The weak conjecture on spherical classes

Abstract.

Let $\mathcal{A}$ be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, \[Tr_k^*: Tor_k^{{\mathcal{A}}}({\bf F}_2,{\bf F}_2) \to{\bf F}_2 \mathop{\otimes}\limits_{\mathcal{A}} {\bf F}_2[x_1,\ldots ,x_k]\; \] which maps Singer's invariant-theoretic model of the dual of the Lambda algebra, $\Gamma_k^{\wedge}$, to ${\bf F}_2[x_1^{\pm 1},\ldots ,x_k^{\pm 1}]$ and is the inclusion of the Dickson algebra, $D_k \subset \Gamma_k^{\wedge}$, into ${\bf F}_2[x_1,\ldots ,x_k]$. 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 $Tor_k^{{\mathcal{A}}}({\bf F}_2,{\bf F}_2)$}, (2) or a conjecture on ${\mathcal{A}}$ -decomposability of the Dickson algebra in $\Gamma_k^{\wedge}$. We prove the conjecture in item (1) for k = 3 and also show a weak form of the conjecture in item (2).