DETERMINATION OF THE FIFTH SINGERALGEBRAIC TRANSFER IN SOMEDEGREES

Abstract

Let
P
k
be the graded polynomial algebra
F
2
[
x
1
, x
2
, . . . , x
k
]
over the
prime field
F
2
with two elements and the degree of each variable
x
i
being
1, and let
GL
k
be the general linear group over
F
2
which acts on
P
k
as the
usual manner. The algebra
P
k
is considered as a module over the mod-2
Steenrod algebra
A
. In 1989, Singer [19] defined the
k
-th homological
algebraic transfer, which is a homomorphism
φ
k
: Tor
A
k,k
+
d
(
F
2
,
F
2
)
→
(
F
2
⊗
A
P
k
)
GL
k
d
from the homological group of the Steenrod algebra Tor
A
k,k
+
d
(
F
2
,
F
2
)
to the subspace
(
F
2
⊗
A
P
k
)
GL
k
d
of
F
2
⊗
A
P
k
consisting of all the
GL
k
-
invariant classes of degree
d
.
In this paper, by using the results of the Peterson hit problem we
present the proof of the fact that the Singer algebraic transfer of rank
five is an isomorphism in the internal degrees
d
= 20
and
d
= 30
. Our
result refutes the proof for the case of
d
= 20
in Phúc [15