What is known about existence and classification of flat spherical orbifolds?. Here I mean orbifolds that admit a flat Riemannian metric (Euclidean orbifolds) and whose underlying topological space (their naive quotient space) is homeomorphic to an $n$-sphere.

In dimension $n = 2$ there is a classical classification (recalled e.g within table 1 of arXiv:1705.08431) listing, up to scale and isomorphism, the pillowcase orbifold and three more.

In higher dimensions it seems that potentially relevant results mostly come from branched covering theory. If we can realize the $n$-torus as a branched cover over the $n$-sphere such that the action of the group of deck transformations is smooth, then the corresponding homotopy quotient of the $n$-torus by the group of deck transformations should be a flat spherical orbifold. And for the action of the group of deck transformations to be smooth, it should be sufficient that the branching locus is smooth. I suppose.

For $n = 4$ there is a result in this direction in

  • Massimiliano Iori, Riccardo Piergallini, 4-manifolds as covers of the 4-sphere branched over non-singular surfaces, Geom. Topol. 6 (2002) 393-401 (arXiv:math/0203087)

I keep feeling a bit unsure about some technical fine print in the definitions there, but that their result implies the desired statement of existence of a smooth branched covering with smooth branching locus of, in particular, the 4-torus over the 4-sphere, is asserted in

  • Ali Chamseddine, Alain Connes, Viatcheslav Mukhanov, Geometry and the Quantum: Basics, JHEP 12 (2014) 098 (arXiv:1411.0977)

  • Alain Connes, Geometry and the Quantum, Foundations of Mathematics and Physics One Century After Hilbert. Springer 2018. 159-196 (arXiv:1703.02470)

on p. 24 and p. 30, respectively (see also footnote 5 in the former).

So it looks like we may conclude that 4-dimensional flat spherical orbifolds exist. (Maybe that's actually wrong, due to fine print I am overlooking, or it's actually trivial, due to general facts that I am missing. If anyone has more, please let me know.) If that is right, can we say anything about the space of available choices?

And how about other dimensions? (I'd be particularly interested in dimensions $\leq 10$.)



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