# Circle-bundles over surfaces are more known than surface-bundles over the circle

Three dimensional $S^1$-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 $N_3$-bundle over $S ^1$ by using the monodromy $-\mathbb{I}$, yes, the minus-identity auto-homeomorphism, of the genus-three-non-orientable-surface $N_3$.

Its Orlik-Raymond presentation is $\{0;(n_2,2,0,1);(1,0)\}$, remember, $n_2$ corresponds to the class $No$ in Seifert symbols

So, in this compact 3-manifold we can see a double structure, as a surface bundle

$N_3\subset \{0;(n_2,2,0,1);(1,0)\}\to S^1$

or as a circle bundle

$S^1\subset \{0;(n_2,2,0,1);(1,0)\}\to \omega$

where $\omega$ is a 2-orbifold with three cone-points and a reflector circle