Global topology of fixed-modulus realization spaces

Characterize the global shape and number of path components of the space of embedded flat configurations on each fixed eight-vertex triangulation with a prescribed marked modulus, and determine whether every path component has a planar configuration in its closure.

Background

For a fixed eight-vertex triangulation and target modulus, the paper considers the space of embedded flat configurations with that modulus. Near the constructed realizations, this space is a smooth fifteen-dimensional manifold before quotienting by similarities, but its global topology is not established. The authors ask how many path components it has and whether every realization can be continuously deformed, through paper tori of the same modulus, toward a planar configuration.

References

Its global shape is unknown to us, and the same is true at every modulus off the two rays, where near universality makes the set nonempty. Two realizations in one path component deform into one another through paper tori of the same shape. We ask how many path components there are, and whether every one of them has a planar configuration in its closure, so that every realization can be driven into the plane through paper tori of its own modulus.

— The two most symmetric flat tori as eight-vertex paper tori  (2609.10703 - Lander, 9 Sep 2026) in Section 'Open questions' (Section 6), fourth paragraph