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Torsion exponents in stable homotopy and the Hurewicz homomorphism

Akhil Mathew

Algebraic & Geometric Topology 16 (2016) 1025–1041

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.

Adams spectral sequence, vanishing lines, Hurewicz homomorphism, exponent theorems
Mathematical Subject Classification 2010
Primary: 55P42, 55Q10
Received: 26 March 2015
Revised: 29 July 2015
Accepted: 4 August 2015
Published: 26 April 2016
Akhil Mathew
Department of Mathematics
University of California
970 Evans Hall
Berkeley, CA 94720