Abstract

A symplectic form on a smooth manifold is a differential 2-form on pairs of tangent vectors of the manifold which is closed and non-degenerate. A polygon in threedimensional space is a closed polygonal line, or, more precisely, a polygon of m sides is a map ρ from the set of the first m integers into Euclidean vector space R3, such that the sum ρ1 + ρ2 + · · · + ρm equals to the zero vector. The vectors ρ1, ρ2, . . . , ρm are called the sides of the polygon ρ, and their lengths are called its side lengths. All polygons of fixed side lengths make up a space, and one may put symplectic structures on this space. In this text we shall describe two ways to do this; these ways, making use of a method called symplectic reduction, are due to Haussmann-Knutson [5] and independently to Kapovich-Millson [7], and have been shown to be equivalent by Hausmann-Knutson [5]. We begin in the first chapter with a compilation of the necessary definitions and results of group action and symplectic manifolds, including Hamiltonian action and symplectic reduction. In the second chapter we define precisely the space of polygons and describe the aforementioned symplectic forms on them.

There may be some group action on a symplectic manifold. If the group is a torus whose dimension is half of that of the manifold, and if the action is an effective Hamiltonian action, then the manifold corresponds to a figure in three-dimensional space called a Delzant polytope; in fact, Delzant polytopes completely classify such manifolds. By a result of Kapovich-Milllson [7], on the space of polygons of m sides of fixed side lengths, one may construct such a torus action if the m−3 diagonals of the polygons, namely the lengths |ρ1+ρ2|, . . . , |ρ1+· · ·+ρm−2|, do no vanish. The space of polygons then corresponds to a Delzant polytope, and may then be identified with other symplectic manifolds corresponding to the same polytope. We shall describe in this text the case where polygons have 4 sides or 5 sides. Thus, in continuation of the second chapter, we start in the third chapter a compilation of necessary definitons and results on torus action and Delzant polytopes. Because in the case of polygons of 5 sides there appears an operation on the Delzant polytopes called blow-up, we describe briefly this concept in the same chapter. In the fourth chapter we describe torus action on the space of polygons due to Kapovich-Millson [7]; this action means geometrically rotations about the diagonals, and this idea is also described in the same section. We then apply the results to the case of polygons of 4 or 5 sides.
Finally, we compile in the appendix a more detailed list of certain definitions and results that appear in the text.