# The weak conjecture on spherical classes

## Authors

DOI: 10.1007/PL00004750

- Cite this article as:
- Hung, N. Math Z (1999) 231: 727. doi:10.1007/PL00004750

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## Abstract.

Let \(\mathcal{A}\) be the mod 2 Steenrod algebra. We construct a chain-level representation of the dual of Singer's algebraic transfer, \(\) 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).