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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: michaelsfarber@googlemail.com
JEAN-CLAUDE HAUSMANN
Affiliation:
Mathematics section, University of Geneva, 2-4 rue du Liévre, Geneva, Switzerland. e-mail: Jean-Claude.Hausmann@unige.ch
DIRK SCHÜTZ
Affiliation:
Department of Mathematical Sciences, University of Durham, Durham DH1 3LE. e-mail: dirk.schuetz@durham.ac.uk
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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

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