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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 annihilatedby $p^{n/(2p-2) + O(1)}$. This leads to explicit bounds on the exponents of thekernel and cokernel of the Hurewicz map $\pi_*(X) \to H_*(X; \mathbb{Z})$ for aconnective spectrum $X$. Such bounds were first considered by Arlettaz,although our estimates are tighter and we prove that they are the best possibleup to a constant factor. As applications, we sharpen existing bounds on theorders of $k$-invariants in a connective spectrum, sharpen bounds on theunstable Hurewicz of an infinite loop space, and prove an exponent theorem forthe equivariant stable stems.

Torsion exponents in stable homotopy and the Hurewicz homomorphism