Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you for your helping.



Is it true that all sphere bundles are some double of disk bundle?
No. If a sphere bundle is the double of a disk bundle, then it has a section. You get counterexamples by considering unit sphere bundles of vector bundles with nonvanishing Euler class.


