Title of Paper: The cohomology of the Steenrod algebra and
representations of the general linear groups

Author: Nguy\^{e}n H. V. Hung

2000 Mathematics Subject Classification:
Primary 55P47, 55Q45,
55S10, 55T15.


Address of Author:
Current Address: Department of Mathematics,
Wayne State University 656 W. Kirby Street, Detroit, MI 48202
(USA) E-mail address: nhvhung@@math.wayne.edu

Permanent Address: Department of Mathematics, Vietnam National
University, Hanoi 334 Nguyen Trai Street, Hanoi, Vietnam E-mail
address: nhvhung@@vnu.edu.vn


Abstract:
Let $Tr_k$ be the algebraic transfer that maps from the
coinvariants of certain $GL_k$-representation to the cohomology of
the Steenrod algebra. This transfer was defined by W. Singer as an
algebraic version of the geometrical transfer $tr_k: \pi_*^S((B\V
_k)_+) \to \pi_*^S(S^0)$. 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 $Tr= \oplus_k Tr_k$ 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
$GL_k$-representation. As a consequence, every finite
$Sq^0$-family in the coinvariants has at most $(k-2)$ non zero
elements. Two applications  are exploited.

The first main theorem is that $Tr_k$ is not an isomorphism for
$k\geq 5$. Furthermore, $Tr_k$ is not an isomorphism in infinitely
many degrees for each $k > 5$. 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,
$Tr_k$ is not a monomorphism in infinitely many degrees for each
$k>\ell$.

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 $g$, $D_3$, $p'$ and $Tr_5$ does not detect the family
$\{h_{n+1}g_n |\; n\geq 1\}$.