In 1985 Kevin Walker in his study of topology of polygon spaces raised an interesting conjecture in the spirit of the well-known question “Can you hear the shape of a drum?” of Marc Kac. Roughly, Walker’s conjecture asks if one can recover relative lengths of the bars of a linkage from intrinsic algebraic properties of the cohomology algebra of its configuration space. I will describe results of a recent joint work with J-Cl. Hausmann and D. Schuetz in which we prove that the conjecture is true for polygon spaces in R3. We also prove that for planar polygon spaces the conjecture holds is several modified forms:(a) if one takes into account the action of a natural involution on cohomology, (b) if the cohomology algebra of the involution’s orbit space is known, or© if the length vector is normal. Some of our results allow the length vector to be non-generic (the corresponding polygon space has singularities).

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