#### Volume 16, issue 2 (2016)

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Torsion exponents in stable homotopy and the Hurewicz homomorphism

### Akhil Mathew

Algebraic & Geometric Topology 16 (2016) 1025–1041
##### Abstract

We give estimates for the torsion in the Postnikov sections ${\tau }_{\left[1,n\right]}{S}^{0}$ of the sphere spectrum, and we show that the $p$–localization is annihilated by ${p}^{n∕\left(2p-2\right)+O\left(1\right)}$. This leads to explicit bounds on the exponents of the kernel and cokernel of the Hurewicz map ${\pi }_{\ast }\left(X\right)\to {H}_{\ast }\left(X;ℤ\right)$ 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.

##### Keywords
Adams spectral sequence, vanishing lines, Hurewicz homomorphism, exponent theorems
##### Mathematical Subject Classification 2010
Primary: 55P42, 55Q10
##### Publication
Received: 26 March 2015
Revised: 29 July 2015
Accepted: 4 August 2015
Published: 26 April 2016
##### Authors
 Akhil Mathew Department of Mathematics University of California 970 Evans Hall Berkeley, CA 94720 USA http://math.harvard.edu/~amathew/