# Surfaces, braids, Stokes matrices, and points on spheres

@article{Fan2020SurfacesBS, title={Surfaces, braids, Stokes matrices, and points on spheres}, author={Yu-Wei Fan and Junho Peter Whang}, journal={arXiv: Algebraic Geometry}, year={2020} }

Moduli spaces of points on $n$-spheres carry natural actions of braid groups. For $n=0$, $1$, and $3$, we prove that these symmetries extend to actions of mapping class groups of positive genus surfaces, through isomorphisms with certain moduli of local systems. This relies on the existence of group structure for spheres in these dimensions. We also use the connection to demonstrate that the space of rank 4 Stokes matrices with fixed Coxeter invariant of nonzero discriminant contains only…

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