Mathematische Zeitschrift

, Volume 231, Issue 4, pp 727–743

The weak conjecture on spherical classes


  • Nguyn H.V. Hung
    • Department of Mathematics, National University of Hanoi, 334 Nguyn Trãi Street, Hanoi, Vietnam (e-mail address: and

DOI: 10.1007/PL00004750

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


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 leastk = 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).

Mathematics Subject Classification (1991):55P47, 55Q45, 55S10, 55T15

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© Springer-Verlag Berlin Heidelberg 1999