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# The Symplectic Geometry of Polygons in the $3$-Sphere

Published:2002-02-01
Printed: Feb 2002
• Thomas Treloar
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## 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 $\SU(2)$ by the diagonal conjugation action of $\SU(2)$. Here the fusion product of $n$ conjugacy classes is a Hamiltonian quasi-Poisson $\SU(2)$-manifold in the sense of \cite{AKSM}. 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 $\E^3$ \cite{KM1}.
 MSC Classifications: 53D - unknown classification 53D