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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.

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Additional Information

Robert R. Bruner
Affiliation: Department of Mathematics, Wayne State University, 656 W. Kirby Street, Detroit, Michigan 48202

Lê M. Hà
Affiliation: Université de Lille I, UFR de Mathématiques, UMR 8524, 59655 Villeneuve d’Ascq Cédex, France

Nguyên H. V. Hung
Affiliation: Department of Mathematics, Vietnam National University, 334 Nguyên Trãi Street, Hanoi, Vietnam

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

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