The topology of spaces of polygons
Michael Farber, Viktor Fromm

TL;DR
This paper classifies the topological types of spaces of closed polygons in higher dimensions, showing they correspond to combinatorial objects related to hyperplane arrangements, extending known results from low dimensions.
Contribution
It establishes a complete classification of the diffeomorphism types of polygon spaces in dimensions three and higher, linking them to hyperplane complement components.
Findings
Diffeomorphism types correspond to hyperplane arrangement components
Classification extends low-dimensional results to higher dimensions
Provides a combinatorial description of polygon space topology
Abstract
Let denote the space of all closed -gons in (where ) with sides of length , viewed up to translations. The spaces are parameterized by their length vectors encoding the length parameters. Generically, is a closed smooth manifold of dimension supporting an obvious action of the orthogonal group . However, the quotient space (the moduli space of shapes of -gons) has singularities for a generic , assuming that ; this quotient is well understood in the low dimensional cases and . Our main result in this paper states that for fixed and , the diffeomorphism types of the manifolds for varying generic vectors are in one-to-one correspondence with some combinatorial…
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Combinatorial Mathematics · Computational Geometry and Mesh Generation
