The isomorphism problem of planar polygon spaces

TL;DR

This paper proves Walker's conjecture that the lengths of bars in a circular linkage can be recovered from the cohomology ring of its configuration space, using Morse theory and algebraic invariants.

Contribution

It completes the proof of Walker's conjecture for all cases by employing Morse theory and the fundamental group to analyze cohomology invariants.

Findings

Confirmed that the cohomology ring determines bar lengths in circular linkages.

Extended the proof to cases previously unresolved by using algebraic and topological methods.

Provided a method to recover linkage parameters from topological invariants.

Abstract

We give a proof of a Conjecture of Walker which states that one can recover the lengths of the bars of a circular linkage from the cohomology ring of the configuration space. For a large class of length vectors, this has been shown by Farber, Hausmann and Schuetz. In the remaining cases, we use Morse theory and the fundamental group to describe a subring of the cohomology invariant under graded ring isomorphism. From this subring the conjecture can be derived by applying a result of Gubeladze on the isomorphism problem of monoidal rings.