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Mathematical Proceedings of the Cambridge Philosophical Society Mathematical Proceedings of the Cambridge Philosophical Society

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The squaring operation on -generators of the Dickson algebra†.

Published online by Cambridge University Press:  03 December 2009

NGUYỄN H. V. HƯNG  and
VÕ T. N. QUỲNH
NGUYỄN H. V. HƯNG
Affiliation:
Department of Mathematics, Vietnam National University, Hanoi 334 Nguyễn Trãi Street, Hanoi, Vietnam e-mail: nhvhung@vnu.edu.vn e-mail: quynhvtn@vnu.edu.vn
VÕ T. N. QUỲNH
Affiliation:
Department of Mathematics, Vietnam National University, Hanoi 334 Nguyễn Trãi Street, Hanoi, Vietnam e-mail: nhvhung@vnu.edu.vn e-mail: quynhvtn@vnu.edu.vn
Corresponding
E-mail address:
nhvhung@vnu.edu.vn
quynhvtn@vnu.edu.vn
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Abstract

We study the squaring operation Sq0 on the dual of the minimal -generators of the Dickson algebra. We show that this squaring operation is isomorphic on its image. We also give vanishing results for this operation in some cases. As a consequence, we prove that the Lannes–Zarati homomorphism vanishes (1) on every element in any finite Sq0-family in except possibly the family initial element, and (2) on almost all known elements in the Ext group. This verifies a part of the algebraic version of the classical conjecture on spherical classes.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 2 , March 2010 , pp. 267 - 288
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

