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 R 3 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 CP 1 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.

On the symplectic volume of the moduli space of spherical and Euclidean polygons

“…Moreover, we can deduce from a result of L. Jeffrey [6], Theorem 6.6 (see also the discussion at the end of [13]) that the moduli space of polygons in S 3 and E 3 with the same side-lengths r are symplectomorphic provided that r i are sufficiently small. So the volume of the moduli space of polygons in S 3 and E 3 are the same if the side-lengths are sufficiently small.…”

“…On the other hand, following [13], we may identify R(S n , r) with the configuration space P of based polygons, i.e. polygons having the first vertex…”

“…. , r n ) be a tuple of real numbers such that 0 < r i < π ∀i, following [13], we will denote by P In E 3 , similarly, if r = (r 1 , . .…”