On the vanishing of the Lannes–Zarati homomorphism

Abstract The conjecture on spherical classes states that the Hopf invariant one and the Kervaire invariant one classes are the only elements in H ⁎ ( Q 0 S 0 ) belonging to the image of the Hurewicz homomorphism. The Lannes–Zarati homomorphism is a map that corresponds to an associated graded (with a certain filtration) of the Hurewicz map. The algebraic version of the conjecture predicts that the s-th Lannes–Zarati homomorphism vanishes in any positive stems for s > 2 . In the article, we prove the conjecture for the fifth Lannes–Zarati homomorphism.