Stability and classification of polygon spaces
Date
2019
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Abstract
We study the spaces of closed linkages of line segments in $\R^d$, called polygon spaces, and the action on them by the orthogonal and special orthogonal groups of matrices. A polygon space $V_d(\ell)$ is determined by an ordered list of edge lengths $\ell=(l_1, \ldots, l_n)$ and the dimension $d\geq 2$ of the ambient space. It is well-known \cite{millson} that the space of admissible edge lengths, given by a generalization of the triangle inequalities, is a combinatorial object whose components determine certain features of $V_d(\ell)$ and of the moduli space $M_d(\ell)=V_d(\ell)/SO(d)$. We expand upon this classification program by describing explicitly the variety $V_d(\ell)$ in terms of those components. We define the ``dimension'' of a polygon to be the dimension of the smallest affine space containing the polygon's edges. The interplay between dimension of polygons and the dimension of the ambient space gives a new approach to the study of the moduli spaces $M_d(\ell)$. In particular, we show that these spaces form a directed system for increasing $d$, and that this system stabilizes at $d=n$, where $n$ is the number of edges of the polygons in $V_d(\ell)$. As a tool toward this end we use a ``diagonals'' map that sends a polygon to its ordered list of diagonal lengths, and show that this map is injective on polygons of relatively small dimension. We also take a detailed look at $4$-gons, and construct the spaces $M_d(\ell)$ as $CW$-complexes for all possible $\ell$ and $d$. These constructions expand upon known constructions for low dimension. They also serve as an example of results presented earlier in the paper, and as evidence for conjectures presented later.