Around Polygons in ℝ3 and S3

Abstract:

We survey certain moduli spaces in low dimensions and some of the geometric structures that they carry, and then construct identifications among all of these spaces. In particular, we identify the moduli spaces of polygons in ℝ³ and S ³, the moduli space of restricted representations of the fundamental group of a punctured 2-sphere, the moduli space of flat connections on a punctured sphere, the moduli space of parabolic bundles on a sphere, the moduli space of weighted points on ℂℙ¹ and the symplectic quotient of SO(3) acting diagonally on (S ²)ⁿ. All of these spaces depend on parameters and some the above identifications require the parameters to be small. One consequence of this work is that these spaces are all biholomorphic with respect to the most natural complex structures they can each be given.