Let script A sign be the mod 2 Steenrod algebra and D_k the Dickson algebra of k variables. We study the Lannes-Zarati homomorphisms φ_k: Ext_{script A sign}^k,k+i(script F sign₂, script F sign₂ → (script F sign₂ ⊗_{script A sign} D_k)*_i, which correspond to an associated graded of the Hurewicz map H : π*^s (S⁰) ≅ π* (Q₀S⁰) → H*(Q₀S⁰). An algebraic version of the long-standing conjecture on spherical classes predicts that φ_k = 0 in positive stems, for k > 2. That the conjecture is no longer valid for k = 1 and 2 is respectively an exposition of the existence of Hopf invariant one classes and Kervaire invariant one classes. This conjecture has been proved for k = 3 by Hu'ng. It has been shown that φ_k vanishes on decomposable elements for k > 2 and on the image of Singer's algebraic transfer for k > 2. in this paper, we establish the conjecture for k = 4. To this end, our main tools include (1) an explicit chain-level representation of φ_k and (2) a squaring operation Sq⁰ on (script F sign₂ ⊗_{script A sign} D_k)*, which commutes with the classical Sq⁰ on Ext_{script A sign}^k(script F sign₂, script F sign₂) through the Lannes-Zarati homomorphism.

Hu'ng N.H.V.

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