On the conjecture of Kevin Walker

TL;DR

This paper proves Kevin Walker's conjecture that the relative lengths of linkage bars can be recovered from the cohomology algebra of their configuration space, with results for both 3D and planar polygon spaces.

Contribution

The paper confirms Walker's conjecture for polygon spaces in R^3 and provides several modified versions for planar spaces, utilizing involution actions and cohomology algebra analysis.

Findings

Walker’s conjecture holds for polygon spaces in R^3.

Modified conjecture versions are valid for planar polygon spaces.

The study employs involution actions and Gubeladze's isomorphism solution.

Abstract

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. In this paper we prove that the conjecture is true for polygon spaces in R^3. 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 (c) if the length vector is normal. Some of our results allow the length vector to be non-generic, the corresponding polygon spaces have singularities. Our main tool is the study of the natural involution and its action on cohomology. A crucial role in our proof plays the solution of the isomorphism problem for monoidal rings due to J. Gubeladze.