Torsion exponents in stable homotopy and the Hurewicz homomorphism (Q276591)

The present paper is concerned with the theory of \(p\)-exponents in stable homotopy theory. Here, we say that an abelian group or a spectrum has \(p\)-exponent \(r\) if the \(p^r\)-multiplication map is null (homotopic). The main result gives new upper bounds for the \(p\)-exponents of the kernel and cokernel of the Hurewicz map of a connective spectrum in a given degree and hence also of the stable Hurewicz map for a space. Moreover, the author shows that these bounds are quite sharp by using K-theory applied to skeleta of classfiying spaces.NEWLINENEWLINEThe starting observation is that the fiber of the Hurewicz map \(\mathbb{S} \to H\mathbb{Z}\) to the Eilenberg-MacLane spectrum is the Postnikov section \(\tau_{[1,\infty]}\mathbb{S}\) of the sphere spectrum. The author gives a bound for the \(p\)-exponent of the spectrum \(\tau_{[1,n]}\mathbb{S}\) by using classical vanishing line results for the Adams spectral sequence of the sphere spectrum. This is enough to deduce the above result about the (co)kernel of the Hurewicz map.NEWLINENEWLINEAs an application the author proves also an explicit bound for the \(p\)-exponent of the equivariant homotopy groups of the (genuine) \(G\)-sphere spectrum for a finite group \(G\).