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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The topology of spaces of polygons

Authors: Michael Farber and Viktor Fromm
Journal: Trans. Amer. Math. Soc. 365 (2013), 3097-3114
MSC (2010): Primary 55R80
Published electronically: September 19, 2012
MathSciNet review: 3034460
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Abstract: Let $E_{d}(\ell )$ denote the space of all closed $n$-gons in $\mathbb {R}^{d}$ (where $d\ge 2$) with sides of length $\ell _1, \dots , \ell _n$, viewed up to translations. The spaces $E_d(\ell )$ are parameterized by their length vectors $\ell =(\ell _1, \dots , \ell _n)\in \mathbb {R}^n_{>}$ encoding the length parameters. Generically, $E_{d}(\ell )$ is a closed smooth manifold of dimension $(n-1)(d-1)-1$ supporting an obvious action of the orthogonal group ${ {O}}(d)$. However, the quotient space $E_{d}(\ell )/{{O}}(d)$ (the moduli space of shapes of $n$-gons) has singularities for a generic $\ell$, assuming that $d>3$; this quotient is well understood in the low-dimensional cases $d=2$ and $d=3$. Our main result in this paper states that for fixed $d\ge 3$ and $n\ge 3$, the diffeomorphism types of the manifolds $E_{d}(\ell )$ for varying generic vectors $\ell$ are in one-to-one correspondence with some combinatorial objects – connected components of the complement of a finite collection of hyperplanes. This result is in the spirit of a conjecture of K. Walker who raised a similar problem in the planar case $d=2$.

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Additional Information

Michael Farber
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Viktor Fromm
Affiliation: Institut für Mathematik, Humboldt-Universität Berlin, Rudower Chaussee 25, D-12489 Berlin, Germany

Received by editor(s): April 11, 2011
Received by editor(s) in revised form: October 8, 2011
Published electronically: September 19, 2012
Article copyright: © Copyright 2012 American Mathematical Society