Universality theorems for configuration spaces of planar linkages (2002)
| Venue: | Topology |
| Citations: | 18 - 1 self |
BibTeX
@ARTICLE{Kapovich02universalitytheorems,
author = {Michael Kapovich and John J. Millson},
title = {Universality theorems for configuration spaces of planar linkages},
journal = {Topology},
year = {2002},
volume = {41},
pages = {1051--1107}
}
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Abstract
We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of mechanical linkages and projective arrangements. 1. Introduction This paper deals with moduli spaces of planar linkages. An abstract linkage (L; `) is a graph L with a positive real number `(e) assigned to each edge e. We assume that we have chosen a distinguished oriented edge e = [v 1 v 2 ] in L. The moduli space M(L) of planar realizations of L := (L; `; e ) is the set 1 of maps OE from the vertex set of L into the Euclidean plane R 2 (which will be identified with the complex plane C ) such that ffl jOE(v) \Gamma OE(w)j 2 = (`[vw]) 2 for each edge [vw] of L. ffl OE(v 1 ) = (0; 0). ffl OE(v 2 ) = (`(e ); 0). Clearly these conditions give M(L) a natural structure of a real-algebraic set in R 2r where r is the number of vertices...
