_5 = S^5$ and $M_4 = \mathbb{CP}^2$ with standard Hopf fibration.Now there is a principal G-bundle (or associated bundle $adP$) over $M_5$.

My question is, how to characterize $P$ over $M_5$, and is there a canonical induced bundle over $M_4$?

Let me explain my question a little bit.

I know that, for even dimensional manifold $X$, characteristic numbers (integrating over $X$ the characteristic class built up by curvature) can efficiently characterize principal. But in odd dimensional cases, I'm not sure how to do similar thing. One thing I can imagine is to integrate characteristic classes over all even-dimensional cycles.

Secondly, suppose there is a way to characterize $P\to M_5$. Then I would like to know if there is a natural principal induced bundle $P'$ over $M_4$, and how to use the data of $P\to M_5$ to describe the induced bundle $P' \to M_4$.

If possible, we can take some simple cases. Say, the Hopf fibration. Principal $G$-bundles over $S^5$ are characterized by $\pi_4G$, and they are $0$ or $\mathbb{Z}_2$, as I check a list of homotopy group for Lie groups. If I take trivial $G$-bundle over $S^5$, do I get a trivial bundle over $\mathbb{CP}^2$ (do I get new "twist" in the process of inducing bundle)? And what about other situation?

Thanks.

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You should separate the two questions –  Daniele Zuddas May 15 '13 at 11:58
    
@Daniele Zuddas: Ok. Let me edit the question. Thanks. –  Lelouch May 15 '13 at 15:50

1 Answer 1

If $G$ is compact, then $P\to M_4$ is a locally trivial fiber bundle by a theorem of Ehresmann (see 17.2 in here). Take a principal $S^1$-connection of the the $S^1$-bundle $M_5\to M_4$, and a principal $G$-connection of $P\to M_5$. Then smooth curves in $M_4$ have horizontal lifts first to $M_5$, unique by choosing an initial point, and one can lift this further horizontally to $P$ by choosing another initial point. So you have a complete Ehresmann connection for the fiber bundle $P\to M_4$. Now we have to compute the holonomy Lie algebra of this Ehresmann connection; if it is finite dimensional, then we have a principal bundle by 17.11 loc.cit, whose structure group has the holonomy Lie algebra as Lie algebra. It seems to me that the holonomy group is an extension of $G$ with kernel $S^1$, but I did not check the details.

The whole construction should also work without the assumption that $G$ is compact.

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So a natural principal bundle on $M_4$ is just the one by composing two projection of $P$ onto $M_4$, with fiber loosely speaking $G \times S^1$. And when $P$ over $M_5$ is trivial, the induced bundle $P$ over $M_4$ will not be trivial if $M_5$ is a non-trivial $S^1$-bundle over $M_4$. –  Lelouch May 16 '13 at 18:53
    
Is there convenient invariants like Chern numbers (invariants expressed as integral of local quantities) that could be used to characterize a bundle (vector of principal) over an odd dimensional manifold? –  Lelouch May 16 '13 at 20:38

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