Torsion exponents in stable homotopy and the Hurewicz homomorphism

TL;DR

This paper provides improved estimates for torsion in the Postnikov sections of the sphere spectrum and derives explicit bounds on the kernel and cokernel exponents of the Hurewicz map, with applications to k-invariants, unstable Hurewicz, and equivariant stems.

Contribution

It offers the tightest known bounds on torsion exponents in stable homotopy, improving upon previous estimates and demonstrating their optimality up to a constant factor.

Findings

Derived explicit bounds on torsion in Postnikov sections

Established bounds on kernel and cokernel exponents of the Hurewicz map

Proved an exponent theorem for equivariant stable stems

Abstract

We give estimates for the torsion in the Postnikov sections $\tau_{[1, n]} S^0$ of the sphere spectrum, and show that the $p$-localization is annihilated by $p^{n/(2p-2) + O(1)}$. This leads to explicit bounds on the exponents of the kernel and cokernel of the Hurewicz map $\pi_*(X) \to H_*(X; \mathbb{Z})$ for a connective spectrum $X$. Such bounds were first considered by Arlettaz, although our estimates are tighter and we prove that they are the best possible up to a constant factor. As applications, we sharpen existing bounds on the orders of $k$-invariants in a connective spectrum, sharpen bounds on the unstable Hurewicz of an infinite loop space, and prove an exponent theorem for the equivariant stable stems.