Torsion exponents in stable homotopy and the Hurewicz homomorphism

摘要

We give estimates for the torsion in the Postnikov sections tau S-[1,n](0) of the sphere spectrum, and we 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(*) (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.