BIEQUIVARIANT MAPS ON SPHERES AND TOPOLOGICAL COMPLEXITY OF LENS SPACES

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 connective complex 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 for a rank-2 abelian 2-group.