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

   

 

The cohomology of the Steenrod algebra and representations of the general linear groups


Author: Nguyên H. V. Hung
Journal: Trans. Amer. Math. Soc. 357 (2005), 4065-4089
MSC (2000): Primary 55P47, 55Q45, 55S10, 55T15
Published electronically: May 20, 2005
MathSciNet review: 2159700
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Abstract: Let $Tr_k$ be the algebraic transfer that maps from the coinvariants of certain $GL_k$-representations to the cohomology of the Steenrod algebra. This transfer was 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, more precisely, that $Tr_k$ is an isomorphism for $k=1, 2, 3$ and that $Tr= \bigoplus_k Tr_k$ is a homomorphism of algebras.

In this paper, we first recognize the phenomenon that if we start from any degree $d$ and apply $Sq^0$ repeatedly at most $(k-2)$ times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the $GL_k$-representations. As a consequence, every finite $Sq^0$-family in the coinvariants has at most $(k-2)$ nonzero elements. Two applications are exploited.

The first main theorem is that $Tr_k$ is not an isomorphism for $k\geq 5$. Furthermore, for every $k>5$, there are infinitely many degrees in which $Tr_k$ is not an isomorphism. We also show that if $Tr_{\ell}$ detects a nonzero element in certain degrees of $\text{Ker}(Sq^0)$, then it is not a monomorphism and further, for each $k>\ell$, $Tr_k$ is not a monomorphism in infinitely many degrees.

The second main theorem is that the elements of any $Sq^0$-family in the cohomology of the Steenrod algebra, except at most its first $(k-2)$ elements, are either all detected or all not detected by $Tr_k$, for every $k$. Applications of this study to the cases $k=4$ and $5$ show that $Tr_4$ does not detect the three families $g$, $D_3$ and $p'$, and that $Tr_5$ does not detect the family $\{h_{n+1}g_n \vert\; n\geq 1\}$.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03889-4
PII: S 0002-9947(05)03889-4
Keywords: Adams spectral sequences, Steenrod algebra, modular representations, invariant theory
Received by editor(s): November 13, 2003
Published electronically: May 20, 2005
Additional Notes: This work was supported in part by the National Research Program, Grant No. 140 804
Dedicated: Dedicated to Professor Nguyễn Hữu Anh on the occasion of his sixtieth birthday
Article copyright: © Copyright 2005 by Nguy\ecirti n H. V. H\uhorn ng, Nguy\ecirti n H. V. Khu\^e and Nguy\ecirti n My Trang