On triviality of Dickson invariants in the homology of the Steenrod algebra

Let A be the Steenrod algebra and let Dk denote the Dickson algebra of k variables. J. E. Lannes and S. Zarati [Math. Z. 194 (1987), no. 1, 25–59; MR0871217 (88j:55014)] defined homomorphisms 'k: Extk,k+i A (F2,F2) ! (F2 A Dk) i , which correspond to an associated graded of the Hurewicz homomorphism H: s (S0) = (Q0S0)!H (Q0S0;F2). The long-standing geometric conjecture, that only Hopf invariant 1 and Kervaire invariant 1 classes are detected by H, has the following algebraic version: 'k = 0 in positive stems, for k > 2. It has been proved that '3 = 0 [Nguy˜ˆen H˜u’u Vi.ˆet Hu’ng, Trans. Amer. Math. Soc. 349 (1997), no. 10, 3893–3910; MR1433119 (98e:55020)] and that 'k vanishes on the image of Singer’s algebraic transfer for k > 2 [Nguy˜ˆen H˜u’u Vi.ˆet Hu’ng and Trˆan Ngoc Nam, Trans. Amer. Math. Soc. 353 (2001), no. 12, 5029–5040 (electronic); MR1852092 (2002f:55041)]. Further, by [Nguy˜ˆen H˜u’u Vi.ˆet Hu’ng and F. P. Peterson, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 2, 253–264; MR1631123 (99i:55021)] we know that 'k vanishes on decomposable elements for k > 2 and that '4 = 0 in positive stems < 89. In the paper under review the author completes this last result, establishing the conjecture for k = 4. The key step in the proof of his result is to show that the squaring operation Sq0 on (F2 ADk) , defined in [Nguy˜ˆen H˜u’u Vi.ˆet Hu’ng, op. cit.], commutes with the classical squaring operation Sq0 on Extk A(F2,F2) through the 'k. To this end, the explicit chain-level representation of the Lannes-Zarati dual map ' k [see Nguy˜ˆen H˜u’u Vi.ˆet Hu’ng, Trans. Amer. Math. Soc. 353 (2001), no. 11, 4447–4460 (electronic); MR1851178 (2002g:55032)] plays a key role and leads to the following equivalent formulation of the conjecture taken into account: the Dickson invariants are homologically trivial in TorAk (F2,F2).