[1]Adams, J. F.On the non-existence of elements of Hopf invariant one. Ann. of Math. 72 (1960), 20–104.CrossRefGoogle Scholar
[2]Boardman, J. M.Modular representations on the homology of powers of real projective space, in Algebraic Topology: Oaxtepec 1991. Contemp. Math. 146 (1993), pp. 49–70.Google Scholar
[3]Bousfield, A. K., Curtis, E. B., Kan, D. M., Quillen, D. G., Rector, D. L. and Schlesinger, J. W.The mod p lower central series and the Adams spectral sequence. Topology 5 (1966), 331–342.CrossRefGoogle Scholar
[4]Browder, W.The Kervaire invariant of a framed manifold and its generalization. Ann. of Math. 90 (1969), 157–186.CrossRefGoogle Scholar
[5]Bruner, R. R.The cohomology of the mod 2 Steenrod algebra: A computer calculation. WSU Research Report 37 (1997), 217 pages.Google Scholar
[6]Bruner, R. R., Hà, Lê M. and Hưng, Nguyễn H. V.On behavior of the algebraic transfer. Trans. Amer. Math. Soc. 357 (2005), 473–487 .CrossRefGoogle Scholar
[7]Curtis, E. B.The Dyer–Lashof algebra and the Λ-algebra. Illinois J. Math. 19 (1975), 231–246.Google Scholar
[8]Dickson, L. E.A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans. Amer. Math. Soc. 12 (1911), 75–98.CrossRefGoogle Scholar
[9]Giambalvo, V. and Peterson, F. P.-generators for ideals in the Dickson algebra. J. Pure Appl. Algebra 158 (2001), 161–182.CrossRefGoogle Scholar
[10]Goerss, P. G.Unstable projectives and stable Ext: with applications. Proc. London Math. Soc. 53 (1986), 539–561.CrossRefGoogle Scholar
[11]Hà, Lê M.Sub-Hopf algebras of the Steenrod algebra and the Singer transfer. in Proceedings of the School and Conference in Algebraic Topology, 11 (2007), Geom. Topol. Publ. Conventry, pp. 81–105.Google Scholar
[12]Hưng, Nguyễn H. V.Spherical classes and the algebraic transfer. Trans. Amer. Math. Soc. 349 (1997), 3893–3910.CrossRefGoogle Scholar
[13]Hưng, Nguyễn H. V.The weak conjecture on spherical classes. Math. Zeit. 231 (1999), 727–743.CrossRefGoogle Scholar
[14]Hưng, Nguyễn H. V.Spherical classes and the lambda algebra. Trans. Amer. Math. Soc. 353 (2001), 4447–4460.CrossRefGoogle Scholar
[15]Hưng, Nguyễn H. V.On triviality of Dickson invariants in the homology of the Steenrod algebra. Math. Proc. Camb. Phil. Soc. 134 (2003), 103–113.Google Scholar
[16]Hưng, Nguyễn H. V.The cohomology of the Steenrod algebra and representations of the general linear groups. Trans. Amer. Math. Soc. 357 (2005), 4065–4089.CrossRefGoogle Scholar
[17]Hưng, Nguyễn H. V. and Nam, Trần N.The hit problem for the Dickson algebra. Trans. Amer. Math. Soc. 353 (2001), 5029–5040.CrossRefGoogle Scholar
[18]Hưng, Nguyễn H. V. and Peterson, F. P.–generators for the Dickson algebra. Trans. Amer. Math. Soc. 347 (1995), 4687–4728.Google Scholar
[19]Hưng, Nguyễn H. V. and Peterson, F. P.Spherical classes and the Dickson algebra. Math. Proc. Camb. Phil. Soc. 124 (1998), 253–264.CrossRefGoogle Scholar
[20]Hưng, Nguyễn H. V. and Quỳnh, Võ T. N.The squaring operation on -generators of the Dickson algebra. Proc. Japan. Acad., Ser. A, 85 (2009), 67–70.CrossRefGoogle Scholar
[21]Hưng, Nguyễn H. V. and Quỳnh, Võ T. N. The image of Singer's fourth transfer. C. R. Acad. Sci. Paris, Ser. I (2009), to appear.Google Scholar
[22]Kameko, M. Products of projective spaces as Steenrod modules. Thesis. (Johns Hopkins University 1990).Google Scholar
[23]Lannes, J. and Zarati, S.Sur les foncteurs dérivés de la déstabilisation. Math. Zeit. 194 (1987), 25–59.CrossRefGoogle Scholar
[24]Lin, W. H. Some differentials in Adams spectral sequence for spheres. Trans. Amer. Math. Soc., to appear.Google Scholar
[25]Lin, W. H. and Mahowald, M.The Adams spectral sequence for Minami's theorem. in Homotopy theory via algebraic geometry and group representations (Evanston, II., 1997) Contemp. Math. 220 (1998), pp. 143–177.CrossRefGoogle Scholar
[26]Minami, N.The iterated transfer analogue of the new doomsday conjecture. Trans. Amer. Math. Soc. 351 (1999), 2325–2351.CrossRefGoogle Scholar
[27]Mùi, HuỳnhModular invariant theory and cohomology algebras of symmetric groups. J. Fac. Sci. Univ. Tokyo, 22 (1975), 310–369.Google Scholar
[28]Nam, Trân N.Transfert algébrique et représentation modulaire du groupe linéare. Ann. Inst. Fourier, 58 (2008), 1785–1837.CrossRefGoogle Scholar
[29]Singer, W. M.Invariant theory and the lambda algebra. Trans. Amer. Math. Soc. 280 (1983), 673–693.CrossRefGoogle Scholar
[30]Singer, W. M.The transfer in homological algebra. Math. Zeit. 202 (1989), 493–523.CrossRefGoogle Scholar
[31]Snaith, V. and Tornehave, J.On π*S(BO) and the Arf invariant of framed manifolds. Amer. Math. Soc. Contemp. Math. 12 (1982), 299–313.CrossRefGoogle Scholar
[32]Tangora, M. C.On the cohomology of the Steenrod algebra. Math. Zeit. 116 (1970), 18–64.CrossRefGoogle Scholar
[33]Wang, J. S. P.On the cohomology of the mod-2 Steenrod algebra and the non-existence of elements of Hopf invariant one. Illinois J. Math. 11 (1967), 480–490.Google Scholar
[34]Wellington, R. J.The unstable Adams spectral sequence of free iterated loop spaces. Mem. Amer. Math. Soc. 258 (1982).Google Scholar

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