Torsion exponents in stable homotopy and the Hurewicz homomorphism

Abstract

We give estimates for the torsion in the Postnikov sections ${\tau }_{[1,n]}{S}^0$ of the sphere spectrum, and show that the $p$-localization is annihilated by $p^{n/(2p-2)+O(1)}$. This leads to explicit bounds on the exponents of the kernel and cokernel of the Hurewicz map ${\pi }_{\ast }(X)\to H_{\ast }(X;\mathbb{Z})$ 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 of an infinite loop space, and prove an exponent theorem for the equivariant stable stems.

arXiv comment13 pages. To appear in Algebraic and Geometric Topology

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