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The Walker conjecture for chains in ℝd

Published online by Cambridge University Press:  05 May 2011

MICHAEL FARBER ,
JEAN-CLAUDE HAUSMANN  and
DIRK SCHÜTZ
MICHAEL FARBER
Affiliation:
Department of Mathematical Sciences, University of Durham, Durham DH1 3LE. e-mail: [email protected]
JEAN-CLAUDE HAUSMANN
Affiliation:
Mathematics section, University of Geneva, 2-4 rue du Liévre, Geneva, Switzerland. e-mail: [email protected]
DIRK SCHÜTZ
Affiliation:
Department of Mathematical Sciences, University of Durham, Durham DH1 3LE. e-mail: [email protected]
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Abstract

A chain is a configuration in ℝd of segments of length ℓ1, . . ., ℓn−1 consecutively joined to each other such that the resulting broken line connects two given points at a distance ℓn. For a fixed generic set of length parameters the space of all chains in ℝd is a closed smooth manifold of dimension (n − 2)(d − 1) − 1. In this paper we study cohomology algebras of spaces of chains. We give a complete classification of these spaces (up to equivariant diffeomorphism) in terms of linear inequalities of a special kind which are satisfied by the length parameters ℓ1, . . ., ℓn. This result is analogous to the conjecture of K. Walker which concerns the special case d=2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 2 , September 2011 , pp. 283 - 292
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Browder, W.Surgery on Simply Connected Manifolds. (Springer-Verlag, 1972).CrossRefGoogle Scholar
[2]Farber, M. and Fromm, V., Homology of planar telescopic linkages. Algebr. Geom. Topol. 10 (2010), 1063–1087.CrossRefGoogle Scholar
[3]Farber, M., Hausmann, J-Cl. and Schütz, D.On the conjecture of Kevin Walker. Journal of Topology and Analysis 1 (2009), 65–86.CrossRefGoogle Scholar
[4]Farber, M., Hausmann, J-Cl. and Schütz, D. On the cohomology ring of chains in ℝd ArXiv:0903.0472v2 [math.AT].Google Scholar
[5]Farber, M. and Schütz, D.Homology of planar polygon spaces. Geom. Dedicata 125 (2007), 75–92.CrossRefGoogle Scholar
[6]Gubeladze, J.The isomorphism problem for commutative monoid rings. J. Pure Appl. Algebra 129 (1998), 35–65.CrossRefGoogle Scholar
[7]Hausmann, J.-C.Sur la topologie des bras articulé. in Algebraic Topology Poznań 1989, Springer Lectures Notes 1474 (1989), 146–159.Google Scholar
[8]Hausmann, J.-C.Geometric descriptions of polygon and chain spaces. In Topology and Robotics Contemp. Math. Amer. Math. Soc. 438 (2007), 47–57.CrossRefGoogle Scholar
[9]Hausmann, J.-C. and Rodriguez, E.The space of clouds in an Euclidean space. Experiment. Math. 13 (2004), 31–47.CrossRefGoogle Scholar
[10]Schütz, D.The isomorphism problem for planar polygon spaces. Journal of Topology 3 (2010), 713–742.CrossRefGoogle Scholar
[11]Walker, K. Configuration spaces of linkages. Bachelor's thesis. Princeton (1985).Google Scholar