On the zero-divisor-cup-length of spaces of oriented isometry classes of planar polygons

TL;DR

This paper investigates the topological complexity of spaces of planar polygons with fixed side lengths by analyzing their rational cohomology rings, providing bounds and exact values for the zero-divisor-cup-length.

Contribution

It offers new bounds and exact calculations for the zero-divisor-cup-length of these spaces, linking cohomology to topological complexity.

Findings

Bounds for zero-divisor-cup-length are established.

Exact zero-divisor-cup-length values are determined in many cases.

Significant gaps are identified between bounds and dimensional estimates of topological complexity.

Abstract

Using information about the rational cohomology ring of the space of oriented isometry classes of planar n-gons with specified side lengths, we obtain bounds for the zero-divisor-cup-length (zcl) of these spaces, which provide lower bounds for their topological complexity (TC). In many cases our result about the cohomology ring is complete and we determine the precise zcl. We find that there will usually be a significant gap between the bounds for TC implied by zcl and dimensional considerations.