Torsion exponents in stable homotopy and the Hurewicz homomorphism

Mathew, Akhil¹

¹Department of Mathematics, University of California, 970 Evans Hall, Berkeley, CA 94720, USA

Algebraic and Geometric Topology, Tome 16 (2016) no. 2, pp. 1025-1041

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.1025

Classification : 55P42, 55Q10
Keywords: Adams spectral sequence, vanishing lines, Hurewicz homomorphism, exponent theorems

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