On the Unstable Hurewicz Homomorphism and Toda Brackets

Abstract

The aim of this note is to examine the Curtis conjecture in the light of existing structural results about the 2-primary part of the stable homotopy ring. Motivated by Joel Cohen’s result on generating stable stems using higher Toda brackets, we obtain sufficient conditions for vanishing of the unstable Hurewicz homomorphism $${_2\pi _*^s}\simeq {_2\pi _*}QS^0\rightarrow H_*(QS^0;\mathbb {Z}/2)$$ . We also record some partial results on the relation between EHP sequence and the behaviour of Hurewicz homomorphism.