THE HIT PROBLEM AND THE ALGEBRAICTRANSFER IN SOME DEGREES

Abstract

Denote by
P
k
the graded polynomial algebra
F
2
[
x
1
, x
2
, . . . , x
k
], with
the degree of each generator
x
i
being 1, and let
GL
k
be the general linear
group over the prime field
F
2
of two elements which acts naturally on
P
k
by matrix substitution.
We study the
Peterson hit problem
of determining a minimal set of
generators for
P
k
as a module over the mod-2 Steenrod algebra,
A
. In this
paper, we study the hit problem in terms of the admissible monomials
at the degree (
k
−
1)(2
d
−
1). These results are used to verify Singer’s
conjecture for the algebraic transfer, which is a homomorphism from
the homology of the mod-2 Steenrod algebra, Tor
A
k,k
+
n
(
F
2
,
F
2
), to the
subspace of
F
2
⊗
A
P
k
consisting of all the
GL
k
-invariant classes of degree
n
. More precisely, using the results on the hit problem, we prove that
Singer’s conjecture for the algebraic transfer is true in the case
k
= 5 and
the degree 4(2
d
−
1) with
d
an arbitrary positive integer