Some asymptotic formulae for torsion in homotopy groups

TL;DR

This paper derives asymptotic formulas for the torsion ranks in homotopy groups, providing explicit lower bounds for various spaces using homology and K-theory, advancing understanding of torsion growth in topology.

Contribution

It introduces two new asymptotic formulas for torsion in homotopy groups, extending previous work on rational homotopy and local hyperbolicity to torsion ranks.

Findings

Derived asymptotic formulas for torsion ranks.

Provided explicit lower bounds for specific spaces.

Connected torsion growth to homology and K-theory.

Abstract

Inspired by a remarkable work of F\'{e}lix, Halperin and Thomas on the asymptotic estimation of the ranks of rational homotopy groups, and more recent works of Wu and the authors on local hyperbolicity, we prove two asymptotic formulae for torsion rank of homotopy groups, one using ordinary homology and one using $K$-theory. We use these to obtain explicit quantitative asymptotic lower bounds on the torsion rank of the homotopy groups for many interesting spaces after suspension, including Moore spaces, Eilenberg-MacLane spaces, complex projective spaces, complex Grassmannians, Milnor hypersurfaces and unitary groups.