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), 473487
MSC (2000):
Primary 55P47, 55Q45, 55S10, 55T15
Published electronically:
May 28, 2004
MathSciNet review:
2095619
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: Let be the algebraic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer . It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that is an isomorphism for . However, Singer showed that is not an epimorphism. In this paper, we prove that does not detect the nonzero element for every . As a consequence, the localized given by inverting the squaring operation is not an epimorphism. This gives a negative answer to a prediction by Minami.
 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), 4970. MR 95a:55041
 2.
E. Brown and F. P. Peterson, as an algebra over the Steenrod algebra, Notas Mat. Simpos. 1 (1975), 1121. 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, 143154. MR 97h:55018
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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), 38933910. 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 , J. Math. Kyoto Univ. 38 (1998), 587593. 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), 143177. 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), 211264. MR 30:4259
 13.
N. Minami, The iterated transfer analogue of the new doomsday conjecture, Trans. Amer. Math. Soc. 351 (1999), 23252351. MR 99i:55023
 14.
W. M. Singer, The transfer in homological algebra, Math. Zeit. 202 (1989), 493523.MR 90i:55035
 15.
R. M. W. Wood, Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Cambridge Phil. Soc. 105 (1989), 307309.MR 90a:55030
 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), 4970. MR 95a:55041
 2.
 E. Brown and F. P. Peterson, as an algebra over the Steenrod algebra, Notas Mat. Simpos. 1 (1975), 1121. 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, 143154. 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), 38933910. 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 , J. Math. Kyoto Univ. 38 (1998), 587593. 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), 143177. 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), 211264. MR 30:4259
 13.
 N. Minami, The iterated transfer analogue of the new doomsday conjecture, Trans. Amer. Math. Soc. 351 (1999), 23252351. MR 99i:55023
 14.
 W. M. Singer, The transfer in homological algebra, Math. Zeit. 202 (1989), 493523.MR 90i:55035
 15.
 R. M. W. Wood, Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Cambridge Phil. Soc. 105 (1989), 307309.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:
MinhHa.Le@math.univlille1.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/S000299470403661X
PII:
S 00029947(04)03661X
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 DMS0079743
Dedicated:
Dedicated to Professor Huỳnh Mùi on the occasion of his sixtieth birthday
Article copyright:
© Copyright 2004 American Mathematical Society
