Torsion exponents in stable homotopy and the Hurewicz homomorphism

Mathew, Akhil

doi:10.2140/agt.2016.16.1025

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Torsion exponents in stable homotopy and the Hurewicz homomorphism

Akhil Mathew

Algebraic & Geometric Topology 16 (2016) 1025–1041

DOI: 10.2140/agt.2016.16.1025

Abstract

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

Keywords

Adams spectral sequence, vanishing lines, Hurewicz homomorphism, exponent theorems

Mathematical Subject Classification 2010

Primary: 55P42, 55Q10

References

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/

Volume 16, issue 2 (2016)