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Transactions of the American Mathematical Society
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
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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
Email: MichaelSFarber@googlemail.com

Viktor Fromm
Affiliation: Institut für Mathematik, Humboldt-Universität Berlin, Rudower Chaussee 25, D-12489 Berlin, Germany
Email: frommv@mathematik.hu-berlin.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05722-9
PII: S 0002-9947(2012)05722-9
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