Homology of planar polygon spaces

2006/09/05 by Michael Färber, Farber, Michael, Dirk Schuetz +1 · 3 citations
Computer Science · Mathematics · #58Exx #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.math/0609140

openalex publication_date 2006/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study topology of the variety of closed planar polygons with given side lengths. We describe the Betti numbers of the moduli spaces as functions of the length vector. We also find sharp upper bounds on the sum of Betti numbers of the moduli space depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.

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• Morse theory and moduli spaces of self-avoiding polygonal linkages
• Smooth critical points of eigenvalues on the torus of magnetic perturbations of graphs
• Optimization and the Topology of Spaces of Parseval Frames

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