Three dimensional -bundles which must be based on a two-manifolds were studied since 1930’s by H. Seifert. Commonly dubbed Seifer fiber spaces, they only include bundles by using singular fibers above cone points in their Zerlegungfläschen, that is 2D-cone-orbifolds
A most general point of view is considered when we ask for 3D-manifolds foliated by circles, these must be called Seifert-Scott fiber spaces and it is needed to consider reflector lines or circles in the orbit-surface
Recently, it was unveiled a -bundle over by using the monodromy , yes, the minus-identity auto-homeomorphism, of the genus-three-non-orientable-surface .
Its Orlik-Raymond presentation is , remember, corresponds to the class in Seifert symbols
So, in this compact 3-manifold we can see a double structure, as a surface bundle:
or as a circle bundle
where is a 2-orbifold with three cone-points and a reflector circle