Let A be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, Tr*_k : Tor^A_k(F₂, F₂) → F₂⊗_AF₂[x₁ , . . . , x_k] which maps Singer's invariant-theoretic model of the dual of the Lambda algebra, Γ^∧_k, to F₂[x^±1₁ , . . . , x^±1_k] and is the inclusion of the Dickson algebra, D_k ⊂ Γ^∧_k, into F₂[x₁ , . . . , x_k]. This chain-level representation allows us to confirm the weak conjecture on spherical classes (see [9]), assuming the truth of (1) either the conjecture that the Dickson invariants of at least k = 3 variables are homologically zero in Tor^A_k(F₂, F₂), (2) or a conjecture on A-decomposability of the Dickson algebra in Γ^∧_k. We prove the conjecture in item (1) for k = 3 and also show a weak form of the conjecture in item (2).

Hung N.H.V.

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