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The topology of spaces of polygons
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Farber, Michael and Fromm, Viktor (2012) The topology of spaces of polygons. Transactions of the American Mathematical Society, Volume 365 (Number 6). pp. 3097-3114. doi:10.1090/S0002-9947-2012-05722-9 ISSN 0002-9947.
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Official URL: http://dx.doi.org/10.1090/S0002-9947-2012-05722-9
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$.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Transactions of the American Mathematical Society | ||||
Publisher: | American Mathematical Society | ||||
ISSN: | 0002-9947 | ||||
Official Date: | 2012 | ||||
Dates: |
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Volume: | Volume 365 | ||||
Number: | Number 6 | ||||
Page Range: | pp. 3097-3114 | ||||
DOI: | 10.1090/S0002-9947-2012-05722-9 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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