Abstract
We study the symplectic geometry of the moduli spaces $M_r=M_r(\s^3)$ of
closed n-gons with fixed side-lengths in the 3-sphere. We prove that these
moduli spaces have symplectic structures obtained by reduction of the fusion
product of $n$ conjugacy classes in SU(2), denoted $C_r^n$, by the diagonal
conjugation action of SU(2). Here $C_r^n$ is a quasi-Hamiltonian SU(2)-space.
An integrable Hamiltonian system is constructed on $M_r$ in which the
Hamiltonian flows are given by bending polygons along a maximal collection of
nonintersecting diagonals. Finally, we show the symplectic structure on $M_r$
relates to the symplectic structure obtained from gauge-theoretic description
of $M_r$. The results of this paper are analogues for the 3-sphere of results
obtained for $M_r(\h^3)$, the moduli space of n-gons with fixed side-lengths in
hyperbolic 3-space \cite{KMT}, and for $M_r(\E^3)$, the moduli space of n-gons
with fixed side-lengths in $\E^3$