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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the behavior of the algebraic transfer


Authors: Robert R. Bruner, Lê M. Hà and Nguyên H. V. Hung
Journal: Trans. Amer. Math. Soc. 357 (2005), 473-487
MSC (2000): Primary 55P47, 55Q45, 55S10, 55T15
Published electronically: May 28, 2004
MathSciNet review: 2095619
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Abstract: Let $Tr_k:\mathbb{F}_2\underset{GL_k}{\otimes} PH_i(B\mathbb{V}_k)\to Ext_{\mathcal{A}}^{k,k+i}(\mathbb{F}_2, \mathbb{F}_2) $ be the algebraic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer $tr_k: \pi_*^S((B\mathbb{V} _k)_+) \to \pi_*^S(S^0)$. It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that $Tr_k$ is an isomorphism for $k=1, 2, 3$. However, Singer showed that $Tr_5$ is not an epimorphism. In this paper, we prove that $Tr_4$does not detect the nonzero element $g_s\in Ext_{\mathcal{A}}^{4,12\cdot 2^s}(\mathbb{F}_2, \mathbb{F}_2)$ for every $s\geq 1$. As a consequence, the localized $(Sq^0)^{-1}Tr_4$ given by inverting the squaring operation $Sq^0$ is not an epimorphism. This gives a negative answer to a prediction by Minami.


References [Enhancements On Off] (What's this?)

  • 1. J. M. Boardman, Modular representations on the homology of powers of real projective space, Algebraic Topology: Oaxtepec 1991, M. C. Tangora (ed.), Contemp. Math. 146 (1993), 49-70. MR 95a:55041
  • 2. E. Brown and F. P. Peterson, $H^*(MO)$ as an algebra over the Steenrod algebra, Notas Mat. Simpos. 1 (1975), 11-21. MR 86b:55001
  • 3. M. C. Crabb and J. R. Hubbuck, Representations of the homology of BV and the Steenrod algebra II, Algebraic Topology: New Trends in Localization and Periodicity (Sant Feliu de Guíxols, 1994; C. Broto et al., eds.), Progr. Math. 136, Birkhäuser, 1996, 143-154. MR 97h:55018
  • 4. Nguy"E e~n H. V. Hu width .12em height .0ex depth .075ex .ng, Spherical classes and the algebraic transfer, Trans. Amer. Math. Soc. 349 (1997), 3893-3910. MR 98e:55020
  • 5. C. Jansen, K. Lux, R. Parker, R. Wilson, An atlas of Brauer characters (Appendix 2 by T. Breuer and S. Norton). London Mathematical Society Monographs. New Series, 11. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. MR 96k:20016
  • 6. M. Kameko, Products of projective spaces as Steenrod modules, Thesis, Johns Hopkins University 1990.
  • 7. M. Kameko, Generators of the cohomology of $BV_3$, J. Math. Kyoto Univ. 38 (1998), 587-593. MR 2000b:55015
  • 8. W. H. Lin, Some differentials in Adams spectral sequence for spheres, Trans. Amer. Math. Soc., to appear.
  • 9. W. H. Lin and M. Mahowald, The Adams spectral sequence for Minami's theorem, Contemp. Math. 220 (1998), 143-177. MR 99f:55023
  • 10. A. Liulevicius, The factorization of cyclic reduced powers by secondary cohomology operations, Mem. Amer. Math. Soc. 42 (1962).MR 31:6226
  • 11. J. Peter May, The cohomology of restricted Lie algebras and of Hopf algebras; applications to the Steenrod algebra, Ph. D. thesis, Princeton University, 1964.
  • 12. J. Milnor and J. Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965), 211-264. MR 30:4259
  • 13. N. Minami, The iterated transfer analogue of the new doomsday conjecture, Trans. Amer. Math. Soc. 351 (1999), 2325-2351. MR 99i:55023
  • 14. W. M. Singer, The transfer in homological algebra, Math. Zeit. 202 (1989), 493-523.MR 90i:55035
  • 15. R. M. W. Wood, Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Cambridge Phil. Soc. 105 (1989), 307-309.MR 90a:55030

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Additional Information

Robert R. Bruner
Affiliation: Department of Mathematics, Wayne State University, 656 W. Kirby Street, Detroit, Michigan 48202
Email: rrb@math.wayne.edu

Lê M. Hà
Affiliation: Université de Lille I, UFR de Mathématiques, UMR 8524, 59655 Villeneuve d’Ascq Cédex, France
Email: Minh-Ha.Le@math.univ-lille1.fr

Nguyên H. V. Hung
Affiliation: Department of Mathematics, Vietnam National University, 334 Nguyên Trãi Street, Hanoi, Vietnam
Email: nhvhung@vnu.edu.vn

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03661-X
PII: S 0002-9947(04)03661-X
Keywords: Adams spectral sequences, Steenrod algebra, invariant theory, algebraic transfer
Received by editor(s): June 18, 2003
Published electronically: May 28, 2004
Additional Notes: The third author was supported in part by the Vietnam National Research Program, Grant N$^{0} 140 801$. The computer calculations herein were done on equipment supplied by NSF grant DMS-0079743
Dedicated: Dedicated to Professor Huỳnh Mùi on the occasion of his sixtieth birthday
Article copyright: © Copyright 2004 American Mathematical Society