Biequivariant Maps on Spheres and Topological Complexity of Lens Spaces

TL;DR

This paper improves lower bounds for the topological complexity of lens spaces by using K-theory and biequivariant maps, advancing previous singular cohomology methods with novel algebraic and topological techniques.

Contribution

It introduces a new approach replacing singular cohomology with K-theory and uses biequivariant maps to significantly enhance lower bounds on topological complexity.

Findings

Improved lower bounds for lens space topological complexity.

Identification of key elements in ku-homology related to the annihilator ideal.

Application of biequivariant maps to topological complexity problems.

Abstract

Weighted cup-length calculations in singular cohomology led Farber and Grant in 2008 to general lower bounds for the topological complexity of lens spaces. We replace singular cohomology by K-theory, and weighted cup-length arguments by considerations with biequivariant maps on spheres to improve on Farber-Grant's bounds by arbitrarily large amounts. Our calculations are based on the identification of key elements conjectured to generate the annihilator ideal of the toral bottom class in the ku-homology of the classifying space of a rank-2 abelian 2-group.