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

2 Comments

Filed under fiber bundle, math, topology

2 responses to “Circle-bundles over surfaces are more known than surface-bundles over the circle

  1. keith

    What is it a zerlegunflasch?

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