## Algebraic & Geometric Topology

### Torsion exponents in stable homotopy and the Hurewicz homomorphism

Akhil Mathew

#### Abstract

We give estimates for the torsion in the Postnikov sections $τ[1,n]S0$ of the sphere spectrum, and we show that the $p$–localization is annihilated by $pn∕(2p−2)+O(1)$. This leads to explicit bounds on the exponents of the kernel and cokernel of the Hurewicz map $π∗(X) → H∗(X; ℤ)$ 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 map of an infinite loop space, and prove an exponent theorem for the equivariant stable stems.

#### Article information

Source
Algebr. Geom. Topol., Volume 16, Number 2 (2016), 1025-1041.

Dates
Revised: 29 July 2015
Accepted: 4 August 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841159

Digital Object Identifier
doi:10.2140/agt.2016.16.1025

Mathematical Reviews number (MathSciNet)
MR3493414

Zentralblatt MATH identifier
1353.55003

#### Citation

Mathew, Akhil. Torsion exponents in stable homotopy and the Hurewicz homomorphism. Algebr. Geom. Topol. 16 (2016), no. 2, 1025--1041. doi:10.2140/agt.2016.16.1025. https://projecteuclid.org/euclid.agt/1510841159

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