Growth of homotopy groups of spheres via the Goodwillie-EHP Sequence

TL;DR

This paper refines bounds on the homotopy groups of spheres using Goodwillie calculus, specifically the Goodwillie-EHP sequence, linking 2-torsion in stable stems to approximations of spheres.

Contribution

It provides a Goodwillie-theoretic refinement of existing bounds on homotopy groups, connecting 2-torsion and excisive approximations through the Goodwillie-EHP sequence.

Findings

Bound the volume of homotopy groups in terms of 2-torsion

At the 2^k-excisive approximation, bounds are polynomially scaled

Utilizes Behrens' Goodwillie-EHP Long Exact Sequence

Abstract

We bound the volume of the homotopy groups of the 2-local Goodwillie approximations of a sphere in terms of the amount of $2$-torsion in the stable stems, providing a Goodwillie-theoretic refinement of a result of Burklund and Senger (arXiv:2203.00670). At the $2^k$-excisive approximation, this bound is obtained by `multiplying the stable answer by a polynomial of degree $k$'. The main tool is Behrens' Goodwillie-EHP Long Exact Sequence.