Algebraic & Geometric Topology

Torsion exponents in stable homotopy and the Hurewicz homomorphism

Akhil Mathew

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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(2p2)+O(1). This leads to explicit bounds on the exponents of the kernel and cokernel of the Hurewicz map π(X) H(X; ) 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

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

Received: 26 March 2015
Revised: 29 July 2015
Accepted: 4 August 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P42: Stable homotopy theory, spectra 55Q10: Stable homotopy groups

Adams spectral sequence vanishing lines Hurewicz homomorphism exponent theorems


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.

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