The moduli space of singular great circle fibrations of three-sphere and their dynamics

How can a fiber bundle develop singularities? Perhaps some fibers move around and bump intoone another so that they are no longer disjoint, while other fibers develop their own individual singularities. To separate these two phenomena from one another in a low-dimensional setting, we focus on fibrations of the three-sphere by great circles, insist that the fibers remain rigid, but allow them to drift and collide. To proceed quantitatively, we take advantage of a known moduli space for these fibrations, two copies of the family of strictly distance-decreasing mappings of a two-sphere to itself. The constant maps correspond to the Hopf fibrations and provide a homotopy-equivalent core consisting of two copies of a two-sphere. Then we view the singular fibrations as providing a boundary for this moduli space, in which the corresponding mappings are now only weakly distance-decreasing, though still of degree zero. We then • Prove that this enlarged moduli space retains its original homotopy type. • Provide a dynamic model for singularity formation, in which all the fibers bump into one another simultaneously. • Finally, we rely on the work of Kirszbraun, on the extension of Lipschitz maps between Hilbert spaces while preserving the Lipschitz constant, to prove that the smooth non-singular fibrations are dense in the enlarged space of possibly singular continuous fibrations.